-"""
-Euclidean Jordan Algebras. These are formally-real Jordan Algebras;
-specifically those where u^2 + v^2 = 0 implies that u = v = 0. They
-are used in optimization, and have some additional nice methods beyond
-what can be supported in a general Jordan Algebra.
-
+r"""
+Representations and constructions for Euclidean Jordan algebras.
+
+A Euclidean Jordan algebra is a Jordan algebra that has some
+additional properties:
+
+ 1. It is finite-dimensional.
+ 2. Its scalar field is the real numbers.
+ 3a. An inner product is defined on it, and...
+ 3b. That inner product is compatible with the Jordan product
+ in the sense that `<x*y,z> = <y,x*z>` for all elements
+ `x,y,z` in the algebra.
+
+Every Euclidean Jordan algebra is formally-real: for any two elements
+`x` and `y` in the algebra, `x^{2} + y^{2} = 0` implies that `x = y =
+0`. Conversely, every finite-dimensional formally-real Jordan algebra
+can be made into a Euclidean Jordan algebra with an appropriate choice
+of inner-product.
+
+Formally-real Jordan algebras were originally studied as a framework
+for quantum mechanics. Today, Euclidean Jordan algebras are crucial in
+symmetric cone optimization, since every symmetric cone arises as the
+cone of squares in some Euclidean Jordan algebra.
+
+It is known that every Euclidean Jordan algebra decomposes into an
+orthogonal direct sum (essentially, a Cartesian product) of simple
+algebras, and that moreover, up to Jordan-algebra isomorphism, there
+are only five families of simple algebras. We provide constructions
+for these simple algebras:
+
+ * :class:`BilinearFormEJA`
+ * :class:`RealSymmetricEJA`
+ * :class:`ComplexHermitianEJA`
+ * :class:`QuaternionHermitianEJA`
+ * :class:`OctonionHermitianEJA`
+
+In addition to these, we provide a few other example constructions,
+
+ * :class:`JordanSpinEJA`
+ * :class:`HadamardEJA`
+ * :class:`AlbertEJA`
+ * :class:`TrivialEJA`
+ * :class:`ComplexSkewSymmetricEJA`
+
+The Jordan spin algebra is a bilinear form algebra where the bilinear
+form is the identity. The Hadamard EJA is simply a Cartesian product
+of one-dimensional spin algebras. The Albert EJA is simply a special
+case of the :class:`OctonionHermitianEJA` where the matrices are
+three-by-three and the resulting space has dimension 27. And
+last/least, the trivial EJA is exactly what you think it is; it could
+also be obtained by constructing a dimension-zero instance of any of
+the other algebras. Cartesian products of these are also supported
+using the usual ``cartesian_product()`` function; as a result, we
+support (up to isomorphism) all Euclidean Jordan algebras.
+
+At a minimum, the following are required to construct a Euclidean
+Jordan algebra:
+
+ * A basis of matrices, column vectors, or MatrixAlgebra elements
+ * A Jordan product defined on the basis
+ * Its inner product defined on the basis
+
+The real numbers form a Euclidean Jordan algebra when both the Jordan
+and inner products are the usual multiplication. We use this as our
+example, and demonstrate a few ways to construct an EJA.
+
+First, we can use one-by-one SageMath matrices with algebraic real
+entries to represent real numbers. We define the Jordan and inner
+products to be essentially real-number multiplication, with the only
+difference being that the Jordan product again returns a one-by-one
+matrix, whereas the inner product must return a scalar. Our basis for
+the one-by-one matrices is of course the set consisting of a single
+matrix with its sole entry non-zero::
+
+ sage: from mjo.eja.eja_algebra import EJA
+ sage: jp = lambda X,Y: X*Y
+ sage: ip = lambda X,Y: X[0,0]*Y[0,0]
+ sage: b1 = matrix(AA, [[1]])
+ sage: J1 = EJA((b1,), jp, ip)
+ sage: J1
+ Euclidean Jordan algebra of dimension 1 over Algebraic Real Field
+
+In fact, any positive scalar multiple of that inner-product would work::
+
+ sage: ip2 = lambda X,Y: 16*ip(X,Y)
+ sage: J2 = EJA((b1,), jp, ip2)
+ sage: J2
+ Euclidean Jordan algebra of dimension 1 over Algebraic Real Field
+
+But beware that your basis will be orthonormalized _with respect to the
+given inner-product_ unless you pass ``orthonormalize=False`` to the
+constructor. For example::
+
+ sage: J3 = EJA((b1,), jp, ip2, orthonormalize=False)
+ sage: J3
+ Euclidean Jordan algebra of dimension 1 over Algebraic Real Field
+
+To see the difference, you can take the first and only basis element
+of the resulting algebra, and ask for it to be converted back into
+matrix form::
+
+ sage: J1.basis()[0].to_matrix()
+ [1]
+ sage: J2.basis()[0].to_matrix()
+ [1/4]
+ sage: J3.basis()[0].to_matrix()
+ [1]
+
+Since square roots are used in that process, the default scalar field
+that we use is the field of algebraic real numbers, ``AA``. You can
+also Use rational numbers, but only if you either pass
+``orthonormalize=False`` or know that orthonormalizing your basis
+won't stray beyond the rational numbers. The example above would
+have worked only because ``sqrt(16) == 4`` is rational.
+
+Another option for your basis is to use elemebts of a
+:class:`MatrixAlgebra`::
+
+ sage: from mjo.matrix_algebra import MatrixAlgebra
+ sage: A = MatrixAlgebra(1,AA,AA)
+ sage: J4 = EJA(A.gens(), jp, ip)
+ sage: J4
+ Euclidean Jordan algebra of dimension 1 over Algebraic Real Field
+ sage: J4.basis()[0].to_matrix()
+ +---+
+ | 1 |
+ +---+
+
+An easier way to view the entire EJA basis in its original (but
+perhaps orthonormalized) matrix form is to use the ``matrix_basis``
+method::
+
+ sage: J4.matrix_basis()
+ (+---+
+ | 1 |
+ +---+,)
+
+In particular, a :class:`MatrixAlgebra` is needed to work around the
+fact that matrices in SageMath must have entries in the same
+(commutative and associative) ring as its scalars. There are many
+Euclidean Jordan algebras whose elements are matrices that violate
+those assumptions. The complex, quaternion, and octonion Hermitian
+matrices all have entries in a ring (the complex numbers, quaternions,
+or octonions...) that differs from the algebra's scalar ring (the real
+numbers). Quaternions are also non-commutative; the octonions are
+neither commutative nor associative.
SETUP::
sage: random_eja()
Euclidean Jordan algebra of dimension...
-
"""
-from itertools import repeat
-
from sage.algebras.quatalg.quaternion_algebra import QuaternionAlgebra
from sage.categories.magmatic_algebras import MagmaticAlgebras
+from sage.categories.sets_cat import cartesian_product
from sage.combinat.free_module import CombinatorialFreeModule
from sage.matrix.constructor import matrix
from sage.matrix.matrix_space import MatrixSpace
from sage.rings.all import (ZZ, QQ, AA, QQbar, RR, RLF, CLF,
PolynomialRing,
QuadraticField)
-from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement
-from mjo.eja.eja_operator import FiniteDimensionalEuclideanJordanAlgebraOperator
-from mjo.eja.eja_utils import _mat2vec
+from mjo.eja.eja_element import (CartesianProductEJAElement,
+ EJAElement)
+from mjo.eja.eja_operator import EJAOperator
+from mjo.eja.eja_utils import _all2list
+
+def EuclideanJordanAlgebras(field):
+ r"""
+ The category of Euclidean Jordan algebras over ``field``, which
+ must be a subfield of the real numbers. For now this is just a
+ convenient wrapper around all of the other category axioms that
+ apply to all EJAs.
+ """
+ category = MagmaticAlgebras(field).FiniteDimensional()
+ category = category.WithBasis().Unital().Commutative()
+ return category
+
+class EJA(CombinatorialFreeModule):
+ r"""
+ A finite-dimensional Euclidean Jordan algebra.
+
+ INPUT:
+
+ - ``basis`` -- a tuple; a tuple of basis elements in "matrix
+ form," which must be the same form as the arguments to
+ ``jordan_product`` and ``inner_product``. In reality, "matrix
+ form" can be either vectors, matrices, or a Cartesian product
+ (ordered tuple) of vectors or matrices. All of these would
+ ideally be vector spaces in sage with no special-casing
+ needed; but in reality we turn vectors into column-matrices
+ and Cartesian products `(a,b)` into column matrices
+ `(a,b)^{T}` after converting `a` and `b` themselves.
+
+ - ``jordan_product`` -- a function; afunction of two ``basis``
+ elements (in matrix form) that returns their jordan product,
+ also in matrix form; this will be applied to ``basis`` to
+ compute a multiplication table for the algebra.
+
+ - ``inner_product`` -- a function; a function of two ``basis``
+ elements (in matrix form) that returns their inner
+ product. This will be applied to ``basis`` to compute an
+ inner-product table (basically a matrix) for this algebra.
+
+ - ``matrix_space`` -- the space that your matrix basis lives in,
+ or ``None`` (the default). So long as your basis does not have
+ length zero you can omit this. But in trivial algebras, it is
+ required.
+
+ - ``field`` -- a subfield of the reals (default: ``AA``); the scalar
+ field for the algebra.
+
+ - ``orthonormalize`` -- boolean (default: ``True``); whether or
+ not to orthonormalize the basis. Doing so is expensive and
+ generally rules out using the rationals as your ``field``, but
+ is required for spectral decompositions.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import random_eja
+
+ TESTS:
+
+ We should compute that an element subalgebra is associative even
+ if we circumvent the element method::
+
+ sage: J = random_eja(field=QQ,orthonormalize=False)
+ sage: x = J.random_element()
+ sage: A = x.subalgebra_generated_by(orthonormalize=False)
+ sage: basis = tuple(b.superalgebra_element() for b in A.basis())
+ sage: J.subalgebra(basis, orthonormalize=False).is_associative()
+ True
+ """
+ Element = EJAElement
+
+ @staticmethod
+ def _check_input_field(field):
+ if not field.is_subring(RR):
+ # Note: this does return true for the real algebraic
+ # field, the rationals, and any quadratic field where
+ # we've specified a real embedding.
+ raise ValueError("scalar field is not real")
+
+ @staticmethod
+ def _check_input_axioms(basis, jordan_product, inner_product):
+ if not all( jordan_product(bi,bj) == jordan_product(bj,bi)
+ for bi in basis
+ for bj in basis ):
+ raise ValueError("Jordan product is not commutative")
+
+ if not all( inner_product(bi,bj) == inner_product(bj,bi)
+ for bi in basis
+ for bj in basis ):
+ raise ValueError("inner-product is not commutative")
+
+ def __init__(self,
+ basis,
+ jordan_product,
+ inner_product,
+ field=AA,
+ matrix_space=None,
+ orthonormalize=True,
+ associative=None,
+ check_field=True,
+ check_axioms=True,
+ prefix="b"):
+
+ n = len(basis)
+
+ if check_field:
+ self._check_input_field(field)
+
+ if check_axioms:
+ # Check commutativity of the Jordan and inner-products.
+ # This has to be done before we build the multiplication
+ # and inner-product tables/matrices, because we take
+ # advantage of symmetry in the process.
+ self._check_input_axioms(basis, jordan_product, inner_product)
+
+ if n <= 1:
+ # All zero- and one-dimensional algebras are just the real
+ # numbers with (some positive multiples of) the usual
+ # multiplication as its Jordan and inner-product.
+ associative = True
+ if associative is None:
+ # We should figure it out. As with check_axioms, we have to do
+ # this without the help of the _jordan_product_is_associative()
+ # method because we need to know the category before we
+ # initialize the algebra.
+ associative = all( jordan_product(jordan_product(bi,bj),bk)
+ ==
+ jordan_product(bi,jordan_product(bj,bk))
+ for bi in basis
+ for bj in basis
+ for bk in basis)
+
+ category = EuclideanJordanAlgebras(field)
+
+ if associative:
+ # Element subalgebras can take advantage of this.
+ category = category.Associative()
+
+ # Call the superclass constructor so that we can use its from_vector()
+ # method to build our multiplication table.
+ CombinatorialFreeModule.__init__(self,
+ field,
+ range(n),
+ prefix=prefix,
+ category=category,
+ bracket=False)
+
+ # Now comes all of the hard work. We'll be constructing an
+ # ambient vector space V that our (vectorized) basis lives in,
+ # as well as a subspace W of V spanned by those (vectorized)
+ # basis elements. The W-coordinates are the coefficients that
+ # we see in things like x = 1*b1 + 2*b2.
+
+ degree = 0
+ if n > 0:
+ degree = len(_all2list(basis[0]))
+
+ # Build an ambient space that fits our matrix basis when
+ # written out as "long vectors."
+ V = VectorSpace(field, degree)
+
+ # The matrix that will hold the orthonormal -> unorthonormal
+ # coordinate transformation. Default to an identity matrix of
+ # the appropriate size to avoid special cases for None
+ # everywhere.
+ self._deortho_matrix = matrix.identity(field,n)
+
+ if orthonormalize:
+ # Save a copy of the un-orthonormalized basis for later.
+ # Convert it to ambient V (vector) coordinates while we're
+ # at it, because we'd have to do it later anyway.
+ deortho_vector_basis = tuple( V(_all2list(b)) for b in basis )
+
+ from mjo.eja.eja_utils import gram_schmidt
+ basis = tuple(gram_schmidt(basis, inner_product))
+
+ # Save the (possibly orthonormalized) matrix basis for
+ # later, as well as the space that its elements live in.
+ # In most cases we can deduce the matrix space, but when
+ # n == 0 (that is, there are no basis elements) we cannot.
+ self._matrix_basis = basis
+ if matrix_space is None:
+ self._matrix_space = self._matrix_basis[0].parent()
+ else:
+ self._matrix_space = matrix_space
+
+ # Now create the vector space for the algebra, which will have
+ # its own set of non-ambient coordinates (in terms of the
+ # supplied basis).
+ vector_basis = tuple( V(_all2list(b)) for b in basis )
+
+ # Save the span of our matrix basis (when written out as long
+ # vectors) because otherwise we'll have to reconstruct it
+ # every time we want to coerce a matrix into the algebra.
+ self._matrix_span = V.span_of_basis( vector_basis, check=check_axioms)
+
+ if orthonormalize:
+ # Now "self._matrix_span" is the vector space of our
+ # algebra coordinates. The variables "X0", "X1",... refer
+ # to the entries of vectors in self._matrix_span. Thus to
+ # convert back and forth between the orthonormal
+ # coordinates and the given ones, we need to stick the
+ # original basis in self._matrix_span.
+ U = V.span_of_basis( deortho_vector_basis, check=check_axioms)
+ self._deortho_matrix = matrix.column( U.coordinate_vector(q)
+ for q in vector_basis )
+
+
+ # Now we actually compute the multiplication and inner-product
+ # tables/matrices using the possibly-orthonormalized basis.
+ self._inner_product_matrix = matrix.identity(field, n)
+ zed = self.zero()
+ self._multiplication_table = [ [zed for j in range(i+1)]
+ for i in range(n) ]
+
+ # Note: the Jordan and inner-products are defined in terms
+ # of the ambient basis. It's important that their arguments
+ # are in ambient coordinates as well.
+ for i in range(n):
+ for j in range(i+1):
+ # ortho basis w.r.t. ambient coords
+ q_i = basis[i]
+ q_j = basis[j]
+
+ # The jordan product returns a matrixy answer, so we
+ # have to convert it to the algebra coordinates.
+ elt = jordan_product(q_i, q_j)
+ elt = self._matrix_span.coordinate_vector(V(_all2list(elt)))
+ self._multiplication_table[i][j] = self.from_vector(elt)
+
+ if not orthonormalize:
+ # If we're orthonormalizing the basis with respect
+ # to an inner-product, then the inner-product
+ # matrix with respect to the resulting basis is
+ # just going to be the identity.
+ ip = inner_product(q_i, q_j)
+ self._inner_product_matrix[i,j] = ip
+ self._inner_product_matrix[j,i] = ip
+
+ self._inner_product_matrix._cache = {'hermitian': True}
+ self._inner_product_matrix.set_immutable()
+
+ if check_axioms:
+ if not self._is_jordanian():
+ raise ValueError("Jordan identity does not hold")
+ if not self._inner_product_is_associative():
+ raise ValueError("inner product is not associative")
-class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule):
def _coerce_map_from_base_ring(self):
"""
TESTS::
- sage: set_random_seed()
sage: J = random_eja()
sage: J(1)
Traceback (most recent call last):
"""
return None
- def __init__(self,
- field,
- mult_table,
- prefix='e',
- category=None,
- matrix_basis=None,
- check_field=True,
- check_axioms=True):
+
+ def product_on_basis(self, i, j):
+ r"""
+ Returns the Jordan product of the `i` and `j`th basis elements.
+
+ This completely defines the Jordan product on the algebra, and
+ is used direclty by our superclass machinery to implement
+ :meth:`product`.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import random_eja
+
+ TESTS::
+
+ sage: J = random_eja()
+ sage: n = J.dimension()
+ sage: bi = J.zero()
+ sage: bj = J.zero()
+ sage: bi_bj = J.zero()*J.zero()
+ sage: if n > 0:
+ ....: i = ZZ.random_element(n)
+ ....: j = ZZ.random_element(n)
+ ....: bi = J.monomial(i)
+ ....: bj = J.monomial(j)
+ ....: bi_bj = J.product_on_basis(i,j)
+ sage: bi*bj == bi_bj
+ True
+
+ """
+ # We only stored the lower-triangular portion of the
+ # multiplication table.
+ if j <= i:
+ return self._multiplication_table[i][j]
+ else:
+ return self._multiplication_table[j][i]
+
+ def inner_product(self, x, y):
"""
+ The inner product associated with this Euclidean Jordan algebra.
+
+ Defaults to the trace inner product, but can be overridden by
+ subclasses if they are sure that the necessary properties are
+ satisfied.
+
SETUP::
- sage: from mjo.eja.eja_algebra import (
- ....: FiniteDimensionalEuclideanJordanAlgebra,
- ....: JordanSpinEJA,
- ....: random_eja)
+ sage: from mjo.eja.eja_algebra import (random_eja,
+ ....: HadamardEJA,
+ ....: BilinearFormEJA)
EXAMPLES:
- By definition, Jordan multiplication commutes::
+ Our inner product is "associative," which means the following for
+ a symmetric bilinear form::
- sage: set_random_seed()
sage: J = random_eja()
+ sage: x,y,z = J.random_elements(3)
+ sage: (x*y).inner_product(z) == y.inner_product(x*z)
+ True
+
+ TESTS:
+
+ Ensure that this is the usual inner product for the algebras
+ over `R^n`::
+
+ sage: J = HadamardEJA.random_instance()
sage: x,y = J.random_elements(2)
- sage: x*y == y*x
+ sage: actual = x.inner_product(y)
+ sage: expected = x.to_vector().inner_product(y.to_vector())
+ sage: actual == expected
+ True
+
+ Ensure that this is one-half of the trace inner-product in a
+ BilinearFormEJA that isn't just the reals (when ``n`` isn't
+ one). This is in Faraut and Koranyi, and also my "On the
+ symmetry..." paper::
+
+ sage: J = BilinearFormEJA.random_instance()
+ sage: n = J.dimension()
+ sage: x = J.random_element()
+ sage: y = J.random_element()
+ sage: (n == 1) or (x.inner_product(y) == (x*y).trace()/2)
+ True
+
+ """
+ B = self._inner_product_matrix
+ return (B*x.to_vector()).inner_product(y.to_vector())
+
+
+ def is_associative(self):
+ r"""
+ Return whether or not this algebra's Jordan product is associative.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
+
+ EXAMPLES::
+
+ sage: J = ComplexHermitianEJA(3, field=QQ, orthonormalize=False)
+ sage: J.is_associative()
+ False
+ sage: x = sum(J.gens())
+ sage: A = x.subalgebra_generated_by(orthonormalize=False)
+ sage: A.is_associative()
+ True
+
+ """
+ return "Associative" in self.category().axioms()
+
+ def _is_commutative(self):
+ r"""
+ Whether or not this algebra's multiplication table is commutative.
+
+ This method should of course always return ``True``, unless
+ this algebra was constructed with ``check_axioms=False`` and
+ passed an invalid multiplication table.
+ """
+ return all( x*y == y*x for x in self.gens() for y in self.gens() )
+
+ def _is_jordanian(self):
+ r"""
+ Whether or not this algebra's multiplication table respects the
+ Jordan identity `(x^{2})(xy) = x(x^{2}y)`.
+
+ We only check one arrangement of `x` and `y`, so for a
+ ``True`` result to be truly true, you should also check
+ :meth:`_is_commutative`. This method should of course always
+ return ``True``, unless this algebra was constructed with
+ ``check_axioms=False`` and passed an invalid multiplication table.
+ """
+ return all( (self.monomial(i)**2)*(self.monomial(i)*self.monomial(j))
+ ==
+ (self.monomial(i))*((self.monomial(i)**2)*self.monomial(j))
+ for i in range(self.dimension())
+ for j in range(self.dimension()) )
+
+ def _jordan_product_is_associative(self):
+ r"""
+ Return whether or not this algebra's Jordan product is
+ associative; that is, whether or not `x*(y*z) = (x*y)*z`
+ for all `x,y,x`.
+
+ This method should agree with :meth:`is_associative` unless
+ you lied about the value of the ``associative`` parameter
+ when you constructed the algebra.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (random_eja,
+ ....: RealSymmetricEJA,
+ ....: ComplexHermitianEJA,
+ ....: QuaternionHermitianEJA)
+
+ EXAMPLES::
+
+ sage: J = RealSymmetricEJA(4, orthonormalize=False)
+ sage: J._jordan_product_is_associative()
+ False
+ sage: x = sum(J.gens())
+ sage: A = x.subalgebra_generated_by()
+ sage: A._jordan_product_is_associative()
+ True
+
+ ::
+
+ sage: J = ComplexHermitianEJA(2,field=QQ,orthonormalize=False)
+ sage: J._jordan_product_is_associative()
+ False
+ sage: x = sum(J.gens())
+ sage: A = x.subalgebra_generated_by(orthonormalize=False)
+ sage: A._jordan_product_is_associative()
+ True
+
+ ::
+
+ sage: J = QuaternionHermitianEJA(2)
+ sage: J._jordan_product_is_associative()
+ False
+ sage: x = sum(J.gens())
+ sage: A = x.subalgebra_generated_by()
+ sage: A._jordan_product_is_associative()
True
TESTS:
- The ``field`` we're given must be real with ``check_field=True``::
+ The values we've presupplied to the constructors agree with
+ the computation::
- sage: JordanSpinEJA(2,QQbar)
- Traceback (most recent call last):
- ...
- ValueError: scalar field is not real
+ sage: J = random_eja()
+ sage: J.is_associative() == J._jordan_product_is_associative()
+ True
- The multiplication table must be square with ``check_axioms=True``::
+ """
+ R = self.base_ring()
- sage: FiniteDimensionalEuclideanJordanAlgebra(QQ,((),()))
- Traceback (most recent call last):
- ...
- ValueError: multiplication table is not square
+ # Used to check whether or not something is zero.
+ epsilon = R.zero()
+ if not R.is_exact():
+ # I don't know of any examples that make this magnitude
+ # necessary because I don't know how to make an
+ # associative algebra when the element subalgebra
+ # construction is unreliable (as it is over RDF; we can't
+ # find the degree of an element because we can't compute
+ # the rank of a matrix). But even multiplication of floats
+ # is non-associative, so *some* epsilon is needed... let's
+ # just take the one from _inner_product_is_associative?
+ epsilon = 1e-15
+ for i in range(self.dimension()):
+ for j in range(self.dimension()):
+ for k in range(self.dimension()):
+ x = self.monomial(i)
+ y = self.monomial(j)
+ z = self.monomial(k)
+ diff = (x*y)*z - x*(y*z)
+
+ if diff.norm() > epsilon:
+ return False
+
+ return True
+
+ def _inner_product_is_associative(self):
+ r"""
+ Return whether or not this algebra's inner product `B` is
+ associative; that is, whether or not `B(xy,z) = B(x,yz)`.
+
+ This method should of course always return ``True``, unless
+ this algebra was constructed with ``check_axioms=False`` and
+ passed an invalid Jordan or inner-product.
"""
- if check_field:
- if not field.is_subring(RR):
- # Note: this does return true for the real algebraic
- # field, the rationals, and any quadratic field where
- # we've specified a real embedding.
- raise ValueError("scalar field is not real")
-
- # The multiplication table had better be square
- n = len(mult_table)
- if check_axioms:
- if not all( len(l) == n for l in mult_table ):
- raise ValueError("multiplication table is not square")
-
- self._matrix_basis = matrix_basis
-
- if category is None:
- category = MagmaticAlgebras(field).FiniteDimensional()
- category = category.WithBasis().Unital()
-
- fda = super(FiniteDimensionalEuclideanJordanAlgebra, self)
- fda.__init__(field,
- range(n),
- prefix=prefix,
- category=category)
- self.print_options(bracket='')
-
- # The multiplication table we're given is necessarily in terms
- # of vectors, because we don't have an algebra yet for
- # anything to be an element of. However, it's faster in the
- # long run to have the multiplication table be in terms of
- # algebra elements. We do this after calling the superclass
- # constructor so that from_vector() knows what to do.
- self._multiplication_table = [ [ self.vector_space().zero()
- for i in range(n) ]
- for j in range(n) ]
- # take advantage of symmetry
- for i in range(n):
- for j in range(i+1):
- elt = self.from_vector(mult_table[i][j])
- self._multiplication_table[i][j] = elt
- self._multiplication_table[j][i] = elt
+ R = self.base_ring()
- if check_axioms:
- if not self._is_commutative():
- raise ValueError("algebra is not commutative")
- if not self._is_jordanian():
- raise ValueError("Jordan identity does not hold")
- if not self._inner_product_is_associative():
- raise ValueError("inner product is not associative")
+ # Used to check whether or not something is zero.
+ epsilon = R.zero()
+ if not R.is_exact():
+ # This choice is sufficient to allow the construction of
+ # QuaternionHermitianEJA(2, field=RDF) with check_axioms=True.
+ epsilon = 1e-15
+
+ for i in range(self.dimension()):
+ for j in range(self.dimension()):
+ for k in range(self.dimension()):
+ x = self.monomial(i)
+ y = self.monomial(j)
+ z = self.monomial(k)
+ diff = (x*y).inner_product(z) - x.inner_product(y*z)
+
+ if diff.abs() > epsilon:
+ return False
+
+ return True
def _element_constructor_(self, elt):
"""
- Construct an element of this algebra from its vector or matrix
- representation.
+ Construct an element of this algebra or a subalgebra from its
+ EJA element, vector, or matrix representation.
This gets called only after the parent element _call_ method
fails to find a coercion for the argument.
SETUP::
- sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
+ sage: from mjo.eja.eja_algebra import (random_eja,
+ ....: JordanSpinEJA,
....: HadamardEJA,
....: RealSymmetricEJA)
...
ValueError: not an element of this algebra
+ Tuples work as well, provided that the matrix basis for the
+ algebra consists of them::
+
+ sage: J1 = HadamardEJA(3)
+ sage: J2 = RealSymmetricEJA(2)
+ sage: J = cartesian_product([J1,J2])
+ sage: J( (J1.matrix_basis()[1], J2.matrix_basis()[2]) )
+ b1 + b5
+
+ Subalgebra elements are embedded into the superalgebra::
+
+ sage: J = JordanSpinEJA(3)
+ sage: J.one()
+ b0
+ sage: x = sum(J.gens())
+ sage: A = x.subalgebra_generated_by()
+ sage: J(A.one())
+ b0
+
TESTS:
- Ensure that we can convert any element of the two non-matrix
- simple algebras (whose matrix representations are columns)
- back and forth faithfully::
+ Ensure that we can convert any element back and forth
+ faithfully between its matrix and algebra representations::
- sage: set_random_seed()
- sage: J = HadamardEJA.random_instance()
- sage: x = J.random_element()
- sage: J(x.to_vector().column()) == x
- True
- sage: J = JordanSpinEJA.random_instance()
+ sage: J = random_eja()
sage: x = J.random_element()
- sage: J(x.to_vector().column()) == x
+ sage: J(x.to_matrix()) == x
True
+
+ We cannot coerce elements between algebras just because their
+ matrix representations are compatible::
+
+ sage: J1 = HadamardEJA(3)
+ sage: J2 = JordanSpinEJA(3)
+ sage: J2(J1.one())
+ Traceback (most recent call last):
+ ...
+ ValueError: not an element of this algebra
+ sage: J1(J2.zero())
+ Traceback (most recent call last):
+ ...
+ ValueError: not an element of this algebra
+
"""
msg = "not an element of this algebra"
- if elt == 0:
- # The superclass implementation of random_element()
- # needs to be able to coerce "0" into the algebra.
- return self.zero()
- elif elt in self.base_ring():
+ if elt in self.base_ring():
# Ensure that no base ring -> algebra coercion is performed
# by this method. There's some stupidity in sage that would
# otherwise propagate to this method; for example, sage thinks
# that the integer 3 belongs to the space of 2-by-2 matrices.
raise ValueError(msg)
+ if hasattr(elt, 'superalgebra_element'):
+ # Handle subalgebra elements
+ if elt.parent().superalgebra() == self:
+ return elt.superalgebra_element()
+
+ if hasattr(elt, 'sparse_vector'):
+ # Convert a vector into a column-matrix. We check for
+ # "sparse_vector" and not "column" because matrices also
+ # have a "column" method.
+ elt = elt.column()
+
if elt not in self.matrix_space():
raise ValueError(msg)
# element's ring because the basis space might be an algebraic
# closure whereas the base ring of the 3-by-3 identity matrix
# could be QQ instead of QQbar.
- V = VectorSpace(self.base_ring(), elt.nrows()*elt.ncols())
- W = V.span_of_basis( _mat2vec(s) for s in self.matrix_basis() )
-
- try:
- coords = W.coordinate_vector(_mat2vec(elt))
- except ArithmeticError: # vector is not in free module
- raise ValueError(msg)
-
- return self.from_vector(coords)
-
- def _repr_(self):
- """
- Return a string representation of ``self``.
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import JordanSpinEJA
-
- TESTS:
-
- Ensure that it says what we think it says::
-
- sage: JordanSpinEJA(2, field=AA)
- Euclidean Jordan algebra of dimension 2 over Algebraic Real Field
- sage: JordanSpinEJA(3, field=RDF)
- Euclidean Jordan algebra of dimension 3 over Real Double Field
-
- """
- fmt = "Euclidean Jordan algebra of dimension {} over {}"
- return fmt.format(self.dimension(), self.base_ring())
-
- def product_on_basis(self, i, j):
- return self._multiplication_table[i][j]
-
- def _is_commutative(self):
- r"""
- Whether or not this algebra's multiplication table is commutative.
+ #
+ # And, we also have to handle Cartesian product bases (when
+ # the matrix basis consists of tuples) here. The "good news"
+ # is that we're already converting everything to long vectors,
+ # and that strategy works for tuples as well.
+ #
+ elt = self._matrix_span.ambient_vector_space()(_all2list(elt))
- This method should of course always return ``True``, unless
- this algebra was constructed with ``check_axioms=False`` and
- passed an invalid multiplication table.
- """
- return all( self.product_on_basis(i,j) == self.product_on_basis(i,j)
- for i in range(self.dimension())
- for j in range(self.dimension()) )
+ try:
+ coords = self._matrix_span.coordinate_vector(elt)
+ except ArithmeticError: # vector is not in free module
+ raise ValueError(msg)
- def _is_jordanian(self):
- r"""
- Whether or not this algebra's multiplication table respects the
- Jordan identity `(x^{2})(xy) = x(x^{2}y)`.
+ return self.from_vector(coords)
- We only check one arrangement of `x` and `y`, so for a
- ``True`` result to be truly true, you should also check
- :meth:`_is_commutative`. This method should of course always
- return ``True``, unless this algebra was constructed with
- ``check_axioms=False`` and passed an invalid multiplication table.
+ def _repr_(self):
"""
- return all( (self.monomial(i)**2)*(self.monomial(i)*self.monomial(j))
- ==
- (self.monomial(i))*((self.monomial(i)**2)*self.monomial(j))
- for i in range(self.dimension())
- for j in range(self.dimension()) )
+ Return a string representation of ``self``.
- def _inner_product_is_associative(self):
- r"""
- Return whether or not this algebra's inner product `B` is
- associative; that is, whether or not `B(xy,z) = B(x,yz)`.
+ SETUP::
- This method should of course always return ``True``, unless
- this algebra was constructed with ``check_axioms=False`` and
- passed an invalid multiplication table.
- """
+ sage: from mjo.eja.eja_algebra import JordanSpinEJA
- # Used to check whether or not something is zero in an inexact
- # ring. This number is sufficient to allow the construction of
- # QuaternionHermitianEJA(2, RDF) with check_axioms=True.
- epsilon = 1e-16
+ TESTS:
- for i in range(self.dimension()):
- for j in range(self.dimension()):
- for k in range(self.dimension()):
- x = self.monomial(i)
- y = self.monomial(j)
- z = self.monomial(k)
- diff = (x*y).inner_product(z) - x.inner_product(y*z)
+ Ensure that it says what we think it says::
+
+ sage: JordanSpinEJA(2, field=AA)
+ Euclidean Jordan algebra of dimension 2 over Algebraic Real Field
+ sage: JordanSpinEJA(3, field=RDF)
+ Euclidean Jordan algebra of dimension 3 over Real Double Field
- if self.base_ring().is_exact():
- if diff != 0:
- return False
- else:
- if diff.abs() > epsilon:
- return False
+ """
+ fmt = "Euclidean Jordan algebra of dimension {} over {}"
+ return fmt.format(self.dimension(), self.base_ring())
- return True
@cached_method
def characteristic_polynomial_of(self):
sage: J = JordanSpinEJA(3)
sage: p = J.characteristic_polynomial_of(); p
- X1^2 - X2^2 - X3^2 + (-2*t)*X1 + t^2
+ X0^2 - X1^2 - X2^2 + (-2*t)*X0 + t^2
sage: xvec = J.one().to_vector()
sage: p(*xvec)
t^2 - 2*t + 1
sage: J = HadamardEJA(2)
sage: J.coordinate_polynomial_ring()
- Multivariate Polynomial Ring in X1, X2...
- sage: J = RealSymmetricEJA(3,QQ)
+ Multivariate Polynomial Ring in X0, X1...
+ sage: J = RealSymmetricEJA(3,field=QQ,orthonormalize=False)
sage: J.coordinate_polynomial_ring()
- Multivariate Polynomial Ring in X1, X2, X3, X4, X5, X6...
+ Multivariate Polynomial Ring in X0, X1, X2, X3, X4, X5...
"""
- var_names = tuple( "X%d" % z for z in range(1, self.dimension()+1) )
+ var_names = tuple( "X%d" % z for z in range(self.dimension()) )
return PolynomialRing(self.base_ring(), var_names)
def inner_product(self, x, y):
Our inner product is "associative," which means the following for
a symmetric bilinear form::
- sage: set_random_seed()
sage: J = random_eja()
sage: x,y,z = J.random_elements(3)
sage: (x*y).inner_product(z) == y.inner_product(x*z)
Ensure that this is the usual inner product for the algebras
over `R^n`::
- sage: set_random_seed()
sage: J = HadamardEJA.random_instance()
sage: x,y = J.random_elements(2)
sage: actual = x.inner_product(y)
one). This is in Faraut and Koranyi, and also my "On the
symmetry..." paper::
- sage: set_random_seed()
sage: J = BilinearFormEJA.random_instance()
sage: n = J.dimension()
sage: x = J.random_element()
sage: J = JordanSpinEJA(4)
sage: J.multiplication_table()
+----++----+----+----+----+
- | * || e0 | e1 | e2 | e3 |
+ | * || b0 | b1 | b2 | b3 |
+====++====+====+====+====+
- | e0 || e0 | e1 | e2 | e3 |
+ | b0 || b0 | b1 | b2 | b3 |
+----++----+----+----+----+
- | e1 || e1 | e0 | 0 | 0 |
+ | b1 || b1 | b0 | 0 | 0 |
+----++----+----+----+----+
- | e2 || e2 | 0 | e0 | 0 |
+ | b2 || b2 | 0 | b0 | 0 |
+----++----+----+----+----+
- | e3 || e3 | 0 | 0 | e0 |
+ | b3 || b3 | 0 | 0 | b0 |
+----++----+----+----+----+
"""
- M = list(self._multiplication_table) # copy
- for i in range(len(M)):
- # M had better be "square"
- M[i] = [self.monomial(i)] + M[i]
- M = [["*"] + list(self.gens())] + M
+ n = self.dimension()
+ # Prepend the header row.
+ M = [["*"] + list(self.gens())]
+
+ # And to each subsequent row, prepend an entry that belongs to
+ # the left-side "header column."
+ M += [ [self.monomial(i)] + [ self.monomial(i)*self.monomial(j)
+ for j in range(n) ]
+ for i in range(n) ]
+
return table(M, header_row=True, header_column=True, frame=True)
Why implement this for non-matrix algebras? Avoiding special
cases for the :class:`BilinearFormEJA` pays with simplicity in
its own right. But mainly, we would like to be able to assume
- that elements of a :class:`DirectSumEJA` can be displayed
+ that elements of a :class:`CartesianProductEJA` can be displayed
nicely, without having to have special classes for direct sums
one of whose components was a matrix algebra.
sage: J = RealSymmetricEJA(2)
sage: J.basis()
- Finite family {0: e0, 1: e1, 2: e2}
+ Finite family {0: b0, 1: b1, 2: b2}
sage: J.matrix_basis()
(
[1 0] [ 0 0.7071067811865475?] [0 0]
sage: J = JordanSpinEJA(2)
sage: J.basis()
- Finite family {0: e0, 1: e1}
+ Finite family {0: b0, 1: b1}
sage: J.matrix_basis()
(
[1] [0]
[0], [1]
)
"""
- if self._matrix_basis is None:
- M = self.matrix_space()
- return tuple( M(b.to_vector()) for b in self.basis() )
- else:
- return self._matrix_basis
+ return self._matrix_basis
def matrix_space(self):
we think of them as matrices (including column vectors of the
appropriate size).
- Generally this will be an `n`-by-`1` column-vector space,
+ "By default" this will be an `n`-by-`1` column-matrix space,
except when the algebra is trivial. There it's `n`-by-`n`
(where `n` is zero), to ensure that two elements of the matrix
- space (empty matrices) can be multiplied.
+ space (empty matrices) can be multiplied. For algebras of
+ matrices, this returns the space in which their
+ real embeddings live.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
+ ....: JordanSpinEJA,
+ ....: QuaternionHermitianEJA,
+ ....: TrivialEJA)
+
+ EXAMPLES:
+
+ By default, the matrix representation is just a column-matrix
+ equivalent to the vector representation::
+
+ sage: J = JordanSpinEJA(3)
+ sage: J.matrix_space()
+ Full MatrixSpace of 3 by 1 dense matrices over Algebraic
+ Real Field
+
+ The matrix representation in the trivial algebra is
+ zero-by-zero instead of the usual `n`-by-one::
+
+ sage: J = TrivialEJA()
+ sage: J.matrix_space()
+ Full MatrixSpace of 0 by 0 dense matrices over Algebraic
+ Real Field
+
+ The matrix space for complex/quaternion Hermitian matrix EJA
+ is the space in which their real-embeddings live, not the
+ original complex/quaternion matrix space::
+
+ sage: J = ComplexHermitianEJA(2,field=QQ,orthonormalize=False)
+ sage: J.matrix_space()
+ Module of 2 by 2 matrices with entries in Algebraic Field over
+ the scalar ring Rational Field
+ sage: J = QuaternionHermitianEJA(1,field=QQ,orthonormalize=False)
+ sage: J.matrix_space()
+ Module of 1 by 1 matrices with entries in Quaternion
+ Algebra (-1, -1) with base ring Rational Field over
+ the scalar ring Rational Field
- Matrix algebras override this with something more useful.
"""
- if self.is_trivial():
- return MatrixSpace(self.base_ring(), 0)
- elif self._matrix_basis is None or len(self._matrix_basis) == 0:
- return MatrixSpace(self.base_ring(), self.dimension(), 1)
- else:
- return self._matrix_basis[0].matrix_space()
+ return self._matrix_space
@cached_method
sage: from mjo.eja.eja_algebra import (HadamardEJA,
....: random_eja)
- EXAMPLES::
+ EXAMPLES:
+
+ We can compute unit element in the Hadamard EJA::
+
+ sage: J = HadamardEJA(5)
+ sage: J.one()
+ b0 + b1 + b2 + b3 + b4
+
+ The unit element in the Hadamard EJA is inherited in the
+ subalgebras generated by its elements::
sage: J = HadamardEJA(5)
sage: J.one()
- e0 + e1 + e2 + e3 + e4
+ b0 + b1 + b2 + b3 + b4
+ sage: x = sum(J.gens())
+ sage: A = x.subalgebra_generated_by(orthonormalize=False)
+ sage: A.one()
+ c0
+ sage: A.one().superalgebra_element()
+ b0 + b1 + b2 + b3 + b4
TESTS:
- The identity element acts like the identity::
+ The identity element acts like the identity, regardless of
+ whether or not we orthonormalize::
- sage: set_random_seed()
sage: J = random_eja()
sage: x = J.random_element()
sage: J.one()*x == x and x*J.one() == x
True
+ sage: A = x.subalgebra_generated_by(orthonormalize=False)
+ sage: y = A.random_element()
+ sage: A.one()*y == y and y*A.one() == y
+ True
+
+ ::
+
+ sage: J = random_eja(field=QQ, orthonormalize=False)
+ sage: x = J.random_element()
+ sage: J.one()*x == x and x*J.one() == x
+ True
+ sage: A = x.subalgebra_generated_by(orthonormalize=False)
+ sage: y = A.random_element()
+ sage: A.one()*y == y and y*A.one() == y
+ True
- The matrix of the unit element's operator is the identity::
+ The matrix of the unit element's operator is the identity,
+ regardless of the base field and whether or not we
+ orthonormalize::
- sage: set_random_seed()
sage: J = random_eja()
sage: actual = J.one().operator().matrix()
sage: expected = matrix.identity(J.base_ring(), J.dimension())
sage: actual == expected
True
+ sage: x = J.random_element()
+ sage: A = x.subalgebra_generated_by(orthonormalize=False)
+ sage: actual = A.one().operator().matrix()
+ sage: expected = matrix.identity(A.base_ring(), A.dimension())
+ sage: actual == expected
+ True
+
+ ::
+
+ sage: J = random_eja(field=QQ, orthonormalize=False)
+ sage: actual = J.one().operator().matrix()
+ sage: expected = matrix.identity(J.base_ring(), J.dimension())
+ sage: actual == expected
+ True
+ sage: x = J.random_element()
+ sage: A = x.subalgebra_generated_by(orthonormalize=False)
+ sage: actual = A.one().operator().matrix()
+ sage: expected = matrix.identity(A.base_ring(), A.dimension())
+ sage: actual == expected
+ True
Ensure that the cached unit element (often precomputed by
hand) agrees with the computed one::
- sage: set_random_seed()
sage: J = random_eja()
sage: cached = J.one()
sage: J.one.clear_cache()
sage: J.one() == cached
True
+ ::
+
+ sage: J = random_eja(field=QQ, orthonormalize=False)
+ sage: cached = J.one()
+ sage: J.one.clear_cache()
+ sage: J.one() == cached
+ True
+
"""
# We can brute-force compute the matrices of the operators
# that correspond to the basis elements of this algebra.
#
# Of course, matrices aren't vectors in sage, so we have to
# appeal to the "long vectors" isometry.
- oper_vecs = [ _mat2vec(g.operator().matrix()) for g in self.gens() ]
+
+ V = VectorSpace(self.base_ring(), self.dimension()**2)
+ oper_vecs = [ V(g.operator().matrix().list()) for g in self.gens() ]
# Now we use basic linear algebra to find the coefficients,
# of the matrices-as-vectors-linear-combination, which should
# We used the isometry on the left-hand side already, but we
# still need to do it for the right-hand side. Recall that we
# wanted something that summed to the identity matrix.
- b = _mat2vec( matrix.identity(self.base_ring(), self.dimension()) )
+ b = V( matrix.identity(self.base_ring(), self.dimension()).list() )
# Now if there's an identity element in the algebra, this
# should work. We solve on the left to avoid having to
Every algebra decomposes trivially with respect to its identity
element::
- sage: set_random_seed()
sage: J = random_eja()
sage: J0,J5,J1 = J.peirce_decomposition(J.one())
sage: J0.dimension() == 0 and J5.dimension() == 0
elements in the two subalgebras are the projections onto their
respective subspaces of the superalgebra's identity element::
- sage: set_random_seed()
sage: J = random_eja()
sage: x = J.random_element()
sage: if not J.is_trivial():
if not c.is_idempotent():
raise ValueError("element is not idempotent: %s" % c)
- from mjo.eja.eja_subalgebra import FiniteDimensionalEuclideanJordanSubalgebra
-
# Default these to what they should be if they turn out to be
# trivial, because eigenspaces_left() won't return eigenvalues
# corresponding to trivial spaces (e.g. it returns only the
# eigenspace corresponding to lambda=1 if you take the
# decomposition relative to the identity element).
- trivial = FiniteDimensionalEuclideanJordanSubalgebra(self, ())
+ trivial = self.subalgebra((), check_axioms=False)
J0 = trivial # eigenvalue zero
J5 = VectorSpace(self.base_ring(), 0) # eigenvalue one-half
J1 = trivial # eigenvalue one
J5 = eigspace
else:
gens = tuple( self.from_vector(b) for b in eigspace.basis() )
- subalg = FiniteDimensionalEuclideanJordanSubalgebra(self,
- gens,
- check_axioms=False)
+ subalg = self.subalgebra(gens, check_axioms=False)
if eigval == 0:
J0 = subalg
elif eigval == 1:
# For a general base ring... maybe we can trust this to do the
# right thing? Unlikely, but.
V = self.vector_space()
- v = V.random_element()
-
- if self.base_ring() is AA:
- # The "random element" method of the algebraic reals is
- # stupid at the moment, and only returns integers between
- # -2 and 2, inclusive:
- #
- # https://trac.sagemath.org/ticket/30875
- #
- # Instead, we implement our own "random vector" method,
- # and then coerce that into the algebra. We use the vector
- # space degree here instead of the dimension because a
- # subalgebra could (for example) be spanned by only two
- # vectors, each with five coordinates. We need to
- # generate all five coordinates.
- if thorough:
- v *= QQbar.random_element().real()
- else:
- v *= QQ.random_element()
+ if self.base_ring() is AA and not thorough:
+ # Now that AA generates actually random random elements
+ # (post Trac 30875), we only need to de-thorough the
+ # randomness when asked to.
+ V = V.change_ring(QQ)
+ v = V.random_element()
return self.from_vector(V.coordinate_vector(v))
def random_elements(self, count, thorough=False):
for idx in range(count) )
+ def operator_polynomial_matrix(self):
+ r"""
+ Return the matrix of polynomials (over this algebra's
+ :meth:`coordinate_polynomial_ring`) that, when evaluated at
+ the basis coordinates of an element `x`, produces the basis
+ representation of `L_{x}`.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (HadamardEJA,
+ ....: JordanSpinEJA)
+
+ EXAMPLES::
+
+ sage: J = HadamardEJA(4)
+ sage: L_x = J.operator_polynomial_matrix()
+ sage: L_x
+ [X0 0 0 0]
+ [ 0 X1 0 0]
+ [ 0 0 X2 0]
+ [ 0 0 0 X3]
+ sage: x = J.one()
+ sage: d = zip(J.coordinate_polynomial_ring().gens(), x.to_vector())
+ sage: L_x.subs(dict(d))
+ [1 0 0 0]
+ [0 1 0 0]
+ [0 0 1 0]
+ [0 0 0 1]
+
+ ::
+
+ sage: J = JordanSpinEJA(4)
+ sage: L_x = J.operator_polynomial_matrix()
+ sage: L_x
+ [X0 X1 X2 X3]
+ [X1 X0 0 0]
+ [X2 0 X0 0]
+ [X3 0 0 X0]
+ sage: x = J.one()
+ sage: d = zip(J.coordinate_polynomial_ring().gens(), x.to_vector())
+ sage: L_x.subs(dict(d))
+ [1 0 0 0]
+ [0 1 0 0]
+ [0 0 1 0]
+ [0 0 0 1]
+
+ """
+ R = self.coordinate_polynomial_ring()
+
+ def L_x_i_j(i,j):
+ # From a result in my book, these are the entries of the
+ # basis representation of L_x.
+ return sum( v*self.monomial(k).operator().matrix()[i,j]
+ for (k,v) in enumerate(R.gens()) )
+
+ n = self.dimension()
+ return matrix(R, n, n, L_x_i_j)
+
@cached_method
def _charpoly_coefficients(self):
r"""
The `r` polynomial coefficients of the "characteristic polynomial
of" function.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import random_eja
+
+ TESTS:
+
+ The theory shows that these are all homogeneous polynomials of
+ a known degree::
+
+ sage: J = random_eja()
+ sage: all(p.is_homogeneous() for p in J._charpoly_coefficients())
+ True
+
"""
n = self.dimension()
R = self.coordinate_polynomial_ring()
- vars = R.gens()
F = R.fraction_field()
- def L_x_i_j(i,j):
- # From a result in my book, these are the entries of the
- # basis representation of L_x.
- return sum( vars[k]*self.monomial(k).operator().matrix()[i,j]
- for k in range(n) )
-
- L_x = matrix(F, n, n, L_x_i_j)
+ L_x = self.operator_polynomial_matrix()
r = None
if self.rank.is_in_cache():
# The theory says that only the first "r" coefficients are
# nonzero, and they actually live in the original polynomial
- # ring and not the fraction field. We negate them because
- # in the actual characteristic polynomial, they get moved
- # to the other side where x^r lives.
- return -A_rref.solve_right(E*b).change_ring(R)[:r]
+ # ring and not the fraction field. We negate them because in
+ # the actual characteristic polynomial, they get moved to the
+ # other side where x^r lives. We don't bother to trim A_rref
+ # down to a square matrix and solve the resulting system,
+ # because the upper-left r-by-r portion of A_rref is
+ # guaranteed to be the identity matrix, so e.g.
+ #
+ # A_rref.solve_right(Y)
+ #
+ # would just be returning Y.
+ return (-E*b)[:r].change_ring(R)
@cached_method
def rank(self):
positive integer rank, unless the algebra is trivial in
which case its rank will be zero::
- sage: set_random_seed()
sage: J = random_eja()
sage: r = J.rank()
sage: r in ZZ
Ensure that computing the rank actually works, since the ranks
of all simple algebras are known and will be cached by default::
- sage: set_random_seed() # long time
sage: J = random_eja() # long time
- sage: caches = J.rank() # long time
+ sage: cached = J.rank() # long time
sage: J.rank.clear_cache() # long time
sage: J.rank() == cached # long time
True
return len(self._charpoly_coefficients())
+ def subalgebra(self, basis, **kwargs):
+ r"""
+ Create a subalgebra of this algebra from the given basis.
+ """
+ from mjo.eja.eja_subalgebra import EJASubalgebra
+ return EJASubalgebra(self, basis, **kwargs)
+
+
def vector_space(self):
"""
Return the vector space that underlies this algebra.
return self.zero().to_vector().parent().ambient_vector_space()
- Element = FiniteDimensionalEuclideanJordanAlgebraElement
-
-class RationalBasisEuclideanJordanAlgebraNg(FiniteDimensionalEuclideanJordanAlgebra):
- def __init__(self,
- field,
- basis,
- jordan_product,
- inner_product,
- orthonormalize=True,
- prefix='e',
- category=None,
- check_field=True,
- check_axioms=True):
-
- n = len(basis)
- vector_basis = basis
- from sage.structure.element import is_Matrix
- basis_is_matrices = False
+class RationalBasisEJA(EJA):
+ r"""
+ Algebras whose supplied basis elements have all rational entries.
- degree = 0
- if n > 0:
- if is_Matrix(basis[0]):
- basis_is_matrices = True
- vector_basis = tuple( map(_mat2vec,basis) )
- degree = basis[0].nrows()**2
- else:
- degree = basis[0].degree()
+ SETUP::
- V = VectorSpace(field, degree)
+ sage: from mjo.eja.eja_algebra import BilinearFormEJA
- # Compute this from "Q" (obtained from Gram-Schmidt) below as
- # R = Q.solve_right(A), where the rows of "Q" are the
- # orthonormalized vector_basis and and the rows of "A" are the
- # original vector_basis.
- self._deorthonormalization_matrix = None
+ EXAMPLES:
- if orthonormalize:
- from mjo.eja.eja_utils import gram_schmidt
- vector_basis = gram_schmidt(vector_basis, inner_product)
- W = V.span_of_basis( vector_basis )
- if basis_is_matrices:
- from mjo.eja.eja_utils import _vec2mat
- basis = tuple( map(_vec2mat,vector_basis) )
+ The supplied basis is orthonormalized by default::
- W = V.span_of_basis( vector_basis )
+ sage: B = matrix(QQ, [[1, 0, 0], [0, 25, -32], [0, -32, 41]])
+ sage: J = BilinearFormEJA(B)
+ sage: J.matrix_basis()
+ (
+ [1] [ 0] [ 0]
+ [0] [1/5] [32/5]
+ [0], [ 0], [ 5]
+ )
- mult_table = [ [0 for i in range(n)] for j in range(n) ]
- ip_table = [ [0 for i in range(n)] for j in range(n) ]
+ """
+ def __init__(self,
+ basis,
+ jordan_product,
+ inner_product,
+ field=AA,
+ check_field=True,
+ **kwargs):
- for i in range(n):
- for j in range(i+1):
- # do another mat2vec because the multiplication
- # table is in terms of vectors
- elt = _mat2vec(jordan_product(basis[i],basis[j]))
- elt = W.coordinate_vector(elt)
- mult_table[i][j] = elt
- mult_table[j][i] = elt
- ip = inner_product(basis[i],basis[j])
- ip_table[i][j] = ip
- ip_table[j][i] = ip
-
- self._inner_product_matrix = matrix(field,ip_table)
-
- if basis_is_matrices:
- for m in basis:
- m.set_immutable()
+ if check_field:
+ # Abuse the check_field parameter to check that the entries of
+ # out basis (in ambient coordinates) are in the field QQ.
+ # Use _all2list to get the vector coordinates of octonion
+ # entries and not the octonions themselves (which are not
+ # rational).
+ if not all( all(b_i in QQ for b_i in _all2list(b))
+ for b in basis ):
+ raise TypeError("basis not rational")
+
+ super().__init__(basis,
+ jordan_product,
+ inner_product,
+ field=field,
+ check_field=check_field,
+ **kwargs)
+
+ self._rational_algebra = None
+ if field is not QQ:
+ # There's no point in constructing the extra algebra if this
+ # one is already rational.
+ #
+ # Note: the same Jordan and inner-products work here,
+ # because they are necessarily defined with respect to
+ # ambient coordinates and not any particular basis.
+ self._rational_algebra = EJA(
+ basis,
+ jordan_product,
+ inner_product,
+ field=QQ,
+ matrix_space=self.matrix_space(),
+ associative=self.is_associative(),
+ orthonormalize=False,
+ check_field=False,
+ check_axioms=False)
+
+ def rational_algebra(self):
+ # Using None as a flag here (rather than just assigning "self"
+ # to self._rational_algebra by default) feels a little bit
+ # more sane to me in a garbage-collected environment.
+ if self._rational_algebra is None:
+ return self
else:
- basis = tuple( x.column() for x in basis )
+ return self._rational_algebra
- super().__init__(field,
- mult_table,
- prefix,
- category,
- basis, # matrix basis
- check_field,
- check_axioms)
-
-class RationalBasisEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra):
- r"""
- Algebras whose basis consists of vectors with rational
- entries. Equivalently, algebras whose multiplication tables
- contain only rational coefficients.
-
- When an EJA has a basis that can be made rational, we can speed up
- the computation of its characteristic polynomial by doing it over
- ``QQ``. All of the named EJA constructors that we provide fall
- into this category.
- """
@cached_method
def _charpoly_coefficients(self):
r"""
- Override the parent method with something that tries to compute
- over a faster (non-extension) field.
-
SETUP::
- sage: from mjo.eja.eja_algebra import JordanSpinEJA
+ sage: from mjo.eja.eja_algebra import (BilinearFormEJA,
+ ....: JordanSpinEJA)
EXAMPLES:
sage: J = JordanSpinEJA(3)
sage: J._charpoly_coefficients()
- (X1^2 - X2^2 - X3^2, -2*X1)
+ (X0^2 - X1^2 - X2^2, -2*X0)
sage: a0 = J._charpoly_coefficients()[0]
sage: J.base_ring()
Algebraic Real Field
Algebraic Real Field
"""
- if self.base_ring() is QQ:
- # There's no need to construct *another* algebra over the
- # rationals if this one is already over the rationals.
- superclass = super(RationalBasisEuclideanJordanAlgebra, self)
- return superclass._charpoly_coefficients()
-
- mult_table = tuple(
- tuple(map(lambda x: x.to_vector(), ls))
- for ls in self._multiplication_table
- )
+ if self.rational_algebra() is self:
+ # Bypass the hijinks if they won't benefit us.
+ return super()._charpoly_coefficients()
+
+ # Do the computation over the rationals.
+ a = ( a_i.change_ring(self.base_ring())
+ for a_i in self.rational_algebra()._charpoly_coefficients() )
- # Do the computation over the rationals. The answer will be
- # the same, because our basis coordinates are (essentially)
- # rational.
- J = FiniteDimensionalEuclideanJordanAlgebra(QQ,
- mult_table,
- check_field=False,
- check_axioms=False)
- a = J._charpoly_coefficients()
- return tuple(map(lambda x: x.change_ring(self.base_ring()), a))
+ # Convert our coordinate variables into deorthonormalized ones
+ # and substitute them into the deorthonormalized charpoly
+ # coefficients.
+ R = self.coordinate_polynomial_ring()
+ from sage.modules.free_module_element import vector
+ X = vector(R, R.gens())
+ BX = self._deortho_matrix*X
+ subs_dict = { X[i]: BX[i] for i in range(len(X)) }
+ return tuple( a_i.subs(subs_dict) for a_i in a )
-class ConcreteEuclideanJordanAlgebra:
+class ConcreteEJA(EJA):
r"""
A class for the Euclidean Jordan algebras that we know by name.
SETUP::
- sage: from mjo.eja.eja_algebra import ConcreteEuclideanJordanAlgebra
+ sage: from mjo.eja.eja_algebra import ConcreteEJA
TESTS:
Our basis is normalized with respect to the algebra's inner
product, unless we specify otherwise::
- sage: set_random_seed()
- sage: J = ConcreteEuclideanJordanAlgebra.random_instance()
+ sage: J = ConcreteEJA.random_instance()
sage: all( b.norm() == 1 for b in J.gens() )
True
natural->EJA basis representation is an isometry and within the
EJA the operator is self-adjoint by the Jordan axiom::
- sage: set_random_seed()
- sage: J = ConcreteEuclideanJordanAlgebra.random_instance()
- sage: x = J.random_element()
- sage: x.operator().is_self_adjoint()
- True
- """
+ sage: J = ConcreteEJA.random_instance()
+ sage: x = J.random_element()
+ sage: x.operator().is_self_adjoint()
+ True
+ """
+
+ @staticmethod
+ def _max_random_instance_dimension():
+ r"""
+ The maximum dimension of any random instance. Ten dimensions seems
+ to be about the point where everything takes a turn for the
+ worse. And dimension ten (but not nine) allows the 4-by-4 real
+ Hermitian matrices, the 2-by-2 quaternion Hermitian matrices,
+ and the 2-by-2 octonion Hermitian matrices.
+ """
+ return 10
+
+ @staticmethod
+ def _max_random_instance_size(max_dimension):
+ """
+ Return an integer "size" that is an upper bound on the size of
+ this algebra when it is used in a random test case. This size
+ (which can be passed to the algebra's constructor) is itself
+ based on the ``max_dimension`` parameter.
+
+ This method must be implemented in each subclass.
+ """
+ raise NotImplementedError
+
+ @classmethod
+ def random_instance(cls, max_dimension=None, *args, **kwargs):
+ """
+ Return a random instance of this type of algebra whose dimension
+ is less than or equal to the lesser of ``max_dimension`` and
+ the value returned by ``_max_random_instance_dimension()``. If
+ the dimension bound is omitted, then only the
+ ``_max_random_instance_dimension()`` is used as a bound.
+
+ This method should be implemented in each subclass.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import ConcreteEJA
+
+ TESTS:
+
+ Both the class bound and the ``max_dimension`` argument are upper
+ bounds on the dimension of the algebra returned::
+
+ sage: from sage.misc.prandom import choice
+ sage: eja_class = choice(ConcreteEJA.__subclasses__())
+ sage: class_max_d = eja_class._max_random_instance_dimension()
+ sage: J = eja_class.random_instance(max_dimension=20,
+ ....: field=QQ,
+ ....: orthonormalize=False)
+ sage: J.dimension() <= class_max_d
+ True
+ sage: J = eja_class.random_instance(max_dimension=2,
+ ....: field=QQ,
+ ....: orthonormalize=False)
+ sage: J.dimension() <= 2
+ True
+
+ """
+ from sage.misc.prandom import choice
+ eja_class = choice(cls.__subclasses__())
+
+ # These all bubble up to the RationalBasisEJA superclass
+ # constructor, so any (kw)args valid there are also valid
+ # here.
+ return eja_class.random_instance(max_dimension, *args, **kwargs)
+
+
+class HermitianMatrixEJA(EJA):
+ @staticmethod
+ def _denormalized_basis(A):
+ """
+ Returns a basis for the given Hermitian matrix space.
+
+ Why do we embed these? Basically, because all of numerical linear
+ algebra assumes that you're working with vectors consisting of `n`
+ entries from a field and scalars from the same field. There's no way
+ to tell SageMath that (for example) the vectors contain complex
+ numbers, while the scalar field is real.
+
+ SETUP::
+
+ sage: from mjo.hurwitz import (ComplexMatrixAlgebra,
+ ....: QuaternionMatrixAlgebra,
+ ....: OctonionMatrixAlgebra)
+ sage: from mjo.eja.eja_algebra import HermitianMatrixEJA
+
+ TESTS::
+
+ sage: n = ZZ.random_element(1,5)
+ sage: A = MatrixSpace(QQ, n)
+ sage: B = HermitianMatrixEJA._denormalized_basis(A)
+ sage: all( M.is_hermitian() for M in B)
+ True
+
+ ::
+
+ sage: n = ZZ.random_element(1,5)
+ sage: A = ComplexMatrixAlgebra(n, scalars=QQ)
+ sage: B = HermitianMatrixEJA._denormalized_basis(A)
+ sage: all( M.is_hermitian() for M in B)
+ True
+
+ ::
+
+ sage: n = ZZ.random_element(1,5)
+ sage: A = QuaternionMatrixAlgebra(n, scalars=QQ)
+ sage: B = HermitianMatrixEJA._denormalized_basis(A)
+ sage: all( M.is_hermitian() for M in B )
+ True
- @staticmethod
- def _max_random_instance_size():
- """
- Return an integer "size" that is an upper bound on the size of
- this algebra when it is used in a random test
- case. Unfortunately, the term "size" is ambiguous -- when
- dealing with `R^n` under either the Hadamard or Jordan spin
- product, the "size" refers to the dimension `n`. When dealing
- with a matrix algebra (real symmetric or complex/quaternion
- Hermitian), it refers to the size of the matrix, which is far
- less than the dimension of the underlying vector space.
+ ::
- This method must be implemented in each subclass.
- """
- raise NotImplementedError
+ sage: n = ZZ.random_element(1,5)
+ sage: A = OctonionMatrixAlgebra(n, scalars=QQ)
+ sage: B = HermitianMatrixEJA._denormalized_basis(A)
+ sage: all( M.is_hermitian() for M in B )
+ True
- @classmethod
- def random_instance(cls, field=AA, **kwargs):
"""
- Return a random instance of this type of algebra.
+ # These work for real MatrixSpace, whose monomials only have
+ # two coordinates (because the last one would always be "1").
+ es = A.base_ring().gens()
+ gen = lambda A,m: A.monomial(m[:2])
- This method should be implemented in each subclass.
- """
- from sage.misc.prandom import choice
- eja_class = choice(cls.__subclasses__())
- return eja_class.random_instance(field)
-
-
-class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra):
-
- def __init__(self, field, basis, normalize_basis=True, **kwargs):
- """
- Compared to the superclass constructor, we take a basis instead of
- a multiplication table because the latter can be computed in terms
- of the former when the product is known (like it is here).
- """
- # Used in this class's fast _charpoly_coefficients() override.
- self._basis_normalizers = None
-
- # We're going to loop through this a few times, so now's a good
- # time to ensure that it isn't a generator expression.
- basis = tuple(basis)
-
- algebra_dim = len(basis)
- degree = 0 # size of the matrices
- if algebra_dim > 0:
- degree = basis[0].nrows()
-
- if algebra_dim > 1 and normalize_basis:
- # We'll need sqrt(2) to normalize the basis, and this
- # winds up in the multiplication table, so the whole
- # algebra needs to be over the field extension.
- R = PolynomialRing(field, 'z')
- z = R.gen()
- p = z**2 - 2
- if p.is_irreducible():
- field = field.extension(p, 'sqrt2', embedding=RLF(2).sqrt())
- basis = tuple( s.change_ring(field) for s in basis )
- self._basis_normalizers = tuple(
- ~(self.matrix_inner_product(s,s).sqrt()) for s in basis )
- basis = tuple(s*c for (s,c) in zip(basis,self._basis_normalizers))
-
- # Now compute the multiplication and inner product tables.
- # We have to do this *after* normalizing the basis, because
- # scaling affects the answers.
- V = VectorSpace(field, degree**2)
- W = V.span_of_basis( _mat2vec(s) for s in basis )
- mult_table = [[W.zero() for j in range(algebra_dim)]
- for i in range(algebra_dim)]
- ip_table = [[W.zero() for j in range(algebra_dim)]
- for i in range(algebra_dim)]
- for i in range(algebra_dim):
- for j in range(algebra_dim):
- mat_entry = (basis[i]*basis[j] + basis[j]*basis[i])/2
- mult_table[i][j] = W.coordinate_vector(_mat2vec(mat_entry))
-
- try:
- # HACK: ignore the error here if we don't need the
- # inner product (as is the case when we construct
- # a dummy QQ-algebra for fast charpoly coefficients.
- ip_table[i][j] = self.matrix_inner_product(basis[i],
- basis[j])
- except:
- pass
+ if hasattr(A, 'entry_algebra_gens'):
+ # We've got a MatrixAlgebra, and its monomials will have
+ # three coordinates.
+ es = A.entry_algebra_gens()
+ gen = lambda A,m: A.monomial(m)
- try:
- # HACK PART DEUX
- self._inner_product_matrix = matrix(field,ip_table)
- except:
- pass
-
- super(MatrixEuclideanJordanAlgebra, self).__init__(field,
- mult_table,
- matrix_basis=basis,
- **kwargs)
-
- if algebra_dim == 0:
- self.one.set_cache(self.zero())
- else:
- n = basis[0].nrows()
- # The identity wrt (A,B) -> (AB + BA)/2 is independent of the
- # details of this algebra.
- self.one.set_cache(self(matrix.identity(field,n)))
+ basis = []
+ for i in range(A.nrows()):
+ for j in range(i+1):
+ if i == j:
+ E_ii = gen(A, (i,j,es[0]))
+ basis.append(E_ii)
+ else:
+ for e in es:
+ E_ij = gen(A, (i,j,e))
+ E_ij += E_ij.conjugate_transpose()
+ basis.append(E_ij)
+ return tuple( basis )
- @cached_method
- def _charpoly_coefficients(self):
+ @staticmethod
+ def jordan_product(X,Y):
+ return (X*Y + Y*X)/2
+
+ @staticmethod
+ def trace_inner_product(X,Y):
r"""
- Override the parent method with something that tries to compute
- over a faster (non-extension) field.
- """
- if self._basis_normalizers is None or self.base_ring() is QQ:
- # We didn't normalize, or the basis we started with had
- # entries in a nice field already. Just compute the thing.
- return super(MatrixEuclideanJordanAlgebra, self)._charpoly_coefficients()
-
- basis = ( (b/n) for (b,n) in zip(self.matrix_basis(),
- self._basis_normalizers) )
-
- # Do this over the rationals and convert back at the end.
- # Only works because we know the entries of the basis are
- # integers. The argument ``check_axioms=False`` is required
- # because the trace inner-product method for this
- # class is a stub and can't actually be checked.
- J = MatrixEuclideanJordanAlgebra(QQ,
- basis,
- normalize_basis=False,
- check_field=False,
- check_axioms=False)
- a = J._charpoly_coefficients()
-
- # Unfortunately, changing the basis does change the
- # coefficients of the characteristic polynomial, but since
- # these are really the coefficients of the "characteristic
- # polynomial of" function, everything is still nice and
- # unevaluated. It's therefore "obvious" how scaling the
- # basis affects the coordinate variables X1, X2, et
- # cetera. Scaling the first basis vector up by "n" adds a
- # factor of 1/n into every "X1" term, for example. So here
- # we simply undo the basis_normalizer scaling that we
- # performed earlier.
- #
- # The a[0] access here is safe because trivial algebras
- # won't have any basis normalizers and therefore won't
- # make it to this "else" branch.
- XS = a[0].parent().gens()
- subs_dict = { XS[i]: self._basis_normalizers[i]*XS[i]
- for i in range(len(XS)) }
- return tuple( a_i.subs(subs_dict) for a_i in a )
+ A trace inner-product for matrices that aren't embedded in the
+ reals. It takes MATRICES as arguments, not EJA elements.
+ SETUP::
- @staticmethod
- def real_embed(M):
- """
- Embed the matrix ``M`` into a space of real matrices.
+ sage: from mjo.eja.eja_algebra import (RealSymmetricEJA,
+ ....: ComplexHermitianEJA,
+ ....: QuaternionHermitianEJA,
+ ....: OctonionHermitianEJA)
- The matrix ``M`` can have entries in any field at the moment:
- the real numbers, complex numbers, or quaternions. And although
- they are not a field, we can probably support octonions at some
- point, too. This function returns a real matrix that "acts like"
- the original with respect to matrix multiplication; i.e.
+ EXAMPLES::
- real_embed(M*N) = real_embed(M)*real_embed(N)
+ sage: J = RealSymmetricEJA(2,field=QQ,orthonormalize=False)
+ sage: I = J.one().to_matrix()
+ sage: J.trace_inner_product(I, -I)
+ -2
- """
- raise NotImplementedError
+ ::
+ sage: J = ComplexHermitianEJA(2,field=QQ,orthonormalize=False)
+ sage: I = J.one().to_matrix()
+ sage: J.trace_inner_product(I, -I)
+ -2
- @staticmethod
- def real_unembed(M):
- """
- The inverse of :meth:`real_embed`.
- """
- raise NotImplementedError
+ ::
- @classmethod
- def matrix_inner_product(cls,X,Y):
- Xu = cls.real_unembed(X)
- Yu = cls.real_unembed(Y)
- tr = (Xu*Yu).trace()
+ sage: J = QuaternionHermitianEJA(2,field=QQ,orthonormalize=False)
+ sage: I = J.one().to_matrix()
+ sage: J.trace_inner_product(I, -I)
+ -2
- try:
- # Works in QQ, AA, RDF, et cetera.
- return tr.real()
- except AttributeError:
- # A quaternion doesn't have a real() method, but does
- # have coefficient_tuple() method that returns the
- # coefficients of 1, i, j, and k -- in that order.
- return tr.coefficient_tuple()[0]
+ ::
+ sage: J = OctonionHermitianEJA(2,field=QQ,orthonormalize=False)
+ sage: I = J.one().to_matrix()
+ sage: J.trace_inner_product(I, -I)
+ -2
-class RealMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra):
- @staticmethod
- def real_embed(M):
- """
- The identity function, for embedding real matrices into real
- matrices.
"""
- return M
+ tr = (X*Y).trace()
+ if hasattr(tr, 'coefficient'):
+ # Works for octonions, and has to come first because they
+ # also have a "real()" method that doesn't return an
+ # element of the scalar ring.
+ return tr.coefficient(0)
+ elif hasattr(tr, 'coefficient_tuple'):
+ # Works for quaternions.
+ return tr.coefficient_tuple()[0]
- @staticmethod
- def real_unembed(M):
- """
- The identity function, for unembedding real matrices from real
- matrices.
- """
- return M
+ # Works for real and complex numbers.
+ return tr.real()
+
+
+ def __init__(self, matrix_space, **kwargs):
+ # We know this is a valid EJA, but will double-check
+ # if the user passes check_axioms=True.
+ if "check_axioms" not in kwargs: kwargs["check_axioms"] = False
+ super().__init__(self._denormalized_basis(matrix_space),
+ self.jordan_product,
+ self.trace_inner_product,
+ field=matrix_space.base_ring(),
+ matrix_space=matrix_space,
+ **kwargs)
-class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra,
- ConcreteEuclideanJordanAlgebra):
+ self.rank.set_cache(matrix_space.nrows())
+ self.one.set_cache( self(matrix_space.one()) )
+
+class RealSymmetricEJA(HermitianMatrixEJA, RationalBasisEJA, ConcreteEJA):
"""
The rank-n simple EJA consisting of real symmetric n-by-n
matrices, the usual symmetric Jordan product, and the trace inner
EXAMPLES::
sage: J = RealSymmetricEJA(2)
- sage: e0, e1, e2 = J.gens()
- sage: e0*e0
- e0
- sage: e1*e1
- 1/2*e0 + 1/2*e2
- sage: e2*e2
- e2
+ sage: b0, b1, b2 = J.gens()
+ sage: b0*b0
+ b0
+ sage: b1*b1
+ 1/2*b0 + 1/2*b2
+ sage: b2*b2
+ b2
In theory, our "field" can be any subfield of the reals::
- sage: RealSymmetricEJA(2, RDF)
+ sage: RealSymmetricEJA(2, field=RDF, check_axioms=True)
Euclidean Jordan algebra of dimension 3 over Real Double Field
- sage: RealSymmetricEJA(2, RR)
+ sage: RealSymmetricEJA(2, field=RR, check_axioms=True)
Euclidean Jordan algebra of dimension 3 over Real Field with
53 bits of precision
The dimension of this algebra is `(n^2 + n) / 2`::
- sage: set_random_seed()
- sage: n_max = RealSymmetricEJA._max_random_instance_size()
- sage: n = ZZ.random_element(1, n_max)
+ sage: d = RealSymmetricEJA._max_random_instance_dimension()
+ sage: n = RealSymmetricEJA._max_random_instance_size(d)
sage: J = RealSymmetricEJA(n)
sage: J.dimension() == (n^2 + n)/2
True
The Jordan multiplication is what we think it is::
- sage: set_random_seed()
sage: J = RealSymmetricEJA.random_instance()
sage: x,y = J.random_elements(2)
sage: actual = (x*y).to_matrix()
Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
"""
- @classmethod
- def _denormalized_basis(cls, n, field):
- """
- Return a basis for the space of real symmetric n-by-n matrices.
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import RealSymmetricEJA
-
- TESTS::
-
- sage: set_random_seed()
- sage: n = ZZ.random_element(1,5)
- sage: B = RealSymmetricEJA._denormalized_basis(n,QQ)
- sage: all( M.is_symmetric() for M in B)
- True
-
- """
- # The basis of symmetric matrices, as matrices, in their R^(n-by-n)
- # coordinates.
- S = []
- for i in range(n):
- for j in range(i+1):
- Eij = matrix(field, n, lambda k,l: k==i and l==j)
- if i == j:
- Sij = Eij
- else:
- Sij = Eij + Eij.transpose()
- S.append(Sij)
- return S
-
-
@staticmethod
- def _max_random_instance_size():
- return 4 # Dimension 10
+ def _max_random_instance_size(max_dimension):
+ # Obtained by solving d = (n^2 + n)/2.
+ # The ZZ-int-ZZ thing is just "floor."
+ return ZZ(int(ZZ(8*max_dimension + 1).sqrt()/2 - 1/2))
@classmethod
- def random_instance(cls, field=AA, **kwargs):
+ def random_instance(cls, max_dimension=None, *args, **kwargs):
"""
Return a random instance of this type of algebra.
"""
- n = ZZ.random_element(cls._max_random_instance_size() + 1)
- return cls(n, field, **kwargs)
+ class_max_d = cls._max_random_instance_dimension()
+ if (max_dimension is None or max_dimension > class_max_d):
+ max_dimension = class_max_d
+ max_size = cls._max_random_instance_size(max_dimension)
+ n = ZZ.random_element(max_size + 1)
+ return cls(n, **kwargs)
def __init__(self, n, field=AA, **kwargs):
- basis = self._denormalized_basis(n, field)
- super(RealSymmetricEJA, self).__init__(field,
- basis,
- check_axioms=False,
- **kwargs)
- self.rank.set_cache(n)
-
-
-class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra):
- @staticmethod
- def real_embed(M):
- """
- Embed the n-by-n complex matrix ``M`` into the space of real
- matrices of size 2n-by-2n via the map the sends each entry `z = a +
- bi` to the block matrix ``[[a,b],[-b,a]]``.
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import \
- ....: ComplexMatrixEuclideanJordanAlgebra
-
- EXAMPLES::
-
- sage: F = QuadraticField(-1, 'I')
- sage: x1 = F(4 - 2*i)
- sage: x2 = F(1 + 2*i)
- sage: x3 = F(-i)
- sage: x4 = F(6)
- sage: M = matrix(F,2,[[x1,x2],[x3,x4]])
- sage: ComplexMatrixEuclideanJordanAlgebra.real_embed(M)
- [ 4 -2| 1 2]
- [ 2 4|-2 1]
- [-----+-----]
- [ 0 -1| 6 0]
- [ 1 0| 0 6]
-
- TESTS:
-
- Embedding is a homomorphism (isomorphism, in fact)::
-
- sage: set_random_seed()
- sage: n = ZZ.random_element(3)
- sage: F = QuadraticField(-1, 'I')
- sage: X = random_matrix(F, n)
- sage: Y = random_matrix(F, n)
- sage: Xe = ComplexMatrixEuclideanJordanAlgebra.real_embed(X)
- sage: Ye = ComplexMatrixEuclideanJordanAlgebra.real_embed(Y)
- sage: XYe = ComplexMatrixEuclideanJordanAlgebra.real_embed(X*Y)
- sage: Xe*Ye == XYe
- True
-
- """
- n = M.nrows()
- if M.ncols() != n:
- raise ValueError("the matrix 'M' must be square")
-
- # We don't need any adjoined elements...
- field = M.base_ring().base_ring()
-
- blocks = []
- for z in M.list():
- a = z.list()[0] # real part, I guess
- b = z.list()[1] # imag part, I guess
- blocks.append(matrix(field, 2, [[a,b],[-b,a]]))
-
- return matrix.block(field, n, blocks)
-
-
- @staticmethod
- def real_unembed(M):
- """
- The inverse of _embed_complex_matrix().
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import \
- ....: ComplexMatrixEuclideanJordanAlgebra
-
- EXAMPLES::
-
- sage: A = matrix(QQ,[ [ 1, 2, 3, 4],
- ....: [-2, 1, -4, 3],
- ....: [ 9, 10, 11, 12],
- ....: [-10, 9, -12, 11] ])
- sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(A)
- [ 2*I + 1 4*I + 3]
- [ 10*I + 9 12*I + 11]
-
- TESTS:
-
- Unembedding is the inverse of embedding::
-
- sage: set_random_seed()
- sage: F = QuadraticField(-1, 'I')
- sage: M = random_matrix(F, 3)
- sage: Me = ComplexMatrixEuclideanJordanAlgebra.real_embed(M)
- sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(Me) == M
- True
-
- """
- n = ZZ(M.nrows())
- if M.ncols() != n:
- raise ValueError("the matrix 'M' must be square")
- if not n.mod(2).is_zero():
- raise ValueError("the matrix 'M' must be a complex embedding")
-
- # If "M" was normalized, its base ring might have roots
- # adjoined and they can stick around after unembedding.
- field = M.base_ring()
- R = PolynomialRing(field, 'z')
- z = R.gen()
- if field is AA:
- # Sage doesn't know how to embed AA into QQbar, i.e. how
- # to adjoin sqrt(-1) to AA.
- F = QQbar
- else:
- F = field.extension(z**2 + 1, 'I', embedding=CLF(-1).sqrt())
- i = F.gen()
-
- # Go top-left to bottom-right (reading order), converting every
- # 2-by-2 block we see to a single complex element.
- elements = []
- for k in range(n/2):
- for j in range(n/2):
- submat = M[2*k:2*k+2,2*j:2*j+2]
- if submat[0,0] != submat[1,1]:
- raise ValueError('bad on-diagonal submatrix')
- if submat[0,1] != -submat[1,0]:
- raise ValueError('bad off-diagonal submatrix')
- z = submat[0,0] + submat[0,1]*i
- elements.append(z)
-
- return matrix(F, n/2, elements)
-
-
- @classmethod
- def matrix_inner_product(cls,X,Y):
- """
- Compute a matrix inner product in this algebra directly from
- its real embedding.
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
+ A = MatrixSpace(field, n)
+ super().__init__(A, **kwargs)
- TESTS:
-
- This gives the same answer as the slow, default method implemented
- in :class:`MatrixEuclideanJordanAlgebra`::
-
- sage: set_random_seed()
- sage: J = ComplexHermitianEJA.random_instance()
- sage: x,y = J.random_elements(2)
- sage: Xe = x.to_matrix()
- sage: Ye = y.to_matrix()
- sage: X = ComplexHermitianEJA.real_unembed(Xe)
- sage: Y = ComplexHermitianEJA.real_unembed(Ye)
- sage: expected = (X*Y).trace().real()
- sage: actual = ComplexHermitianEJA.matrix_inner_product(Xe,Ye)
- sage: actual == expected
- True
+ from mjo.eja.eja_cache import real_symmetric_eja_coeffs
+ a = real_symmetric_eja_coeffs(self)
+ if a is not None:
+ self.rational_algebra()._charpoly_coefficients.set_cache(a)
- """
- return RealMatrixEuclideanJordanAlgebra.matrix_inner_product(X,Y)/2
-class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra,
- ConcreteEuclideanJordanAlgebra):
+class ComplexHermitianEJA(HermitianMatrixEJA, RationalBasisEJA, ConcreteEJA):
"""
The rank-n simple EJA consisting of complex Hermitian n-by-n
matrices over the real numbers, the usual symmetric Jordan product,
EXAMPLES:
- In theory, our "field" can be any subfield of the reals::
+ In theory, our "field" can be any subfield of the reals, but we
+ can't use inexact real fields at the moment because SageMath
+ doesn't know how to convert their elements into complex numbers,
+ or even into algebraic reals::
- sage: ComplexHermitianEJA(2, RDF)
- Euclidean Jordan algebra of dimension 4 over Real Double Field
- sage: ComplexHermitianEJA(2, RR)
- Euclidean Jordan algebra of dimension 4 over Real Field with
- 53 bits of precision
+ sage: QQbar(RDF(1))
+ Traceback (most recent call last):
+ ...
+ TypeError: Illegal initializer for algebraic number
+ sage: AA(RR(1))
+ Traceback (most recent call last):
+ ...
+ TypeError: Illegal initializer for algebraic number
TESTS:
The dimension of this algebra is `n^2`::
- sage: set_random_seed()
- sage: n_max = ComplexHermitianEJA._max_random_instance_size()
- sage: n = ZZ.random_element(1, n_max)
+ sage: d = ComplexHermitianEJA._max_random_instance_dimension()
+ sage: n = ComplexHermitianEJA._max_random_instance_size(d)
sage: J = ComplexHermitianEJA(n)
sage: J.dimension() == n^2
True
The Jordan multiplication is what we think it is::
- sage: set_random_seed()
sage: J = ComplexHermitianEJA.random_instance()
sage: x,y = J.random_elements(2)
sage: actual = (x*y).to_matrix()
Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
"""
-
- @classmethod
- def _denormalized_basis(cls, n, field):
- """
- Returns a basis for the space of complex Hermitian n-by-n matrices.
-
- Why do we embed these? Basically, because all of numerical linear
- algebra assumes that you're working with vectors consisting of `n`
- entries from a field and scalars from the same field. There's no way
- to tell SageMath that (for example) the vectors contain complex
- numbers, while the scalar field is real.
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
-
- TESTS::
-
- sage: set_random_seed()
- sage: n = ZZ.random_element(1,5)
- sage: field = QuadraticField(2, 'sqrt2')
- sage: B = ComplexHermitianEJA._denormalized_basis(n, field)
- sage: all( M.is_symmetric() for M in B)
- True
-
- """
- R = PolynomialRing(field, 'z')
- z = R.gen()
- F = field.extension(z**2 + 1, 'I')
- I = F.gen()
-
- # This is like the symmetric case, but we need to be careful:
- #
- # * We want conjugate-symmetry, not just symmetry.
- # * The diagonal will (as a result) be real.
- #
- S = []
- for i in range(n):
- for j in range(i+1):
- Eij = matrix(F, n, lambda k,l: k==i and l==j)
- if i == j:
- Sij = cls.real_embed(Eij)
- S.append(Sij)
- else:
- # The second one has a minus because it's conjugated.
- Sij_real = cls.real_embed(Eij + Eij.transpose())
- S.append(Sij_real)
- Sij_imag = cls.real_embed(I*Eij - I*Eij.transpose())
- S.append(Sij_imag)
-
- # Since we embedded these, we can drop back to the "field" that we
- # started with instead of the complex extension "F".
- return ( s.change_ring(field) for s in S )
-
-
def __init__(self, n, field=AA, **kwargs):
- basis = self._denormalized_basis(n,field)
- super(ComplexHermitianEJA,self).__init__(field,
- basis,
- check_axioms=False,
- **kwargs)
- self.rank.set_cache(n)
-
- @staticmethod
- def _max_random_instance_size():
- return 3 # Dimension 9
-
- @classmethod
- def random_instance(cls, field=AA, **kwargs):
- """
- Return a random instance of this type of algebra.
- """
- n = ZZ.random_element(cls._max_random_instance_size() + 1)
- return cls(n, field, **kwargs)
-
-class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra):
- @staticmethod
- def real_embed(M):
- """
- Embed the n-by-n quaternion matrix ``M`` into the space of real
- matrices of size 4n-by-4n by first sending each quaternion entry `z
- = a + bi + cj + dk` to the block-complex matrix ``[[a + bi,
- c+di],[-c + di, a-bi]]`, and then embedding those into a real
- matrix.
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import \
- ....: QuaternionMatrixEuclideanJordanAlgebra
-
- EXAMPLES::
-
- sage: Q = QuaternionAlgebra(QQ,-1,-1)
- sage: i,j,k = Q.gens()
- sage: x = 1 + 2*i + 3*j + 4*k
- sage: M = matrix(Q, 1, [[x]])
- sage: QuaternionMatrixEuclideanJordanAlgebra.real_embed(M)
- [ 1 2 3 4]
- [-2 1 -4 3]
- [-3 4 1 -2]
- [-4 -3 2 1]
-
- Embedding is a homomorphism (isomorphism, in fact)::
-
- sage: set_random_seed()
- sage: n = ZZ.random_element(2)
- sage: Q = QuaternionAlgebra(QQ,-1,-1)
- sage: X = random_matrix(Q, n)
- sage: Y = random_matrix(Q, n)
- sage: Xe = QuaternionMatrixEuclideanJordanAlgebra.real_embed(X)
- sage: Ye = QuaternionMatrixEuclideanJordanAlgebra.real_embed(Y)
- sage: XYe = QuaternionMatrixEuclideanJordanAlgebra.real_embed(X*Y)
- sage: Xe*Ye == XYe
- True
-
- """
- quaternions = M.base_ring()
- n = M.nrows()
- if M.ncols() != n:
- raise ValueError("the matrix 'M' must be square")
-
- F = QuadraticField(-1, 'I')
- i = F.gen()
-
- blocks = []
- for z in M.list():
- t = z.coefficient_tuple()
- a = t[0]
- b = t[1]
- c = t[2]
- d = t[3]
- cplxM = matrix(F, 2, [[ a + b*i, c + d*i],
- [-c + d*i, a - b*i]])
- realM = ComplexMatrixEuclideanJordanAlgebra.real_embed(cplxM)
- blocks.append(realM)
-
- # We should have real entries by now, so use the realest field
- # we've got for the return value.
- return matrix.block(quaternions.base_ring(), n, blocks)
-
+ from mjo.hurwitz import ComplexMatrixAlgebra
+ A = ComplexMatrixAlgebra(n, scalars=field)
+ super().__init__(A, **kwargs)
+ from mjo.eja.eja_cache import complex_hermitian_eja_coeffs
+ a = complex_hermitian_eja_coeffs(self)
+ if a is not None:
+ self.rational_algebra()._charpoly_coefficients.set_cache(a)
@staticmethod
- def real_unembed(M):
- """
- The inverse of _embed_quaternion_matrix().
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import \
- ....: QuaternionMatrixEuclideanJordanAlgebra
-
- EXAMPLES::
-
- sage: M = matrix(QQ, [[ 1, 2, 3, 4],
- ....: [-2, 1, -4, 3],
- ....: [-3, 4, 1, -2],
- ....: [-4, -3, 2, 1]])
- sage: QuaternionMatrixEuclideanJordanAlgebra.real_unembed(M)
- [1 + 2*i + 3*j + 4*k]
-
- TESTS:
-
- Unembedding is the inverse of embedding::
-
- sage: set_random_seed()
- sage: Q = QuaternionAlgebra(QQ, -1, -1)
- sage: M = random_matrix(Q, 3)
- sage: Me = QuaternionMatrixEuclideanJordanAlgebra.real_embed(M)
- sage: QuaternionMatrixEuclideanJordanAlgebra.real_unembed(Me) == M
- True
-
- """
- n = ZZ(M.nrows())
- if M.ncols() != n:
- raise ValueError("the matrix 'M' must be square")
- if not n.mod(4).is_zero():
- raise ValueError("the matrix 'M' must be a quaternion embedding")
-
- # Use the base ring of the matrix to ensure that its entries can be
- # multiplied by elements of the quaternion algebra.
- field = M.base_ring()
- Q = QuaternionAlgebra(field,-1,-1)
- i,j,k = Q.gens()
-
- # Go top-left to bottom-right (reading order), converting every
- # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1
- # quaternion block.
- elements = []
- for l in range(n/4):
- for m in range(n/4):
- submat = ComplexMatrixEuclideanJordanAlgebra.real_unembed(
- M[4*l:4*l+4,4*m:4*m+4] )
- if submat[0,0] != submat[1,1].conjugate():
- raise ValueError('bad on-diagonal submatrix')
- if submat[0,1] != -submat[1,0].conjugate():
- raise ValueError('bad off-diagonal submatrix')
- z = submat[0,0].real()
- z += submat[0,0].imag()*i
- z += submat[0,1].real()*j
- z += submat[0,1].imag()*k
- elements.append(z)
-
- return matrix(Q, n/4, elements)
-
+ def _max_random_instance_size(max_dimension):
+ # Obtained by solving d = n^2.
+ # The ZZ-int-ZZ thing is just "floor."
+ return ZZ(int(ZZ(max_dimension).sqrt()))
@classmethod
- def matrix_inner_product(cls,X,Y):
+ def random_instance(cls, max_dimension=None, *args, **kwargs):
"""
- Compute a matrix inner product in this algebra directly from
- its real embedding.
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
-
- TESTS:
-
- This gives the same answer as the slow, default method implemented
- in :class:`MatrixEuclideanJordanAlgebra`::
-
- sage: set_random_seed()
- sage: J = QuaternionHermitianEJA.random_instance()
- sage: x,y = J.random_elements(2)
- sage: Xe = x.to_matrix()
- sage: Ye = y.to_matrix()
- sage: X = QuaternionHermitianEJA.real_unembed(Xe)
- sage: Y = QuaternionHermitianEJA.real_unembed(Ye)
- sage: expected = (X*Y).trace().coefficient_tuple()[0]
- sage: actual = QuaternionHermitianEJA.matrix_inner_product(Xe,Ye)
- sage: actual == expected
- True
-
+ Return a random instance of this type of algebra.
"""
- return RealMatrixEuclideanJordanAlgebra.matrix_inner_product(X,Y)/4
+ class_max_d = cls._max_random_instance_dimension()
+ if (max_dimension is None or max_dimension > class_max_d):
+ max_dimension = class_max_d
+ max_size = cls._max_random_instance_size(max_dimension)
+ n = ZZ.random_element(max_size + 1)
+ return cls(n, **kwargs)
-class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra,
- ConcreteEuclideanJordanAlgebra):
+class QuaternionHermitianEJA(HermitianMatrixEJA, RationalBasisEJA, ConcreteEJA):
r"""
The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
matrices, the usual symmetric Jordan product, and the
In theory, our "field" can be any subfield of the reals::
- sage: QuaternionHermitianEJA(2, RDF)
+ sage: QuaternionHermitianEJA(2, field=RDF, check_axioms=True)
Euclidean Jordan algebra of dimension 6 over Real Double Field
- sage: QuaternionHermitianEJA(2, RR)
+ sage: QuaternionHermitianEJA(2, field=RR, check_axioms=True)
Euclidean Jordan algebra of dimension 6 over Real Field with
53 bits of precision
The dimension of this algebra is `2*n^2 - n`::
- sage: set_random_seed()
- sage: n_max = QuaternionHermitianEJA._max_random_instance_size()
- sage: n = ZZ.random_element(1, n_max)
+ sage: d = QuaternionHermitianEJA._max_random_instance_dimension()
+ sage: n = QuaternionHermitianEJA._max_random_instance_size(d)
sage: J = QuaternionHermitianEJA(n)
sage: J.dimension() == 2*(n^2) - n
True
The Jordan multiplication is what we think it is::
- sage: set_random_seed()
sage: J = QuaternionHermitianEJA.random_instance()
sage: x,y = J.random_elements(2)
sage: actual = (x*y).to_matrix()
Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
"""
- @classmethod
- def _denormalized_basis(cls, n, field):
- """
- Returns a basis for the space of quaternion Hermitian n-by-n matrices.
-
- Why do we embed these? Basically, because all of numerical
- linear algebra assumes that you're working with vectors consisting
- of `n` entries from a field and scalars from the same field. There's
- no way to tell SageMath that (for example) the vectors contain
- complex numbers, while the scalar field is real.
+ def __init__(self, n, field=AA, **kwargs):
+ from mjo.hurwitz import QuaternionMatrixAlgebra
+ A = QuaternionMatrixAlgebra(n, scalars=field)
+ super().__init__(A, **kwargs)
- SETUP::
+ from mjo.eja.eja_cache import quaternion_hermitian_eja_coeffs
+ a = quaternion_hermitian_eja_coeffs(self)
+ if a is not None:
+ self.rational_algebra()._charpoly_coefficients.set_cache(a)
- sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
- TESTS::
- sage: set_random_seed()
- sage: n = ZZ.random_element(1,5)
- sage: B = QuaternionHermitianEJA._denormalized_basis(n,QQ)
- sage: all( M.is_symmetric() for M in B )
- True
+ @staticmethod
+ def _max_random_instance_size(max_dimension):
+ r"""
+ The maximum rank of a random QuaternionHermitianEJA.
+ """
+ # Obtained by solving d = 2n^2 - n.
+ # The ZZ-int-ZZ thing is just "floor."
+ return ZZ(int(ZZ(8*max_dimension + 1).sqrt()/4 + 1/4))
+ @classmethod
+ def random_instance(cls, max_dimension=None, *args, **kwargs):
+ """
+ Return a random instance of this type of algebra.
"""
- Q = QuaternionAlgebra(QQ,-1,-1)
- I,J,K = Q.gens()
+ class_max_d = cls._max_random_instance_dimension()
+ if (max_dimension is None or max_dimension > class_max_d):
+ max_dimension = class_max_d
+ max_size = cls._max_random_instance_size(max_dimension)
+ n = ZZ.random_element(max_size + 1)
+ return cls(n, **kwargs)
- # This is like the symmetric case, but we need to be careful:
- #
- # * We want conjugate-symmetry, not just symmetry.
- # * The diagonal will (as a result) be real.
- #
- S = []
- for i in range(n):
- for j in range(i+1):
- Eij = matrix(Q, n, lambda k,l: k==i and l==j)
- if i == j:
- Sij = cls.real_embed(Eij)
- S.append(Sij)
- else:
- # The second, third, and fourth ones have a minus
- # because they're conjugated.
- Sij_real = cls.real_embed(Eij + Eij.transpose())
- S.append(Sij_real)
- Sij_I = cls.real_embed(I*Eij - I*Eij.transpose())
- S.append(Sij_I)
- Sij_J = cls.real_embed(J*Eij - J*Eij.transpose())
- S.append(Sij_J)
- Sij_K = cls.real_embed(K*Eij - K*Eij.transpose())
- S.append(Sij_K)
-
- # Since we embedded these, we can drop back to the "field" that we
- # started with instead of the quaternion algebra "Q".
- return ( s.change_ring(field) for s in S )
+class OctonionHermitianEJA(HermitianMatrixEJA, RationalBasisEJA, ConcreteEJA):
+ r"""
+ SETUP::
+ sage: from mjo.eja.eja_algebra import (EJA,
+ ....: OctonionHermitianEJA)
+ sage: from mjo.hurwitz import Octonions, OctonionMatrixAlgebra
- def __init__(self, n, field=AA, **kwargs):
- basis = self._denormalized_basis(n,field)
- super(QuaternionHermitianEJA,self).__init__(field,
- basis,
- check_axioms=False,
- **kwargs)
- self.rank.set_cache(n)
+ EXAMPLES:
+ The 3-by-3 algebra satisfies the axioms of an EJA::
+
+ sage: OctonionHermitianEJA(3, # long time
+ ....: field=QQ, # long time
+ ....: orthonormalize=False, # long time
+ ....: check_axioms=True) # long time
+ Euclidean Jordan algebra of dimension 27 over Rational Field
+
+ After a change-of-basis, the 2-by-2 algebra has the same
+ multiplication table as the ten-dimensional Jordan spin algebra::
+
+ sage: A = OctonionMatrixAlgebra(2,Octonions(QQ),QQ)
+ sage: b = OctonionHermitianEJA._denormalized_basis(A)
+ sage: basis = (b[0] + b[9],) + b[1:9] + (b[0] - b[9],)
+ sage: jp = OctonionHermitianEJA.jordan_product
+ sage: ip = OctonionHermitianEJA.trace_inner_product
+ sage: J = EJA(basis,
+ ....: jp,
+ ....: ip,
+ ....: field=QQ,
+ ....: orthonormalize=False)
+ sage: J.multiplication_table()
+ +----++----+----+----+----+----+----+----+----+----+----+
+ | * || b0 | b1 | b2 | b3 | b4 | b5 | b6 | b7 | b8 | b9 |
+ +====++====+====+====+====+====+====+====+====+====+====+
+ | b0 || b0 | b1 | b2 | b3 | b4 | b5 | b6 | b7 | b8 | b9 |
+ +----++----+----+----+----+----+----+----+----+----+----+
+ | b1 || b1 | b0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
+ +----++----+----+----+----+----+----+----+----+----+----+
+ | b2 || b2 | 0 | b0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
+ +----++----+----+----+----+----+----+----+----+----+----+
+ | b3 || b3 | 0 | 0 | b0 | 0 | 0 | 0 | 0 | 0 | 0 |
+ +----++----+----+----+----+----+----+----+----+----+----+
+ | b4 || b4 | 0 | 0 | 0 | b0 | 0 | 0 | 0 | 0 | 0 |
+ +----++----+----+----+----+----+----+----+----+----+----+
+ | b5 || b5 | 0 | 0 | 0 | 0 | b0 | 0 | 0 | 0 | 0 |
+ +----++----+----+----+----+----+----+----+----+----+----+
+ | b6 || b6 | 0 | 0 | 0 | 0 | 0 | b0 | 0 | 0 | 0 |
+ +----++----+----+----+----+----+----+----+----+----+----+
+ | b7 || b7 | 0 | 0 | 0 | 0 | 0 | 0 | b0 | 0 | 0 |
+ +----++----+----+----+----+----+----+----+----+----+----+
+ | b8 || b8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | b0 | 0 |
+ +----++----+----+----+----+----+----+----+----+----+----+
+ | b9 || b9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | b0 |
+ +----++----+----+----+----+----+----+----+----+----+----+
+
+ TESTS:
+
+ We can actually construct the 27-dimensional Albert algebra,
+ and we get the right unit element if we recompute it::
+
+ sage: J = OctonionHermitianEJA(3, # long time
+ ....: field=QQ, # long time
+ ....: orthonormalize=False) # long time
+ sage: J.one.clear_cache() # long time
+ sage: J.one() # long time
+ b0 + b9 + b26
+ sage: J.one().to_matrix() # long time
+ +----+----+----+
+ | e0 | 0 | 0 |
+ +----+----+----+
+ | 0 | e0 | 0 |
+ +----+----+----+
+ | 0 | 0 | e0 |
+ +----+----+----+
+
+ The 2-by-2 algebra is isomorphic to the ten-dimensional Jordan
+ spin algebra, but just to be sure, we recompute its rank::
+
+ sage: J = OctonionHermitianEJA(2, # long time
+ ....: field=QQ, # long time
+ ....: orthonormalize=False) # long time
+ sage: J.rank.clear_cache() # long time
+ sage: J.rank() # long time
+ 2
+
+ """
@staticmethod
- def _max_random_instance_size():
- r"""
- The maximum rank of a random QuaternionHermitianEJA.
- """
- return 2 # Dimension 6
+ def _max_random_instance_size(max_dimension):
+ r"""
+ The maximum rank of a random OctonionHermitianEJA.
+ """
+ # There's certainly a formula for this, but with only four
+ # cases to worry about, I'm not that motivated to derive it.
+ if max_dimension >= 27:
+ return 3
+ elif max_dimension >= 10:
+ return 2
+ elif max_dimension >= 1:
+ return 1
+ else:
+ return 0
@classmethod
- def random_instance(cls, field=AA, **kwargs):
+ def random_instance(cls, max_dimension=None, *args, **kwargs):
"""
Return a random instance of this type of algebra.
"""
- n = ZZ.random_element(cls._max_random_instance_size() + 1)
- return cls(n, field, **kwargs)
+ class_max_d = cls._max_random_instance_dimension()
+ if (max_dimension is None or max_dimension > class_max_d):
+ max_dimension = class_max_d
+ max_size = cls._max_random_instance_size(max_dimension)
+ n = ZZ.random_element(max_size + 1)
+ return cls(n, **kwargs)
+
+ def __init__(self, n, field=AA, **kwargs):
+ if n > 3:
+ # Otherwise we don't get an EJA.
+ raise ValueError("n cannot exceed 3")
+
+ # We know this is a valid EJA, but will double-check
+ # if the user passes check_axioms=True.
+ if "check_axioms" not in kwargs: kwargs["check_axioms"] = False
+
+ from mjo.hurwitz import OctonionMatrixAlgebra
+ A = OctonionMatrixAlgebra(n, scalars=field)
+ super().__init__(A, **kwargs)
+
+ from mjo.eja.eja_cache import octonion_hermitian_eja_coeffs
+ a = octonion_hermitian_eja_coeffs(self)
+ if a is not None:
+ self.rational_algebra()._charpoly_coefficients.set_cache(a)
+
+
+class AlbertEJA(OctonionHermitianEJA):
+ r"""
+ The Albert algebra is the algebra of three-by-three Hermitian
+ matrices whose entries are octonions.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import AlbertEJA
+
+ EXAMPLES::
+
+ sage: AlbertEJA(field=QQ, orthonormalize=False)
+ Euclidean Jordan algebra of dimension 27 over Rational Field
+ sage: AlbertEJA() # long time
+ Euclidean Jordan algebra of dimension 27 over Algebraic Real Field
+
+ """
+ def __init__(self, *args, **kwargs):
+ super().__init__(3, *args, **kwargs)
-class HadamardEJA(RationalBasisEuclideanJordanAlgebraNg,
- ConcreteEuclideanJordanAlgebra):
+class HadamardEJA(RationalBasisEJA, ConcreteEJA):
"""
- Return the Euclidean Jordan Algebra corresponding to the set
- `R^n` under the Hadamard product.
+ Return the Euclidean Jordan algebra on `R^n` with the Hadamard
+ (pointwise real-number multiplication) Jordan product and the
+ usual inner-product.
- Note: this is nothing more than the Cartesian product of ``n``
- copies of the spin algebra. Once Cartesian product algebras
- are implemented, this can go.
+ This is nothing more than the Cartesian product of ``n`` copies of
+ the one-dimensional Jordan spin algebra, and is the most common
+ example of a non-simple Euclidean Jordan algebra.
SETUP::
This multiplication table can be verified by hand::
sage: J = HadamardEJA(3)
- sage: e0,e1,e2 = J.gens()
- sage: e0*e0
- e0
- sage: e0*e1
+ sage: b0,b1,b2 = J.gens()
+ sage: b0*b0
+ b0
+ sage: b0*b1
0
- sage: e0*e2
+ sage: b0*b2
0
- sage: e1*e1
- e1
- sage: e1*e2
+ sage: b1*b1
+ b1
+ sage: b1*b2
0
- sage: e2*e2
- e2
+ sage: b2*b2
+ b2
TESTS:
sage: HadamardEJA(3, prefix='r').gens()
(r0, r1, r2)
-
"""
def __init__(self, n, field=AA, **kwargs):
- V = VectorSpace(field, n)
- basis = V.basis()
-
- def jordan_product(x,y):
- return V([ xi*yi for (xi,yi) in zip(x,y) ])
- def inner_product(x,y):
- return x.inner_product(y)
-
- super(HadamardEJA, self).__init__(field,
- basis,
- jordan_product,
- inner_product,
- **kwargs)
- self.rank.set_cache(n)
+ MS = MatrixSpace(field, n, 1)
if n == 0:
- self.one.set_cache( self.zero() )
+ jordan_product = lambda x,y: x
+ inner_product = lambda x,y: x
else:
- self.one.set_cache( sum(self.gens()) )
+ def jordan_product(x,y):
+ return MS( xi*yi for (xi,yi) in zip(x,y) )
+
+ def inner_product(x,y):
+ return (x.T*y)[0,0]
+
+ # New defaults for keyword arguments. Don't orthonormalize
+ # because our basis is already orthonormal with respect to our
+ # inner-product. Don't check the axioms, because we know this
+ # is a valid EJA... but do double-check if the user passes
+ # check_axioms=True. Note: we DON'T override the "check_field"
+ # default here, because the user can pass in a field!
+ if "orthonormalize" not in kwargs: kwargs["orthonormalize"] = False
+ if "check_axioms" not in kwargs: kwargs["check_axioms"] = False
+
+ column_basis = tuple( MS(b) for b in FreeModule(field, n).basis() )
+ super().__init__(column_basis,
+ jordan_product,
+ inner_product,
+ field=field,
+ matrix_space=MS,
+ associative=True,
+ **kwargs)
+ self.rank.set_cache(n)
+
+ self.one.set_cache( self.sum(self.gens()) )
@staticmethod
- def _max_random_instance_size():
+ def _max_random_instance_dimension():
r"""
- The maximum dimension of a random HadamardEJA.
+ There's no reason to go higher than five here. That's
+ enough to get the point across.
"""
return 5
+ @staticmethod
+ def _max_random_instance_size(max_dimension):
+ r"""
+ The maximum size (=dimension) of a random HadamardEJA.
+ """
+ return max_dimension
+
@classmethod
- def random_instance(cls, field=AA, **kwargs):
+ def random_instance(cls, max_dimension=None, *args, **kwargs):
"""
Return a random instance of this type of algebra.
"""
- n = ZZ.random_element(cls._max_random_instance_size() + 1)
- return cls(n, field, **kwargs)
+ class_max_d = cls._max_random_instance_dimension()
+ if (max_dimension is None or max_dimension > class_max_d):
+ max_dimension = class_max_d
+ max_size = cls._max_random_instance_size(max_dimension)
+ n = ZZ.random_element(max_size + 1)
+ return cls(n, **kwargs)
-class BilinearFormEJA(RationalBasisEuclideanJordanAlgebra,
- ConcreteEuclideanJordanAlgebra):
+class BilinearFormEJA(RationalBasisEJA, ConcreteEJA):
r"""
The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
with the half-trace inner product and jordan product ``x*y =
We can check the multiplication condition given in the Jordan, von
Neumann, and Wigner paper (and also discussed on my "On the
symmetry..." paper). Note that this relies heavily on the standard
- choice of basis, as does anything utilizing the bilinear form matrix::
+ choice of basis, as does anything utilizing the bilinear form
+ matrix. We opt not to orthonormalize the basis, because if we
+ did, we would have to normalize the `s_{i}` in a similar manner::
- sage: set_random_seed()
sage: n = ZZ.random_element(5)
sage: M = matrix.random(QQ, max(0,n-1), algorithm='unimodular')
sage: B11 = matrix.identity(QQ,1)
sage: B22 = M.transpose()*M
sage: B = block_matrix(2,2,[ [B11,0 ],
....: [0, B22 ] ])
- sage: J = BilinearFormEJA(B)
+ sage: J = BilinearFormEJA(B, orthonormalize=False)
sage: eis = VectorSpace(M.base_ring(), M.ncols()).basis()
sage: V = J.vector_space()
- sage: sis = [ J.from_vector(V([0] + (M.inverse()*ei).list()))
+ sage: sis = [ J( V([0] + (M.inverse()*ei).list()).column() )
....: for ei in eis ]
sage: actual = [ sis[i]*sis[j]
....: for i in range(n-1)
....: for j in range(n-1) ]
sage: actual == expected
True
+
"""
def __init__(self, B, field=AA, **kwargs):
+ # The matrix "B" is supplied by the user in most cases,
+ # so it makes sense to check whether or not its positive-
+ # definite unless we are specifically asked not to...
+ if ("check_axioms" not in kwargs) or kwargs["check_axioms"]:
+ if not B.is_positive_definite():
+ raise ValueError("bilinear form is not positive-definite")
+
+ # However, all of the other data for this EJA is computed
+ # by us in manner that guarantees the axioms are
+ # satisfied. So, again, unless we are specifically asked to
+ # verify things, we'll skip the rest of the checks.
+ if "check_axioms" not in kwargs: kwargs["check_axioms"] = False
+
n = B.nrows()
+ MS = MatrixSpace(field, n, 1)
- if not B.is_positive_definite():
- raise ValueError("bilinear form is not positive-definite")
+ def inner_product(x,y):
+ return (y.T*B*x)[0,0]
- V = VectorSpace(field, n)
- mult_table = [[V.zero() for j in range(n)] for i in range(n)]
- for i in range(n):
- for j in range(n):
- x = V.gen(i)
- y = V.gen(j)
- x0 = x[0]
- xbar = x[1:]
- y0 = y[0]
- ybar = y[1:]
- z0 = (B*x).inner_product(y)
- zbar = y0*xbar + x0*ybar
- z = V([z0] + zbar.list())
- mult_table[i][j] = z
-
- # Inner products are real numbers and not algebra
- # elements, so once we turn the algebra element
- # into a vector in inner_product(), we never go
- # back. As a result -- contrary to what we do with
- # self._multiplication_table -- we store the inner
- # product table as a plain old matrix and not as
- # an algebra operator.
- ip_table = B
- self._inner_product_matrix = ip_table
-
- super(BilinearFormEJA, self).__init__(field,
- mult_table,
- check_axioms=False,
- **kwargs)
+ def jordan_product(x,y):
+ x0 = x[0,0]
+ xbar = x[1:,0]
+ y0 = y[0,0]
+ ybar = y[1:,0]
+ z0 = inner_product(y,x)
+ zbar = y0*xbar + x0*ybar
+ return MS([z0] + zbar.list())
+
+ column_basis = tuple( MS(b) for b in FreeModule(field, n).basis() )
+
+ # TODO: I haven't actually checked this, but it seems legit.
+ associative = False
+ if n <= 2:
+ associative = True
+
+ super().__init__(column_basis,
+ jordan_product,
+ inner_product,
+ field=field,
+ matrix_space=MS,
+ associative=associative,
+ **kwargs)
# The rank of this algebra is two, unless we're in a
# one-dimensional ambient space (because the rank is bounded
# by the ambient dimension).
self.rank.set_cache(min(n,2))
-
if n == 0:
self.one.set_cache( self.zero() )
else:
self.one.set_cache( self.monomial(0) )
@staticmethod
- def _max_random_instance_size():
+ def _max_random_instance_dimension():
r"""
- The maximum dimension of a random BilinearFormEJA.
+ There's no reason to go higher than five here. That's
+ enough to get the point across.
"""
return 5
+ @staticmethod
+ def _max_random_instance_size(max_dimension):
+ r"""
+ The maximum size (=dimension) of a random BilinearFormEJA.
+ """
+ return max_dimension
+
@classmethod
- def random_instance(cls, field=AA, **kwargs):
+ def random_instance(cls, max_dimension=None, *args, **kwargs):
"""
Return a random instance of this algebra.
"""
- n = ZZ.random_element(cls._max_random_instance_size() + 1)
+ class_max_d = cls._max_random_instance_dimension()
+ if (max_dimension is None or max_dimension > class_max_d):
+ max_dimension = class_max_d
+ max_size = cls._max_random_instance_size(max_dimension)
+ n = ZZ.random_element(max_size + 1)
+
if n.is_zero():
- B = matrix.identity(field, n)
- return cls(B, field, **kwargs)
+ B = matrix.identity(ZZ, n)
+ return cls(B, **kwargs)
- B11 = matrix.identity(field,1)
- M = matrix.random(field, n-1)
- I = matrix.identity(field, n-1)
- alpha = field.zero()
+ B11 = matrix.identity(ZZ, 1)
+ M = matrix.random(ZZ, n-1)
+ I = matrix.identity(ZZ, n-1)
+ alpha = ZZ.zero()
while alpha.is_zero():
- alpha = field.random_element().abs()
+ alpha = ZZ.random_element().abs()
+
B22 = M.transpose()*M + alpha*I
from sage.matrix.special import block_matrix
B = block_matrix(2,2, [ [B11, ZZ(0) ],
[ZZ(0), B22 ] ])
- return cls(B, field, **kwargs)
+ return cls(B, **kwargs)
class JordanSpinEJA(BilinearFormEJA):
This multiplication table can be verified by hand::
sage: J = JordanSpinEJA(4)
- sage: e0,e1,e2,e3 = J.gens()
- sage: e0*e0
- e0
- sage: e0*e1
- e1
- sage: e0*e2
- e2
- sage: e0*e3
- e3
- sage: e1*e2
+ sage: b0,b1,b2,b3 = J.gens()
+ sage: b0*b0
+ b0
+ sage: b0*b1
+ b1
+ sage: b0*b2
+ b2
+ sage: b0*b3
+ b3
+ sage: b1*b2
0
- sage: e1*e3
+ sage: b1*b3
0
- sage: e2*e3
+ sage: b2*b3
0
We can change the generator prefix::
Ensure that we have the usual inner product on `R^n`::
- sage: set_random_seed()
sage: J = JordanSpinEJA.random_instance()
sage: x,y = J.random_elements(2)
sage: actual = x.inner_product(y)
True
"""
- def __init__(self, n, field=AA, **kwargs):
- # This is a special case of the BilinearFormEJA with the identity
- # matrix as its bilinear form.
- B = matrix.identity(field, n)
- super(JordanSpinEJA, self).__init__(B, field, **kwargs)
+ def __init__(self, n, *args, **kwargs):
+ # This is a special case of the BilinearFormEJA with the
+ # identity matrix as its bilinear form.
+ B = matrix.identity(ZZ, n)
- @staticmethod
- def _max_random_instance_size():
- r"""
- The maximum dimension of a random JordanSpinEJA.
- """
- return 5
+ # Don't orthonormalize because our basis is already
+ # orthonormal with respect to our inner-product.
+ if "orthonormalize" not in kwargs: kwargs["orthonormalize"] = False
+
+ # But also don't pass check_field=False here, because the user
+ # can pass in a field!
+ super().__init__(B, *args, **kwargs)
@classmethod
- def random_instance(cls, field=AA, **kwargs):
+ def random_instance(cls, max_dimension=None, *args, **kwargs):
"""
Return a random instance of this type of algebra.
Needed here to override the implementation for ``BilinearFormEJA``.
"""
- n = ZZ.random_element(cls._max_random_instance_size() + 1)
- return cls(n, field, **kwargs)
+ class_max_d = cls._max_random_instance_dimension()
+ if (max_dimension is None or max_dimension > class_max_d):
+ max_dimension = class_max_d
+ max_size = cls._max_random_instance_size(max_dimension)
+ n = ZZ.random_element(max_size + 1)
+ return cls(n, **kwargs)
-class TrivialEJA(FiniteDimensionalEuclideanJordanAlgebra,
- ConcreteEuclideanJordanAlgebra):
+class TrivialEJA(RationalBasisEJA, ConcreteEJA):
"""
The trivial Euclidean Jordan algebra consisting of only a zero element.
"""
def __init__(self, field=AA, **kwargs):
- mult_table = []
- self._inner_product_matrix = matrix(field,0)
- super(TrivialEJA, self).__init__(field,
- mult_table,
- check_axioms=False,
- **kwargs)
+ jordan_product = lambda x,y: x
+ inner_product = lambda x,y: field.zero()
+ basis = ()
+ MS = MatrixSpace(field,0)
+
+ # New defaults for keyword arguments
+ if "orthonormalize" not in kwargs: kwargs["orthonormalize"] = False
+ if "check_axioms" not in kwargs: kwargs["check_axioms"] = False
+
+ super().__init__(basis,
+ jordan_product,
+ inner_product,
+ associative=True,
+ field=field,
+ matrix_space=MS,
+ **kwargs)
+
# The rank is zero using my definition, namely the dimension of the
# largest subalgebra generated by any element.
self.rank.set_cache(0)
self.one.set_cache( self.zero() )
@classmethod
- def random_instance(cls, field=AA, **kwargs):
+ def random_instance(cls, max_dimension=None, *args, **kwargs):
# We don't take a "size" argument so the superclass method is
- # inappropriate for us.
- return cls(field, **kwargs)
+ # inappropriate for us. The ``max_dimension`` argument is
+ # included so that if this method is called generically with a
+ # ``max_dimension=<whatever>`` argument, we don't try to pass
+ # it on to the algebra constructor.
+ return cls(**kwargs)
+
-class DirectSumEJA(FiniteDimensionalEuclideanJordanAlgebra):
+class CartesianProductEJA(EJA):
r"""
- The external (orthogonal) direct sum of two other Euclidean Jordan
- algebras. Essentially the Cartesian product of its two factors.
- Every Euclidean Jordan algebra decomposes into an orthogonal
- direct sum of simple Euclidean Jordan algebras, so no generality
- is lost by providing only this construction.
+ The external (orthogonal) direct sum of two or more Euclidean
+ Jordan algebras. Every Euclidean Jordan algebra decomposes into an
+ orthogonal direct sum of simple Euclidean Jordan algebras which is
+ then isometric to a Cartesian product, so no generality is lost by
+ providing only this construction.
SETUP::
sage: from mjo.eja.eja_algebra import (random_eja,
+ ....: CartesianProductEJA,
+ ....: ComplexHermitianEJA,
....: HadamardEJA,
- ....: RealSymmetricEJA,
- ....: DirectSumEJA)
+ ....: JordanSpinEJA,
+ ....: RealSymmetricEJA)
- EXAMPLES::
+ EXAMPLES:
+
+ The Jordan product is inherited from our factors and implemented by
+ our CombinatorialFreeModule Cartesian product superclass::
sage: J1 = HadamardEJA(2)
- sage: J2 = RealSymmetricEJA(3)
- sage: J = DirectSumEJA(J1,J2)
- sage: J.dimension()
- 8
+ sage: J2 = RealSymmetricEJA(2)
+ sage: J = cartesian_product([J1,J2])
+ sage: x,y = J.random_elements(2)
+ sage: x*y in J
+ True
+
+ The ability to retrieve the original factors is implemented by our
+ CombinatorialFreeModule Cartesian product superclass::
+
+ sage: J1 = HadamardEJA(2, field=QQ)
+ sage: J2 = JordanSpinEJA(3, field=QQ)
+ sage: J = cartesian_product([J1,J2])
+ sage: J.cartesian_factors()
+ (Euclidean Jordan algebra of dimension 2 over Rational Field,
+ Euclidean Jordan algebra of dimension 3 over Rational Field)
+
+ You can provide more than two factors::
+
+ sage: J1 = HadamardEJA(2)
+ sage: J2 = JordanSpinEJA(3)
+ sage: J3 = RealSymmetricEJA(3)
+ sage: cartesian_product([J1,J2,J3])
+ Euclidean Jordan algebra of dimension 2 over Algebraic Real
+ Field (+) Euclidean Jordan algebra of dimension 3 over Algebraic
+ Real Field (+) Euclidean Jordan algebra of dimension 6 over
+ Algebraic Real Field
+
+ Rank is additive on a Cartesian product::
+
+ sage: J1 = HadamardEJA(1)
+ sage: J2 = RealSymmetricEJA(2)
+ sage: J = cartesian_product([J1,J2])
+ sage: J1.rank.clear_cache()
+ sage: J2.rank.clear_cache()
+ sage: J.rank.clear_cache()
sage: J.rank()
- 5
+ 3
+ sage: J.rank() == J1.rank() + J2.rank()
+ True
+
+ The same rank computation works over the rationals, with whatever
+ basis you like::
+
+ sage: J1 = HadamardEJA(1, field=QQ, orthonormalize=False)
+ sage: J2 = RealSymmetricEJA(2, field=QQ, orthonormalize=False)
+ sage: J = cartesian_product([J1,J2])
+ sage: J1.rank.clear_cache()
+ sage: J2.rank.clear_cache()
+ sage: J.rank.clear_cache()
+ sage: J.rank()
+ 3
+ sage: J.rank() == J1.rank() + J2.rank()
+ True
+
+ The product algebra will be associative if and only if all of its
+ components are associative::
+
+ sage: J1 = HadamardEJA(2)
+ sage: J1.is_associative()
+ True
+ sage: J2 = HadamardEJA(3)
+ sage: J2.is_associative()
+ True
+ sage: J3 = RealSymmetricEJA(3)
+ sage: J3.is_associative()
+ False
+ sage: CP1 = cartesian_product([J1,J2])
+ sage: CP1.is_associative()
+ True
+ sage: CP2 = cartesian_product([J1,J3])
+ sage: CP2.is_associative()
+ False
+
+ Cartesian products of Cartesian products work::
+
+ sage: J1 = JordanSpinEJA(1)
+ sage: J2 = JordanSpinEJA(1)
+ sage: J3 = JordanSpinEJA(1)
+ sage: J = cartesian_product([J1,cartesian_product([J2,J3])])
+ sage: J.multiplication_table()
+ +----++----+----+----+
+ | * || b0 | b1 | b2 |
+ +====++====+====+====+
+ | b0 || b0 | 0 | 0 |
+ +----++----+----+----+
+ | b1 || 0 | b1 | 0 |
+ +----++----+----+----+
+ | b2 || 0 | 0 | b2 |
+ +----++----+----+----+
+ sage: HadamardEJA(3).multiplication_table()
+ +----++----+----+----+
+ | * || b0 | b1 | b2 |
+ +====++====+====+====+
+ | b0 || b0 | 0 | 0 |
+ +----++----+----+----+
+ | b1 || 0 | b1 | 0 |
+ +----++----+----+----+
+ | b2 || 0 | 0 | b2 |
+ +----++----+----+----+
+
+ The "matrix space" of a Cartesian product always consists of
+ ordered pairs (or triples, or...) whose components are the
+ matrix spaces of its factors::
+
+ sage: J1 = HadamardEJA(2)
+ sage: J2 = ComplexHermitianEJA(2)
+ sage: J = cartesian_product([J1,J2])
+ sage: J.matrix_space()
+ The Cartesian product of (Full MatrixSpace of 2 by 1 dense
+ matrices over Algebraic Real Field, Module of 2 by 2 matrices
+ with entries in Algebraic Field over the scalar ring Algebraic
+ Real Field)
+ sage: J.one().to_matrix()[0]
+ [1]
+ [1]
+ sage: J.one().to_matrix()[1]
+ +---+---+
+ | 1 | 0 |
+ +---+---+
+ | 0 | 1 |
+ +---+---+
TESTS:
- The external direct sum construction is only valid when the two factors
- have the same base ring; an error is raised otherwise::
+ All factors must share the same base field::
- sage: set_random_seed()
- sage: J1 = random_eja(AA)
- sage: J2 = random_eja(QQ)
- sage: J = DirectSumEJA(J1,J2)
+ sage: J1 = HadamardEJA(2, field=QQ)
+ sage: J2 = RealSymmetricEJA(2)
+ sage: CartesianProductEJA((J1,J2))
Traceback (most recent call last):
...
- ValueError: algebras must share the same base field
+ ValueError: all factors must share the same base field
+
+ The cached unit element is the same one that would be computed::
+ sage: J1 = random_eja() # long time
+ sage: J2 = random_eja() # long time
+ sage: J = cartesian_product([J1,J2]) # long time
+ sage: actual = J.one() # long time
+ sage: J.one.clear_cache() # long time
+ sage: expected = J.one() # long time
+ sage: actual == expected # long time
+ True
"""
- def __init__(self, J1, J2, **kwargs):
- if J1.base_ring() != J2.base_ring():
- raise ValueError("algebras must share the same base field")
- field = J1.base_ring()
-
- self._factors = (J1, J2)
- n1 = J1.dimension()
- n2 = J2.dimension()
- n = n1+n2
- V = VectorSpace(field, n)
- mult_table = [ [ V.zero() for j in range(n) ]
- for i in range(n) ]
- for i in range(n1):
- for j in range(n1):
- p = (J1.monomial(i)*J1.monomial(j)).to_vector()
- mult_table[i][j] = V(p.list() + [field.zero()]*n2)
-
- for i in range(n2):
- for j in range(n2):
- p = (J2.monomial(i)*J2.monomial(j)).to_vector()
- mult_table[n1+i][n1+j] = V([field.zero()]*n1 + p.list())
-
- super(DirectSumEJA, self).__init__(field,
- mult_table,
- check_axioms=False,
- **kwargs)
- self.rank.set_cache(J1.rank() + J2.rank())
-
-
- def factors(self):
+ Element = CartesianProductEJAElement
+ def __init__(self, factors, **kwargs):
+ m = len(factors)
+ if m == 0:
+ return TrivialEJA()
+
+ self._sets = factors
+
+ field = factors[0].base_ring()
+ if not all( J.base_ring() == field for J in factors ):
+ raise ValueError("all factors must share the same base field")
+
+ # Figure out the category to use.
+ associative = all( f.is_associative() for f in factors )
+ category = EuclideanJordanAlgebras(field)
+ if associative: category = category.Associative()
+ category = category.join([category, category.CartesianProducts()])
+
+ # Compute my matrix space. We don't simply use the
+ # ``cartesian_product()`` functor here because it acts
+ # differently on SageMath MatrixSpaces and our custom
+ # MatrixAlgebras, which are CombinatorialFreeModules. We
+ # always want the result to be represented (and indexed) as an
+ # ordered tuple. This category isn't perfect, but is good
+ # enough for what we need to do.
+ MS_cat = MagmaticAlgebras(field).FiniteDimensional().WithBasis()
+ MS_cat = MS_cat.Unital().CartesianProducts()
+ MS_factors = tuple( J.matrix_space() for J in factors )
+ from sage.sets.cartesian_product import CartesianProduct
+ self._matrix_space = CartesianProduct(MS_factors, MS_cat)
+
+ self._matrix_basis = []
+ zero = self._matrix_space.zero()
+ for i in range(m):
+ for b in factors[i].matrix_basis():
+ z = list(zero)
+ z[i] = b
+ self._matrix_basis.append(z)
+
+ self._matrix_basis = tuple( self._matrix_space(b)
+ for b in self._matrix_basis )
+ n = len(self._matrix_basis)
+
+ # We already have what we need for the super-superclass constructor.
+ CombinatorialFreeModule.__init__(self,
+ field,
+ range(n),
+ prefix="b",
+ category=category,
+ bracket=False)
+
+ # Now create the vector space for the algebra, which will have
+ # its own set of non-ambient coordinates (in terms of the
+ # supplied basis).
+ degree = sum( f._matrix_span.ambient_vector_space().degree()
+ for f in factors )
+ V = VectorSpace(field, degree)
+ vector_basis = tuple( V(_all2list(b)) for b in self._matrix_basis )
+
+ # Save the span of our matrix basis (when written out as long
+ # vectors) because otherwise we'll have to reconstruct it
+ # every time we want to coerce a matrix into the algebra.
+ self._matrix_span = V.span_of_basis( vector_basis, check=False)
+
+ # Since we don't (re)orthonormalize the basis, the FDEJA
+ # constructor is going to set self._deortho_matrix to the
+ # identity matrix. Here we set it to the correct value using
+ # the deortho matrices from our factors.
+ self._deortho_matrix = matrix.block_diagonal(
+ [J._deortho_matrix for J in factors]
+ )
+
+ self._inner_product_matrix = matrix.block_diagonal(
+ [J._inner_product_matrix for J in factors]
+ )
+ self._inner_product_matrix._cache = {'hermitian': True}
+ self._inner_product_matrix.set_immutable()
+
+ # Building the multiplication table is a bit more tricky
+ # because we have to embed the entries of the factors'
+ # multiplication tables into the product EJA.
+ zed = self.zero()
+ self._multiplication_table = [ [zed for j in range(i+1)]
+ for i in range(n) ]
+
+ # Keep track of an offset that tallies the dimensions of all
+ # previous factors. If the second factor is dim=2 and if the
+ # first one is dim=3, then we want to skip the first 3x3 block
+ # when copying the multiplication table for the second factor.
+ offset = 0
+ for f in range(m):
+ phi_f = self.cartesian_embedding(f)
+ factor_dim = factors[f].dimension()
+ for i in range(factor_dim):
+ for j in range(i+1):
+ f_ij = factors[f]._multiplication_table[i][j]
+ e = phi_f(f_ij)
+ self._multiplication_table[offset+i][offset+j] = e
+ offset += factor_dim
+
+ self.rank.set_cache(sum(J.rank() for J in factors))
+ ones = tuple(J.one().to_matrix() for J in factors)
+ self.one.set_cache(self(ones))
+
+ def _sets_keys(self):
r"""
- Return the pair of this algebra's factors.
SETUP::
- sage: from mjo.eja.eja_algebra import (HadamardEJA,
- ....: JordanSpinEJA,
- ....: DirectSumEJA)
+ sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
+ ....: RealSymmetricEJA)
- EXAMPLES::
+ TESTS:
+
+ The superclass uses ``_sets_keys()`` to implement its
+ ``cartesian_factors()`` method::
- sage: J1 = HadamardEJA(2,QQ)
- sage: J2 = JordanSpinEJA(3,QQ)
- sage: J = DirectSumEJA(J1,J2)
- sage: J.factors()
- (Euclidean Jordan algebra of dimension 2 over Rational Field,
- Euclidean Jordan algebra of dimension 3 over Rational Field)
+ sage: J1 = RealSymmetricEJA(2,
+ ....: field=QQ,
+ ....: orthonormalize=False,
+ ....: prefix="a")
+ sage: J2 = ComplexHermitianEJA(2,field=QQ,orthonormalize=False)
+ sage: J = cartesian_product([J1,J2])
+ sage: x = sum(i*J.gens()[i] for i in range(len(J.gens())))
+ sage: x.cartesian_factors()
+ (a1 + 2*a2, 3*b0 + 4*b1 + 5*b2 + 6*b3)
"""
- return self._factors
+ # Copy/pasted from CombinatorialFreeModule_CartesianProduct,
+ # but returning a tuple instead of a list.
+ return tuple(range(len(self.cartesian_factors())))
- def projections(self):
+ def cartesian_factors(self):
+ # Copy/pasted from CombinatorialFreeModule_CartesianProduct.
+ return self._sets
+
+ def cartesian_factor(self, i):
r"""
- Return a pair of projections onto this algebra's factors.
+ Return the ``i``th factor of this algebra.
+ """
+ return self._sets[i]
+ def _repr_(self):
+ # Copy/pasted from CombinatorialFreeModule_CartesianProduct.
+ from sage.categories.cartesian_product import cartesian_product
+ return cartesian_product.symbol.join("%s" % factor
+ for factor in self._sets)
+
+
+ @cached_method
+ def cartesian_projection(self, i):
+ r"""
SETUP::
- sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
- ....: ComplexHermitianEJA,
- ....: DirectSumEJA)
+ sage: from mjo.eja.eja_algebra import (random_eja,
+ ....: JordanSpinEJA,
+ ....: HadamardEJA,
+ ....: RealSymmetricEJA,
+ ....: ComplexHermitianEJA)
- EXAMPLES::
+ EXAMPLES:
+
+ The projection morphisms are Euclidean Jordan algebra
+ operators::
+
+ sage: J1 = HadamardEJA(2)
+ sage: J2 = RealSymmetricEJA(2)
+ sage: J = cartesian_product([J1,J2])
+ sage: J.cartesian_projection(0)
+ Linear operator between finite-dimensional Euclidean Jordan
+ algebras represented by the matrix:
+ [1 0 0 0 0]
+ [0 1 0 0 0]
+ Domain: Euclidean Jordan algebra of dimension 2 over Algebraic
+ Real Field (+) Euclidean Jordan algebra of dimension 3 over
+ Algebraic Real Field
+ Codomain: Euclidean Jordan algebra of dimension 2 over Algebraic
+ Real Field
+ sage: J.cartesian_projection(1)
+ Linear operator between finite-dimensional Euclidean Jordan
+ algebras represented by the matrix:
+ [0 0 1 0 0]
+ [0 0 0 1 0]
+ [0 0 0 0 1]
+ Domain: Euclidean Jordan algebra of dimension 2 over Algebraic
+ Real Field (+) Euclidean Jordan algebra of dimension 3 over
+ Algebraic Real Field
+ Codomain: Euclidean Jordan algebra of dimension 3 over Algebraic
+ Real Field
+
+ The projections work the way you'd expect on the vector
+ representation of an element::
sage: J1 = JordanSpinEJA(2)
sage: J2 = ComplexHermitianEJA(2)
- sage: J = DirectSumEJA(J1,J2)
- sage: (pi_left, pi_right) = J.projections()
- sage: J.one().to_vector()
- (1, 0, 1, 0, 0, 1)
+ sage: J = cartesian_product([J1,J2])
+ sage: pi_left = J.cartesian_projection(0)
+ sage: pi_right = J.cartesian_projection(1)
sage: pi_left(J.one()).to_vector()
(1, 0)
sage: pi_right(J.one()).to_vector()
(1, 0, 0, 1)
+ sage: J.one().to_vector()
+ (1, 0, 1, 0, 0, 1)
+
+ TESTS:
+
+ The answer never changes::
+
+ sage: J1 = random_eja()
+ sage: J2 = random_eja()
+ sage: J = cartesian_product([J1,J2])
+ sage: P0 = J.cartesian_projection(0)
+ sage: P1 = J.cartesian_projection(0)
+ sage: P0 == P1
+ True
"""
- (J1,J2) = self.factors()
- m = J1.dimension()
- n = J2.dimension()
- V_basis = self.vector_space().basis()
- # Need to specify the dimensions explicitly so that we don't
- # wind up with a zero-by-zero matrix when we want e.g. a
- # zero-by-two matrix (important for composing things).
- P1 = matrix(self.base_ring(), m, m+n, V_basis[:m])
- P2 = matrix(self.base_ring(), n, m+n, V_basis[m:])
- pi_left = FiniteDimensionalEuclideanJordanAlgebraOperator(self,J1,P1)
- pi_right = FiniteDimensionalEuclideanJordanAlgebraOperator(self,J2,P2)
- return (pi_left, pi_right)
-
- def inclusions(self):
- r"""
- Return the pair of inclusion maps from our factors into us.
+ offset = sum( self.cartesian_factor(k).dimension()
+ for k in range(i) )
+ Ji = self.cartesian_factor(i)
+ Pi = self._module_morphism(lambda j: Ji.monomial(j - offset),
+ codomain=Ji)
+
+ return EJAOperator(self,Ji,Pi.matrix())
+ @cached_method
+ def cartesian_embedding(self, i):
+ r"""
SETUP::
sage: from mjo.eja.eja_algebra import (random_eja,
....: JordanSpinEJA,
- ....: RealSymmetricEJA,
- ....: DirectSumEJA)
+ ....: HadamardEJA,
+ ....: RealSymmetricEJA)
- EXAMPLES::
+ EXAMPLES:
+
+ The embedding morphisms are Euclidean Jordan algebra
+ operators::
+
+ sage: J1 = HadamardEJA(2)
+ sage: J2 = RealSymmetricEJA(2)
+ sage: J = cartesian_product([J1,J2])
+ sage: J.cartesian_embedding(0)
+ Linear operator between finite-dimensional Euclidean Jordan
+ algebras represented by the matrix:
+ [1 0]
+ [0 1]
+ [0 0]
+ [0 0]
+ [0 0]
+ Domain: Euclidean Jordan algebra of dimension 2 over
+ Algebraic Real Field
+ Codomain: Euclidean Jordan algebra of dimension 2 over
+ Algebraic Real Field (+) Euclidean Jordan algebra of
+ dimension 3 over Algebraic Real Field
+ sage: J.cartesian_embedding(1)
+ Linear operator between finite-dimensional Euclidean Jordan
+ algebras represented by the matrix:
+ [0 0 0]
+ [0 0 0]
+ [1 0 0]
+ [0 1 0]
+ [0 0 1]
+ Domain: Euclidean Jordan algebra of dimension 3 over
+ Algebraic Real Field
+ Codomain: Euclidean Jordan algebra of dimension 2 over
+ Algebraic Real Field (+) Euclidean Jordan algebra of
+ dimension 3 over Algebraic Real Field
+
+ The embeddings work the way you'd expect on the vector
+ representation of an element::
sage: J1 = JordanSpinEJA(3)
sage: J2 = RealSymmetricEJA(2)
- sage: J = DirectSumEJA(J1,J2)
- sage: (iota_left, iota_right) = J.inclusions()
+ sage: J = cartesian_product([J1,J2])
+ sage: iota_left = J.cartesian_embedding(0)
+ sage: iota_right = J.cartesian_embedding(1)
sage: iota_left(J1.zero()) == J.zero()
True
sage: iota_right(J2.zero()) == J.zero()
TESTS:
+ The answer never changes::
+
+ sage: J1 = random_eja()
+ sage: J2 = random_eja()
+ sage: J = cartesian_product([J1,J2])
+ sage: E0 = J.cartesian_embedding(0)
+ sage: E1 = J.cartesian_embedding(0)
+ sage: E0 == E1
+ True
+
Composing a projection with the corresponding inclusion should
produce the identity map, and mismatching them should produce
the zero map::
- sage: set_random_seed()
sage: J1 = random_eja()
sage: J2 = random_eja()
- sage: J = DirectSumEJA(J1,J2)
- sage: (iota_left, iota_right) = J.inclusions()
- sage: (pi_left, pi_right) = J.projections()
+ sage: J = cartesian_product([J1,J2])
+ sage: iota_left = J.cartesian_embedding(0)
+ sage: iota_right = J.cartesian_embedding(1)
+ sage: pi_left = J.cartesian_projection(0)
+ sage: pi_right = J.cartesian_projection(1)
sage: pi_left*iota_left == J1.one().operator()
True
sage: pi_right*iota_right == J2.one().operator()
True
"""
- (J1,J2) = self.factors()
- m = J1.dimension()
- n = J2.dimension()
- V_basis = self.vector_space().basis()
- # Need to specify the dimensions explicitly so that we don't
- # wind up with a zero-by-zero matrix when we want e.g. a
- # two-by-zero matrix (important for composing things).
- I1 = matrix.column(self.base_ring(), m, m+n, V_basis[:m])
- I2 = matrix.column(self.base_ring(), n, m+n, V_basis[m:])
- iota_left = FiniteDimensionalEuclideanJordanAlgebraOperator(J1,self,I1)
- iota_right = FiniteDimensionalEuclideanJordanAlgebraOperator(J2,self,I2)
- return (iota_left, iota_right)
+ offset = sum( self.cartesian_factor(k).dimension()
+ for k in range(i) )
+ Ji = self.cartesian_factor(i)
+ Ei = Ji._module_morphism(lambda j: self.monomial(j + offset),
+ codomain=self)
+ return EJAOperator(Ji,self,Ei.matrix())
- def inner_product(self, x, y):
+
+ def subalgebra(self, basis, **kwargs):
r"""
- The standard Cartesian inner-product.
+ Create a subalgebra of this algebra from the given basis.
- We project ``x`` and ``y`` onto our factors, and add up the
- inner-products from the subalgebras.
+ Only overridden to allow us to use a special Cartesian product
+ subalgebra class.
SETUP::
-
sage: from mjo.eja.eja_algebra import (HadamardEJA,
- ....: QuaternionHermitianEJA,
- ....: DirectSumEJA)
-
- EXAMPLE::
-
- sage: J1 = HadamardEJA(3,QQ)
- sage: J2 = QuaternionHermitianEJA(2,QQ,normalize_basis=False)
- sage: J = DirectSumEJA(J1,J2)
- sage: x1 = J1.one()
- sage: x2 = x1
- sage: y1 = J2.one()
- sage: y2 = y1
- sage: x1.inner_product(x2)
- 3
- sage: y1.inner_product(y2)
- 2
- sage: J.one().inner_product(J.one())
- 5
+ ....: QuaternionHermitianEJA)
+
+ EXAMPLES:
+
+ Subalgebras of Cartesian product EJAs have a different class
+ than those of non-Cartesian-product EJAs::
+
+ sage: J1 = HadamardEJA(2,field=QQ,orthonormalize=False)
+ sage: J2 = QuaternionHermitianEJA(0,field=QQ,orthonormalize=False)
+ sage: J = cartesian_product([J1,J2])
+ sage: K1 = J1.subalgebra((J1.one(),), orthonormalize=False)
+ sage: K = J.subalgebra((J.one(),), orthonormalize=False)
+ sage: K1.__class__ is K.__class__
+ False
+
+ """
+ from mjo.eja.eja_subalgebra import CartesianProductEJASubalgebra
+ return CartesianProductEJASubalgebra(self, basis, **kwargs)
+
+EJA.CartesianProduct = CartesianProductEJA
+
+class RationalBasisCartesianProductEJA(CartesianProductEJA,
+ RationalBasisEJA):
+ r"""
+ A separate class for products of algebras for which we know a
+ rational basis.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (EJA,
+ ....: HadamardEJA,
+ ....: JordanSpinEJA,
+ ....: RealSymmetricEJA)
+
+ EXAMPLES:
+
+ This gives us fast characteristic polynomial computations in
+ product algebras, too::
+
+
+ sage: J1 = JordanSpinEJA(2)
+ sage: J2 = RealSymmetricEJA(3)
+ sage: J = cartesian_product([J1,J2])
+ sage: J.characteristic_polynomial_of().degree()
+ 5
+ sage: J.rank()
+ 5
+
+ TESTS:
+
+ The ``cartesian_product()`` function only uses the first factor to
+ decide where the result will live; thus we have to be careful to
+ check that all factors do indeed have a ``rational_algebra()`` method
+ before we construct an algebra that claims to have a rational basis::
+
+ sage: J1 = HadamardEJA(2)
+ sage: jp = lambda X,Y: X*Y
+ sage: ip = lambda X,Y: X[0,0]*Y[0,0]
+ sage: b1 = matrix(QQ, [[1]])
+ sage: J2 = EJA((b1,), jp, ip)
+ sage: cartesian_product([J2,J1]) # factor one not RationalBasisEJA
+ Euclidean Jordan algebra of dimension 1 over Algebraic Real
+ Field (+) Euclidean Jordan algebra of dimension 2 over Algebraic
+ Real Field
+ sage: cartesian_product([J1,J2]) # factor one is RationalBasisEJA
+ Traceback (most recent call last):
+ ...
+ ValueError: factor not a RationalBasisEJA
+
+ """
+ def __init__(self, algebras, **kwargs):
+ if not all( hasattr(r, "rational_algebra") for r in algebras ):
+ raise ValueError("factor not a RationalBasisEJA")
+
+ CartesianProductEJA.__init__(self, algebras, **kwargs)
+
+ @cached_method
+ def rational_algebra(self):
+ if self.base_ring() is QQ:
+ return self
+
+ return cartesian_product([
+ r.rational_algebra() for r in self.cartesian_factors()
+ ])
+
+
+RationalBasisEJA.CartesianProduct = RationalBasisCartesianProductEJA
+
+def random_eja(max_dimension=None, *args, **kwargs):
+ r"""
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import random_eja
+
+ TESTS::
+
+ sage: n = ZZ.random_element(1,5)
+ sage: J = random_eja(max_dimension=n, field=QQ, orthonormalize=False)
+ sage: J.dimension() <= n
+ True
+
+ """
+ # Use the ConcreteEJA default as the total upper bound (regardless
+ # of any whether or not any individual factors set a lower limit).
+ if max_dimension is None:
+ max_dimension = ConcreteEJA._max_random_instance_dimension()
+ J1 = ConcreteEJA.random_instance(max_dimension, *args, **kwargs)
+
+
+ # Roll the dice to see if we attempt a Cartesian product.
+ dice_roll = ZZ.random_element(len(ConcreteEJA.__subclasses__()) + 1)
+ new_max_dimension = max_dimension - J1.dimension()
+ if new_max_dimension == 0 or dice_roll != 0:
+ # If it's already as big as we're willing to tolerate, just
+ # return it and don't worry about Cartesian products.
+ return J1
+ else:
+ # Use random_eja() again so we can get more than two factors
+ # if the sub-call also Decides on a cartesian product.
+ J2 = random_eja(new_max_dimension, *args, **kwargs)
+ return cartesian_product([J1,J2])
+
+
+class ComplexSkewSymmetricEJA(RationalBasisEJA, ConcreteEJA):
+ r"""
+ The skew-symmetric EJA of order `2n` described in Faraut and
+ Koranyi's Exercise III.1.b. It has dimension `2n^2 - n`.
+
+ It is (not obviously) isomorphic to the QuaternionHermitianEJA of
+ order `n`, as can be inferred by comparing rank/dimension or
+ explicitly from their "characteristic polynomial of" functions,
+ which just so happen to align nicely.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (ComplexSkewSymmetricEJA,
+ ....: QuaternionHermitianEJA)
+ sage: from mjo.eja.eja_operator import EJAOperator
+
+ EXAMPLES:
+
+ This EJA is isomorphic to the quaternions::
+
+ sage: J = ComplexSkewSymmetricEJA(2, field=QQ, orthonormalize=False)
+ sage: K = QuaternionHermitianEJA(2, field=QQ, orthonormalize=False)
+ sage: jordan_isom_matrix = matrix.diagonal(QQ,[-1,1,1,1,1,-1])
+ sage: phi = EJAOperator(J,K,jordan_isom_matrix)
+ sage: all( phi(x*y) == phi(x)*phi(y)
+ ....: for x in J.gens()
+ ....: for y in J.gens() )
+ True
+ sage: x,y = J.random_elements(2)
+ sage: phi(x*y) == phi(x)*phi(y)
+ True
+
+ TESTS:
+
+ Random elements should satisfy the same conditions that the basis
+ elements do::
+
+ sage: K = ComplexSkewSymmetricEJA.random_instance(field=QQ,
+ ....: orthonormalize=False)
+ sage: x,y = K.random_elements(2)
+ sage: z = x*y
+ sage: x = x.to_matrix()
+ sage: y = y.to_matrix()
+ sage: z = z.to_matrix()
+ sage: all( e.is_skew_symmetric() for e in (x,y,z) )
+ True
+ sage: J = -K.one().to_matrix()
+ sage: all( e*J == J*e.conjugate() for e in (x,y,z) )
+ True
+
+ The power law in Faraut & Koranyi's II.7.a is satisfied.
+ We're in a subalgebra of theirs, but powers are still
+ defined the same::
+
+ sage: K = ComplexSkewSymmetricEJA.random_instance(field=QQ,
+ ....: orthonormalize=False)
+ sage: x = K.random_element()
+ sage: k = ZZ.random_element(5)
+ sage: actual = x^k
+ sage: J = -K.one().to_matrix()
+ sage: expected = K(-J*(J*x.to_matrix())^k)
+ sage: actual == expected
+ True
+
+ """
+ @staticmethod
+ def _max_random_instance_size(max_dimension):
+ # Obtained by solving d = 2n^2 - n, which comes from noticing
+ # that, in 2x2 block form, any element of this algebra has a
+ # free skew-symmetric top-left block, a Hermitian top-right
+ # block, and two bottom blocks that are determined by the top.
+ # The ZZ-int-ZZ thing is just "floor."
+ return ZZ(int(ZZ(8*max_dimension + 1).sqrt()/4 + 1/4))
+
+ @classmethod
+ def random_instance(cls, max_dimension=None, *args, **kwargs):
+ """
+ Return a random instance of this type of algebra.
+ """
+ class_max_d = cls._max_random_instance_dimension()
+ if (max_dimension is None or max_dimension > class_max_d):
+ max_dimension = class_max_d
+ max_size = cls._max_random_instance_size(max_dimension)
+ n = ZZ.random_element(max_size + 1)
+ return cls(n, **kwargs)
+
+ @staticmethod
+ def _denormalized_basis(A):
+ """
+ SETUP::
+
+ sage: from mjo.hurwitz import ComplexMatrixAlgebra
+ sage: from mjo.eja.eja_algebra import ComplexSkewSymmetricEJA
+
+ TESTS:
+
+ The basis elements are all skew-Hermitian::
+
+ sage: d_max = ComplexSkewSymmetricEJA._max_random_instance_dimension()
+ sage: n_max = ComplexSkewSymmetricEJA._max_random_instance_size(d_max)
+ sage: n = ZZ.random_element(n_max + 1)
+ sage: A = ComplexMatrixAlgebra(2*n, scalars=QQ)
+ sage: B = ComplexSkewSymmetricEJA._denormalized_basis(A)
+ sage: all( M.is_skew_symmetric() for M in B)
+ True
+
+ The basis elements ``b`` all satisfy ``b*J == J*b.conjugate()``,
+ as in the definition of the algebra::
+
+ sage: d_max = ComplexSkewSymmetricEJA._max_random_instance_dimension()
+ sage: n_max = ComplexSkewSymmetricEJA._max_random_instance_size(d_max)
+ sage: n = ZZ.random_element(n_max + 1)
+ sage: A = ComplexMatrixAlgebra(2*n, scalars=QQ)
+ sage: I_n = matrix.identity(ZZ, n)
+ sage: J = matrix.block(ZZ, 2, 2, (0, I_n, -I_n, 0), subdivide=False)
+ sage: J = A.from_list(J.rows())
+ sage: B = ComplexSkewSymmetricEJA._denormalized_basis(A)
+ sage: all( b*J == J*b.conjugate() for b in B )
+ True
"""
- (pi_left, pi_right) = self.projections()
- x1 = pi_left(x)
- x2 = pi_right(x)
- y1 = pi_left(y)
- y2 = pi_right(y)
+ es = A.entry_algebra_gens()
+ gen = lambda A,m: A.monomial(m)
+
+ basis = []
+
+ # The size of the blocks. We're going to treat these thing as
+ # 2x2 block matrices,
+ #
+ # [ x1 x2 ]
+ # [ -x2-conj x1-conj ]
+ #
+ # where x1 is skew-symmetric and x2 is Hermitian.
+ #
+ m = A.nrows()/2
+
+ # We only loop through the top half of the matrix, because the
+ # bottom can be constructed from the top.
+ for i in range(m):
+ # First do the top-left block, which is skew-symmetric.
+ # We can compute the bottom-right block in the process.
+ for j in range(i+1):
+ if i != j:
+ # Skew-symmetry implies zeros for (i == j).
+ for e in es:
+ # Top-left block's entry.
+ E_ij = gen(A, (i,j,e))
+ E_ij -= gen(A, (j,i,e))
+
+ # Bottom-right block's entry.
+ F_ij = gen(A, (i+m,j+m,e)).conjugate()
+ F_ij -= gen(A, (j+m,i+m,e)).conjugate()
+
+ basis.append(E_ij + F_ij)
+
+ # Now do the top-right block, which is Hermitian, and compute
+ # the bottom-left block along the way.
+ for j in range(m,i+m+1):
+ if (i+m) == j:
+ # Hermitian matrices have real diagonal entries.
+ # Top-right block's entry.
+ E_ii = gen(A, (i,j,es[0]))
+
+ # Bottom-left block's entry. Don't conjugate
+ # 'cause it's real.
+ E_ii -= gen(A, (i+m,j-m,es[0]))
+ basis.append(E_ii)
+ else:
+ for e in es:
+ # Top-right block's entry. BEWARE! We're not
+ # reflecting across the main diagonal as in
+ # (i,j)~(j,i). We're only reflecting across
+ # the diagonal for the top-right block.
+ E_ij = gen(A, (i,j,e))
+
+ # Shift it back to non-offset coords, transpose,
+ # conjugate, and put it back:
+ #
+ # (i,j) -> (i,j-m) -> (j-m, i) -> (j-m, i+m)
+ E_ij += gen(A, (j-m,i+m,e)).conjugate()
+
+ # Bottom-left's block's below-diagonal entry.
+ # Just shift the top-right coords down m and
+ # left m.
+ F_ij = -gen(A, (i+m,j-m,e)).conjugate()
+ F_ij += -gen(A, (j,i,e)) # double-conjugate cancels
- return (x1.inner_product(y1) + x2.inner_product(y2))
+ basis.append(E_ij + F_ij)
+ return tuple( basis )
+ @staticmethod
+ @cached_method
+ def _J_matrix(matrix_space):
+ n = matrix_space.nrows() // 2
+ F = matrix_space.base_ring()
+ I_n = matrix.identity(F, n)
+ J = matrix.block(F, 2, 2, (0, I_n, -I_n, 0), subdivide=False)
+ return matrix_space.from_list(J.rows())
-random_eja = ConcreteEuclideanJordanAlgebra.random_instance
+ def J_matrix(self):
+ return ComplexSkewSymmetricEJA._J_matrix(self.matrix_space())
+
+ def __init__(self, n, field=AA, **kwargs):
+ # New code; always check the axioms.
+ #if "check_axioms" not in kwargs: kwargs["check_axioms"] = False
+
+ from mjo.hurwitz import ComplexMatrixAlgebra
+ A = ComplexMatrixAlgebra(2*n, scalars=field)
+ J = ComplexSkewSymmetricEJA._J_matrix(A)
+
+ def jordan_product(X,Y):
+ return (X*J*Y + Y*J*X)/2
+
+ def inner_product(X,Y):
+ return (X.conjugate_transpose()*Y).trace().real()
+
+ super().__init__(self._denormalized_basis(A),
+ jordan_product,
+ inner_product,
+ field=field,
+ matrix_space=A,
+ **kwargs)
+
+ # This algebra is conjectured (by me) to be isomorphic to
+ # the quaternion Hermitian EJA of size n, and the rank
+ # would follow from that.
+ #self.rank.set_cache(n)
+ self.one.set_cache( self(-J) )
projects / sage.d.git / blobdiff