from sage.matrix.constructor import matrix
from sage.matrix.matrix_space import MatrixSpace
from sage.misc.cachefunc import cached_method
-from sage.misc.lazy_import import lazy_import
from sage.misc.table import table
from sage.modules.free_module import FreeModule, VectorSpace
from sage.rings.all import (ZZ, QQ, AA, QQbar, RR, RLF, CLF,
PolynomialRing,
QuadraticField)
from mjo.eja.eja_element import FiniteDimensionalEuclideanJordanAlgebraElement
-lazy_import('mjo.eja.eja_subalgebra',
- 'FiniteDimensionalEuclideanJordanSubalgebra')
from mjo.eja.eja_operator import FiniteDimensionalEuclideanJordanAlgebraOperator
from mjo.eja.eja_utils import _mat2vec
sage: J(1)
Traceback (most recent call last):
...
- ValueError: not a naturally-represented algebra element
+ ValueError: not an element of this algebra
"""
return None
mult_table,
prefix='e',
category=None,
- natural_basis=None,
+ matrix_basis=None,
check_field=True,
check_axioms=True):
"""
if not all( len(l) == n for l in mult_table ):
raise ValueError("multiplication table is not square")
- self._natural_basis = natural_basis
+ self._matrix_basis = matrix_basis
if category is None:
category = MagmaticAlgebras(field).FiniteDimensional()
def _element_constructor_(self, elt):
"""
- Construct an element of this algebra from its natural
+ Construct an element of this algebra from its vector or matrix
representation.
This gets called only after the parent element _call_ method
sage: J(A)
Traceback (most recent call last):
...
- ArithmeticError: vector is not in free module
+ ValueError: not an element of this algebra
TESTS:
Ensure that we can convert any element of the two non-matrix
- simple algebras (whose natural representations are their usual
- vector representations) back and forth faithfully::
+ simple algebras (whose matrix representations are columns)
+ back and forth faithfully::
sage: set_random_seed()
sage: J = HadamardEJA.random_instance()
sage: x = J.random_element()
sage: J(x.to_vector().column()) == x
True
-
"""
- msg = "not a naturally-represented algebra element"
+ 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.
# that the integer 3 belongs to the space of 2-by-2 matrices.
raise ValueError(msg)
- natural_basis = self.natural_basis()
- basis_space = natural_basis[0].matrix_space()
- if elt not in basis_space:
+ if elt not in self.matrix_space():
raise ValueError(msg)
# Thanks for nothing! Matrix spaces aren't vector spaces in
- # Sage, so we have to figure out its natural-basis coordinates
+ # Sage, so we have to figure out its matrix-basis coordinates
# ourselves. We use the basis space's ring instead of the
# 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(basis_space.base_ring(), elt.nrows()*elt.ncols())
- W = V.span_of_basis( _mat2vec(s) for s in natural_basis )
- coords = W.coordinate_vector(_mat2vec(elt))
+ 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 (t**r + sum( a[k]*(t**k) for k in range(r) ))
+ def coordinate_polynomial_ring(self):
+ r"""
+ The multivariate polynomial ring in which this algebra's
+ :meth:`characteristic_polynomial_of` lives.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (HadamardEJA,
+ ....: RealSymmetricEJA)
+
+ EXAMPLES::
+
+ sage: J = HadamardEJA(2)
+ sage: J.coordinate_polynomial_ring()
+ Multivariate Polynomial Ring in X1, X2...
+ sage: J = RealSymmetricEJA(3,QQ)
+ sage: J.coordinate_polynomial_ring()
+ Multivariate Polynomial Ring in X1, X2, X3, X4, X5, X6...
+
+ """
+ var_names = tuple( "X%d" % z for z in range(1, self.dimension()+1) )
+ return PolynomialRing(self.base_ring(), var_names)
def inner_product(self, x, y):
"""
SETUP::
- sage: from mjo.eja.eja_algebra import random_eja
+ sage: from mjo.eja.eja_algebra import (random_eja,
+ ....: HadamardEJA,
+ ....: BilinearFormEJA)
EXAMPLES:
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: set_random_seed()
+ sage: J = HadamardEJA.random_instance()
+ sage: x,y = J.random_elements(2)
+ 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: set_random_seed()
+ 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
"""
- X = x.natural_representation()
- Y = y.natural_representation()
- return self.natural_inner_product(X,Y)
+ B = self._inner_product_matrix
+ return (B*x.to_vector()).inner_product(y.to_vector())
def is_trivial(self):
return table(M, header_row=True, header_column=True, frame=True)
- def natural_basis(self):
+ def matrix_basis(self):
"""
- Return a more-natural representation of this algebra's basis.
+ Return an (often more natural) representation of this algebras
+ basis as an ordered tuple of matrices.
+
+ Every finite-dimensional Euclidean Jordan Algebra is a, up to
+ Jordan isomorphism, a direct sum of five simple
+ algebras---four of which comprise Hermitian matrices. And the
+ last type of algebra can of course be thought of as `n`-by-`1`
+ column matrices (ambiguusly called column vectors) to avoid
+ special cases. As a result, matrices (and column vectors) are
+ a natural representation format for Euclidean Jordan algebra
+ elements.
- Every finite-dimensional Euclidean Jordan Algebra is a direct
- sum of five simple algebras, four of which comprise Hermitian
- matrices. This method returns the original "natural" basis
- for our underlying vector space. (Typically, the natural basis
- is used to construct the multiplication table in the first place.)
+ But, when we construct an algebra from a basis of matrices,
+ those matrix representations are lost in favor of coordinate
+ vectors *with respect to* that basis. We could eventually
+ convert back if we tried hard enough, but having the original
+ representations handy is valuable enough that we simply store
+ them and return them from this method.
- Note that this will always return a matrix. The standard basis
- in `R^n` will be returned as `n`-by-`1` column matrices.
+ 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
+ nicely, without having to have special classes for direct sums
+ one of whose components was a matrix algebra.
SETUP::
sage: J = RealSymmetricEJA(2)
sage: J.basis()
Finite family {0: e0, 1: e1, 2: e2}
- sage: J.natural_basis()
+ sage: J.matrix_basis()
(
[1 0] [ 0 0.7071067811865475?] [0 0]
[0 0], [0.7071067811865475? 0], [0 1]
sage: J = JordanSpinEJA(2)
sage: J.basis()
Finite family {0: e0, 1: e1}
- sage: J.natural_basis()
+ sage: J.matrix_basis()
(
[1] [0]
[0], [1]
)
-
"""
- if self._natural_basis is None:
- M = self.natural_basis_space()
+ if self._matrix_basis is None:
+ M = self.matrix_space()
return tuple( M(b.to_vector()) for b in self.basis() )
else:
- return self._natural_basis
+ return self._matrix_basis
- def natural_basis_space(self):
+ def matrix_space(self):
"""
- Return the matrix space in which this algebra's natural basis
- elements live.
+ Return the matrix space in which this algebra's elements live, if
+ we think of them as matrices (including column vectors of the
+ appropriate size).
Generally this will be an `n`-by-`1` column-vector space,
except when the algebra is trivial. There it's `n`-by-`n`
- (where `n` is zero), to ensure that two elements of the
- natural basis space (empty matrices) can be multiplied.
+ (where `n` is zero), to ensure that two elements of the matrix
+ space (empty matrices) can be multiplied.
+
+ Matrix algebras override this with something more useful.
"""
if self.is_trivial():
return MatrixSpace(self.base_ring(), 0)
- elif self._natural_basis is None or len(self._natural_basis) == 0:
+ elif self._matrix_basis is None or len(self._matrix_basis) == 0:
return MatrixSpace(self.base_ring(), self.dimension(), 1)
else:
- return self._natural_basis[0].matrix_space()
-
-
- @staticmethod
- def natural_inner_product(X,Y):
- """
- Compute the inner product of two naturally-represented elements.
-
- For example in the real symmetric matrix EJA, this will compute
- the trace inner-product of two n-by-n symmetric matrices. The
- default should work for the real cartesian product EJA, the
- Jordan spin EJA, and the real symmetric matrices. The others
- will have to be overridden.
- """
- return (X.conjugate_transpose()*Y).trace()
+ return self._matrix_basis[0].matrix_space()
@cached_method
Vector space of degree 6 and dimension 2...
sage: J1
Euclidean Jordan algebra of dimension 3...
- sage: J0.one().natural_representation()
+ sage: J0.one().to_matrix()
[0 0 0]
[0 0 0]
[0 0 1]
sage: orig_df = AA.options.display_format
sage: AA.options.display_format = 'radical'
- sage: J.from_vector(J5.basis()[0]).natural_representation()
+ sage: J.from_vector(J5.basis()[0]).to_matrix()
[ 0 0 1/2*sqrt(2)]
[ 0 0 0]
[1/2*sqrt(2) 0 0]
- sage: J.from_vector(J5.basis()[1]).natural_representation()
+ sage: J.from_vector(J5.basis()[1]).to_matrix()
[ 0 0 0]
[ 0 0 1/2*sqrt(2)]
[ 0 1/2*sqrt(2) 0]
sage: AA.options.display_format = orig_df
- sage: J1.one().natural_representation()
+ sage: J1.one().to_matrix()
[1 0 0]
[0 1 0]
[0 0 0]
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
of" function.
"""
n = self.dimension()
- var_names = [ "X" + str(z) for z in range(1,n+1) ]
- R = PolynomialRing(self.base_ring(), var_names)
+ R = self.coordinate_polynomial_ring()
vars = R.gens()
F = R.fraction_field()
TESTS:
- Our natural basis is normalized with respect to the natural inner
- product unless we specify otherwise::
+ 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: all( b.norm() == 1 for b in J.gens() )
True
- Since our natural basis is normalized with respect to the natural
- inner product, and since we know that this algebra is an EJA, any
+ Since our basis is orthonormal with respect to the algebra's inner
+ product, and since we know that this algebra is an EJA, any
left-multiplication operator's matrix will be symmetric because
- natural->EJA basis representation is an isometry and within the EJA
- the operator is self-adjoint by the Jordan axiom::
+ 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().matrix().is_symmetric()
+ sage: x.operator().is_self_adjoint()
True
-
"""
@staticmethod
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
field = field.extension(p, 'sqrt2', embedding=RLF(2).sqrt())
basis = tuple( s.change_ring(field) for s in basis )
self._basis_normalizers = tuple(
- ~(self.natural_inner_product(s,s).sqrt()) for s in basis )
+ ~(self.matrix_inner_product(s,s).sqrt()) for s in basis )
basis = tuple(s*c for (s,c) in zip(basis,self._basis_normalizers))
- Qs = self.multiplication_table_from_matrix_basis(basis)
+ # 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
+
+ try:
+ # HACK PART DEUX
+ self._inner_product_matrix = matrix(field,ip_table)
+ except:
+ pass
super(MatrixEuclideanJordanAlgebra, self).__init__(field,
- Qs,
- natural_basis=basis,
+ mult_table,
+ matrix_basis=basis,
**kwargs)
if algebra_dim == 0:
# 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.natural_basis(),
+ 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.
return tuple( a_i.subs(subs_dict) for a_i in a )
- @staticmethod
- def multiplication_table_from_matrix_basis(basis):
- """
- At least three of the five simple Euclidean Jordan algebras have the
- symmetric multiplication (A,B) |-> (AB + BA)/2, where the
- multiplication on the right is matrix multiplication. Given a basis
- for the underlying matrix space, this function returns a
- multiplication table (obtained by looping through the basis
- elements) for an algebra of those matrices.
- """
- # In S^2, for example, we nominally have four coordinates even
- # though the space is of dimension three only. The vector space V
- # is supposed to hold the entire long vector, and the subspace W
- # of V will be spanned by the vectors that arise from symmetric
- # matrices. Thus for S^2, dim(V) == 4 and dim(W) == 3.
- if len(basis) == 0:
- return []
-
- field = basis[0].base_ring()
- dimension = basis[0].nrows()
-
- V = VectorSpace(field, dimension**2)
- W = V.span_of_basis( _mat2vec(s) for s in basis )
- n = len(basis)
- mult_table = [[W.zero() for j in range(n)] for i in range(n)]
- for i in range(n):
- for j in range(n):
- mat_entry = (basis[i]*basis[j] + basis[j]*basis[i])/2
- mult_table[i][j] = W.coordinate_vector(_mat2vec(mat_entry))
-
- return mult_table
-
-
@staticmethod
def real_embed(M):
"""
"""
raise NotImplementedError
-
@classmethod
- def natural_inner_product(cls,X,Y):
+ def matrix_inner_product(cls,X,Y):
Xu = cls.real_unembed(X)
Yu = cls.real_unembed(Y)
tr = (Xu*Yu).trace()
sage: set_random_seed()
sage: J = RealSymmetricEJA.random_instance()
sage: x,y = J.random_elements(2)
- sage: actual = (x*y).natural_representation()
- sage: X = x.natural_representation()
- sage: Y = y.natural_representation()
+ sage: actual = (x*y).to_matrix()
+ sage: X = x.to_matrix()
+ sage: Y = y.to_matrix()
sage: expected = (X*Y + Y*X)/2
sage: actual == expected
True
@classmethod
- def natural_inner_product(cls,X,Y):
+ def matrix_inner_product(cls,X,Y):
"""
- Compute a natural inner product in this algebra directly from
+ Compute a matrix inner product in this algebra directly from
its real embedding.
SETUP::
sage: set_random_seed()
sage: J = ComplexHermitianEJA.random_instance()
sage: x,y = J.random_elements(2)
- sage: Xe = x.natural_representation()
- sage: Ye = y.natural_representation()
+ 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.natural_inner_product(Xe,Ye)
+ sage: actual = ComplexHermitianEJA.matrix_inner_product(Xe,Ye)
sage: actual == expected
True
"""
- return RealMatrixEuclideanJordanAlgebra.natural_inner_product(X,Y)/2
+ return RealMatrixEuclideanJordanAlgebra.matrix_inner_product(X,Y)/2
class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra,
sage: set_random_seed()
sage: J = ComplexHermitianEJA.random_instance()
sage: x,y = J.random_elements(2)
- sage: actual = (x*y).natural_representation()
- sage: X = x.natural_representation()
- sage: Y = y.natural_representation()
+ sage: actual = (x*y).to_matrix()
+ sage: X = x.to_matrix()
+ sage: Y = y.to_matrix()
sage: expected = (X*Y + Y*X)/2
sage: actual == expected
True
@classmethod
- def natural_inner_product(cls,X,Y):
+ def matrix_inner_product(cls,X,Y):
"""
- Compute a natural inner product in this algebra directly from
+ Compute a matrix inner product in this algebra directly from
its real embedding.
SETUP::
sage: set_random_seed()
sage: J = QuaternionHermitianEJA.random_instance()
sage: x,y = J.random_elements(2)
- sage: Xe = x.natural_representation()
- sage: Ye = y.natural_representation()
+ 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.natural_inner_product(Xe,Ye)
+ sage: actual = QuaternionHermitianEJA.matrix_inner_product(Xe,Ye)
sage: actual == expected
True
"""
- return RealMatrixEuclideanJordanAlgebra.natural_inner_product(X,Y)/4
+ return RealMatrixEuclideanJordanAlgebra.matrix_inner_product(X,Y)/4
class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra,
sage: set_random_seed()
sage: J = QuaternionHermitianEJA.random_instance()
sage: x,y = J.random_elements(2)
- sage: actual = (x*y).natural_representation()
- sage: X = x.natural_representation()
- sage: Y = y.natural_representation()
+ sage: actual = (x*y).to_matrix()
+ sage: X = x.to_matrix()
+ sage: Y = y.to_matrix()
sage: expected = (X*Y + Y*X)/2
sage: actual == expected
True
mult_table = [ [ V.gen(i)*(i == j) for j in range(n) ]
for i in range(n) ]
+ # 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 = matrix.identity(field,n)
+ self._inner_product_matrix = ip_table
+
super(HadamardEJA, self).__init__(field,
mult_table,
check_axioms=False,
return cls(n, field, **kwargs)
- def inner_product(self, x, y):
- """
- Faster to reimplement than to use natural representations.
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import HadamardEJA
-
- TESTS:
-
- 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: X = x.natural_representation()
- sage: Y = y.natural_representation()
- sage: x.inner_product(y) == J.natural_inner_product(X,Y)
- True
-
- """
- return x.to_vector().inner_product(y.to_vector())
-
-
class BilinearFormEJA(RationalBasisEuclideanJordanAlgebra,
ConcreteEuclideanJordanAlgebra):
r"""
True
"""
def __init__(self, B, field=AA, **kwargs):
- self._B = B
n = B.nrows()
if not B.is_positive_definite():
z = V([z0] + zbar.list())
mult_table[i][j] = z
- # 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).
+ # 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)
+
+ # 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:
return cls(B, field, **kwargs)
- def inner_product(self, x, y):
- r"""
- Half of the trace inner product.
-
- This is defined so that the special case of the Jordan spin
- algebra gets the usual inner product.
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import BilinearFormEJA
-
- TESTS:
-
- Ensure that this is one-half of the trace inner-product when
- the algebra isn't just the reals (when ``n`` isn't 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: y = J.random_element()
- sage: (n == 1) or (x.inner_product(y) == (x*y).trace()/2)
- True
-
- """
- return (self._B*x.to_vector()).inner_product(y.to_vector())
-
class JordanSpinEJA(BilinearFormEJA):
"""
sage: set_random_seed()
sage: J = JordanSpinEJA.random_instance()
sage: x,y = J.random_elements(2)
- sage: X = x.natural_representation()
- sage: Y = y.natural_representation()
- sage: x.inner_product(y) == J.natural_inner_product(X,Y)
+ sage: actual = x.inner_product(y)
+ sage: expected = x.to_vector().inner_product(y.to_vector())
+ sage: actual == expected
True
"""
"""
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,
projects / sage.d.git / blobdiff