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.
+
+
+SETUP::
+
+ sage: from mjo.eja.eja_algebra import random_eja
+
+EXAMPLES::
+
+ sage: random_eja()
+ Euclidean Jordan algebra of dimension...
+
"""
from itertools import repeat
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.prandom import choice
from sage.misc.table import table
from sage.modules.free_module import FreeModule, VectorSpace
-from sage.rings.all import (ZZ, QQ, RR, RLF, CLF,
+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
class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule):
- # This is an ugly hack needed to prevent the category framework
- # from implementing a coercion from our base ring (e.g. the
- # rationals) into the algebra. First of all -- such a coercion is
- # nonsense to begin with. But more importantly, it tries to do so
- # in the category of rings, and since our algebras aren't
- # associative they generally won't be rings.
- _no_generic_basering_coercion = True
+
+ def _coerce_map_from_base_ring(self):
+ """
+ Disable the map from the base ring into the algebra.
+
+ Performing a nonsense conversion like this automatically
+ is counterpedagogical. The fallback is to try the usual
+ element constructor, which should also fail.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import random_eja
+
+ TESTS::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: J(1)
+ Traceback (most recent call last):
+ ...
+ ValueError: not a naturally-represented algebra element
+
+ """
+ return None
def __init__(self,
field,
mult_table,
- rank,
prefix='e',
category=None,
natural_basis=None,
- check=True):
+ check_field=True,
+ check_axioms=True):
"""
SETUP::
- sage: from mjo.eja.eja_algebra import (JordanSpinEJA, random_eja)
+ sage: from mjo.eja.eja_algebra import (
+ ....: FiniteDimensionalEuclideanJordanAlgebra,
+ ....: JordanSpinEJA,
+ ....: random_eja)
EXAMPLES:
TESTS:
- The ``field`` we're given must be real::
+ The ``field`` we're given must be real with ``check_field=True``::
sage: JordanSpinEJA(2,QQbar)
Traceback (most recent call last):
...
- ValueError: field is not real
+ ValueError: scalar field is not real
+
+ The multiplication table must be square with ``check_axioms=True``::
+
+ sage: FiniteDimensionalEuclideanJordanAlgebra(QQ,((),()))
+ Traceback (most recent call last):
+ ...
+ ValueError: multiplication table is not square
"""
- if check:
+ if check_field:
if not field.is_subring(RR):
# Note: this does return true for the real algebraic
- # field, and any quadratic field where we've specified
- # a real embedding.
- raise ValueError('field is not real')
+ # 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._rank = rank
self._natural_basis = natural_basis
if category is None:
fda = super(FiniteDimensionalEuclideanJordanAlgebra, self)
fda.__init__(field,
- range(len(mult_table)),
+ range(n),
prefix=prefix,
category=category)
self.print_options(bracket='')
# 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 = [ map(lambda x: self.from_vector(x), ls)
- for ls in mult_table ]
-
+ self._multiplication_table = [
+ list(map(lambda x: self.from_vector(x), ls))
+ for ls in mult_table
+ ]
+
+ 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")
def _element_constructor_(self, elt):
"""
SETUP::
sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
- ....: RealCartesianProductEJA,
+ ....: HadamardEJA,
....: RealSymmetricEJA)
EXAMPLES:
vector representations) back and forth faithfully::
sage: set_random_seed()
- sage: J = RealCartesianProductEJA.random_instance()
+ sage: J = HadamardEJA.random_instance()
sage: x = J.random_element()
sage: J(x.to_vector().column()) == x
True
True
"""
+ msg = "not a naturally-represented algebra element"
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():
+ # 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)
natural_basis = self.natural_basis()
basis_space = natural_basis[0].matrix_space()
if elt not in basis_space:
- raise ValueError("not a naturally-represented algebra element")
+ raise ValueError(msg)
# Thanks for nothing! Matrix spaces aren't vector spaces in
# Sage, so we have to figure out its natural-basis coordinates
coords = W.coordinate_vector(_mat2vec(elt))
return self.from_vector(coords)
-
def _repr_(self):
"""
Return a string representation of ``self``.
Ensure that it says what we think it says::
- sage: JordanSpinEJA(2, field=QQ)
- Euclidean Jordan algebra of dimension 2 over Rational Field
+ 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
def product_on_basis(self, i, j):
return self._multiplication_table[i][j]
- def _a_regular_element(self):
- """
- Guess a regular element. Needed to compute the basis for our
- characteristic polynomial coefficients.
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import random_eja
-
- TESTS:
-
- Ensure that this hacky method succeeds for every algebra that we
- know how to construct::
-
- sage: set_random_seed()
- sage: J = random_eja()
- sage: J._a_regular_element().is_regular()
- True
-
- """
- gs = self.gens()
- z = self.sum( (i+1)*gs[i] for i in range(len(gs)) )
- if not z.is_regular():
- raise ValueError("don't know a regular element")
- return z
-
-
- @cached_method
- def _charpoly_basis_space(self):
- """
- Return the vector space spanned by the basis used in our
- characteristic polynomial coefficients. This is used not only to
- compute those coefficients, but also any time we need to
- evaluate the coefficients (like when we compute the trace or
- determinant).
- """
- z = self._a_regular_element()
- # Don't use the parent vector space directly here in case this
- # happens to be a subalgebra. In that case, we would be e.g.
- # two-dimensional but span_of_basis() would expect three
- # coordinates.
- V = VectorSpace(self.base_ring(), self.vector_space().dimension())
- basis = [ (z**k).to_vector() for k in range(self.rank()) ]
- V1 = V.span_of_basis( basis )
- b = (V1.basis() + V1.complement().basis())
- return V.span_of_basis(b)
-
-
-
- @cached_method
- def _charpoly_coeff(self, i):
- """
- Return the coefficient polynomial "a_{i}" of this algebra's
- general characteristic polynomial.
-
- Having this be a separate cached method lets us compute and
- store the trace/determinant (a_{r-1} and a_{0} respectively)
- separate from the entire characteristic polynomial.
- """
- (A_of_x, x, xr, detA) = self._charpoly_matrix_system()
- R = A_of_x.base_ring()
-
- if i == self.rank():
- return R.one()
- if i > self.rank():
- # Guaranteed by theory
- return R.zero()
-
- # Danger: the in-place modification is done for performance
- # reasons (reconstructing a matrix with huge polynomial
- # entries is slow), but I don't know how cached_method works,
- # so it's highly possible that we're modifying some global
- # list variable by reference, here. In other words, you
- # probably shouldn't call this method twice on the same
- # algebra, at the same time, in two threads
- Ai_orig = A_of_x.column(i)
- A_of_x.set_column(i,xr)
- numerator = A_of_x.det()
- A_of_x.set_column(i,Ai_orig)
-
- # We're relying on the theory here to ensure that each a_i is
- # indeed back in R, and the added negative signs are to make
- # the whole charpoly expression sum to zero.
- return R(-numerator/detA)
-
-
- @cached_method
- def _charpoly_matrix_system(self):
- """
- Compute the matrix whose entries A_ij are polynomials in
- X1,...,XN, the vector ``x`` of variables X1,...,XN, the vector
- corresponding to `x^r` and the determinent of the matrix A =
- [A_ij]. In other words, all of the fixed (cachable) data needed
- to compute the coefficients of the characteristic polynomial.
- """
- r = self.rank()
- n = self.dimension()
-
- # Turn my vector space into a module so that "vectors" can
- # have multivatiate polynomial entries.
- names = tuple('X' + str(i) for i in range(1,n+1))
- R = PolynomialRing(self.base_ring(), names)
-
- # Using change_ring() on the parent's vector space doesn't work
- # here because, in a subalgebra, that vector space has a basis
- # and change_ring() tries to bring the basis along with it. And
- # that doesn't work unless the new ring is a PID, which it usually
- # won't be.
- V = FreeModule(R,n)
-
- # Now let x = (X1,X2,...,Xn) be the vector whose entries are
- # indeterminates...
- x = V(names)
-
- # And figure out the "left multiplication by x" matrix in
- # that setting.
- lmbx_cols = []
- monomial_matrices = [ self.monomial(i).operator().matrix()
- for i in range(n) ] # don't recompute these!
- for k in range(n):
- ek = self.monomial(k).to_vector()
- lmbx_cols.append(
- sum( x[i]*(monomial_matrices[i]*ek)
- for i in range(n) ) )
- Lx = matrix.column(R, lmbx_cols)
-
- # Now we can compute powers of x "symbolically"
- x_powers = [self.one().to_vector(), x]
- for d in range(2, r+1):
- x_powers.append( Lx*(x_powers[-1]) )
-
- idmat = matrix.identity(R, n)
-
- W = self._charpoly_basis_space()
- W = W.change_ring(R.fraction_field())
-
- # Starting with the standard coordinates x = (X1,X2,...,Xn)
- # and then converting the entries to W-coordinates allows us
- # to pass in the standard coordinates to the charpoly and get
- # back the right answer. Specifically, with x = (X1,X2,...,Xn),
- # we have
- #
- # W.coordinates(x^2) eval'd at (standard z-coords)
- # =
- # W-coords of (z^2)
- # =
- # W-coords of (standard coords of x^2 eval'd at std-coords of z)
- #
- # We want the middle equivalent thing in our matrix, but use
- # the first equivalent thing instead so that we can pass in
- # standard coordinates.
- x_powers = [ W.coordinate_vector(xp) for xp in x_powers ]
- l2 = [idmat.column(k-1) for k in range(r+1, n+1)]
- A_of_x = matrix.column(R, n, (x_powers[:r] + l2))
- return (A_of_x, x, x_powers[r], A_of_x.det())
-
+ 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( self.product_on_basis(i,j) == self.product_on_basis(i,j)
+ for i in range(self.dimension())
+ for j in range(self.dimension()) )
+
+ 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 _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 multiplication table.
+ """
+
+ # 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
+
+ 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 self.base_ring().is_exact():
+ if diff != 0:
+ return False
+ else:
+ if diff.abs() > epsilon:
+ return False
+
+ return True
@cached_method
- def characteristic_polynomial(self):
+ def characteristic_polynomial_of(self):
"""
- Return a characteristic polynomial that works for all elements
- of this algebra.
+ Return the algebra's "characteristic polynomial of" function,
+ which is itself a multivariate polynomial that, when evaluated
+ at the coordinates of some algebra element, returns that
+ element's characteristic polynomial.
The resulting polynomial has `n+1` variables, where `n` is the
dimension of this algebra. The first `n` variables correspond to
Alizadeh, Example 11.11::
sage: J = JordanSpinEJA(3)
- sage: p = J.characteristic_polynomial(); p
+ sage: p = J.characteristic_polynomial_of(); p
X1^2 - X2^2 - X3^2 + (-2*t)*X1 + t^2
sage: xvec = J.one().to_vector()
sage: p(*xvec)
any argument::
sage: J = TrivialEJA()
- sage: J.characteristic_polynomial()
+ sage: J.characteristic_polynomial_of()
1
"""
r = self.rank()
n = self.dimension()
- # The list of coefficient polynomials a_0, a_1, a_2, ..., a_n.
- a = [ self._charpoly_coeff(i) for i in range(r+1) ]
+ # The list of coefficient polynomials a_0, a_1, a_2, ..., a_(r-1).
+ a = self._charpoly_coefficients()
# We go to a bit of trouble here to reorder the
# indeterminates, so that it's easier to evaluate the
# characteristic polynomial at x's coordinates and get back
# something in terms of t, which is what we want.
- R = a[0].parent()
S = PolynomialRing(self.base_ring(),'t')
t = S.gen(0)
- S = PolynomialRing(S, R.variable_names())
- t = S(t)
+ if r > 0:
+ R = a[0].parent()
+ S = PolynomialRing(S, R.variable_names())
+ t = S(t)
- return sum( a[k]*(t**k) for k in xrange(len(a)) )
+ return (t**r + sum( a[k]*(t**k) for k in range(r) ))
def inner_product(self, x, y):
"""
M = list(self._multiplication_table) # copy
- for i in xrange(len(M)):
+ for i in range(len(M)):
# M had better be "square"
M[i] = [self.monomial(i)] + M[i]
M = [["*"] + list(self.gens())] + M
Finite family {0: e0, 1: e1, 2: e2}
sage: J.natural_basis()
(
- [1 0] [ 0 1/2*sqrt2] [0 0]
- [0 0], [1/2*sqrt2 0], [0 1]
+ [1 0] [ 0 0.7071067811865475?] [0 0]
+ [0 0], [0.7071067811865475? 0], [0 1]
)
::
"""
Return the matrix space in which this algebra's natural basis
elements live.
+
+ 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.
"""
- if self._natural_basis is None or len(self._natural_basis) == 0:
+ if self.is_trivial():
+ return MatrixSpace(self.base_ring(), 0)
+ elif self._natural_basis is None or len(self._natural_basis) == 0:
return MatrixSpace(self.base_ring(), self.dimension(), 1)
else:
return self._natural_basis[0].matrix_space()
SETUP::
- sage: from mjo.eja.eja_algebra import (RealCartesianProductEJA,
+ sage: from mjo.eja.eja_algebra import (HadamardEJA,
....: random_eja)
EXAMPLES::
- sage: J = RealCartesianProductEJA(5)
+ sage: J = HadamardEJA(5)
sage: J.one()
e0 + e1 + e2 + e3 + e4
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
+
"""
# We can brute-force compute the matrices of the operators
# that correspond to the basis elements of this algebra.
# appeal to the "long vectors" isometry.
oper_vecs = [ _mat2vec(g.operator().matrix()) for g in self.gens() ]
- # Now we use basis linear algebra to find the coefficients,
+ # Now we use basic linear algebra to find the coefficients,
# of the matrices-as-vectors-linear-combination, which should
# work for the original algebra basis too.
- A = matrix.column(self.base_ring(), oper_vecs)
+ A = matrix(self.base_ring(), oper_vecs)
# 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()) )
- # Now if there's an identity element in the algebra, this should work.
- coeffs = A.solve_right(b)
- return self.linear_combination(zip(self.gens(), coeffs))
+ # Now if there's an identity element in the algebra, this
+ # should work. We solve on the left to avoid having to
+ # transpose the matrix "A".
+ return self.from_vector(A.solve_left(b))
def peirce_decomposition(self, c):
Vector space of degree 6 and dimension 2...
sage: J1
Euclidean Jordan algebra of dimension 3...
+ sage: J0.one().natural_representation()
+ [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()
+ [ 0 0 1/2*sqrt(2)]
+ [ 0 0 0]
+ [1/2*sqrt(2) 0 0]
+ sage: J.from_vector(J5.basis()[1]).natural_representation()
+ [ 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()
+ [1 0 0]
+ [0 1 0]
+ [0 0 0]
TESTS:
sage: J1.superalgebra() == J and J1.dimension() == J.dimension()
True
- The identity elements in the two subalgebras are the
- projections onto their respective subspaces of the
- superalgebra's identity element::
+ The decomposition is into eigenspaces, and its components are
+ therefore necessarily orthogonal. Moreover, the identity
+ 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()
....: x = J.random_element()
sage: c = x.subalgebra_idempotent()
sage: J0,J5,J1 = J.peirce_decomposition(c)
+ sage: ipsum = 0
+ sage: for (w,y,z) in zip(J0.basis(), J5.basis(), J1.basis()):
+ ....: w = w.superalgebra_element()
+ ....: y = J.from_vector(y)
+ ....: z = z.superalgebra_element()
+ ....: ipsum += w.inner_product(y).abs()
+ ....: ipsum += w.inner_product(z).abs()
+ ....: ipsum += y.inner_product(z).abs()
+ sage: ipsum
+ 0
sage: J1(c) == J1.one()
True
sage: J0(J.one() - c) == J0.one()
J5 = VectorSpace(self.base_ring(), 0) # eigenvalue one-half
J1 = trivial # eigenvalue one
- for (eigval, eigspace) in c.operator().matrix().left_eigenspaces():
+ for (eigval, eigspace) in c.operator().matrix().right_eigenspaces():
if eigval == ~(self.base_ring()(2)):
J5 = eigspace
else:
gens = tuple( self.from_vector(b) for b in eigspace.basis() )
- subalg = FiniteDimensionalEuclideanJordanSubalgebra(self, gens)
+ subalg = FiniteDimensionalEuclideanJordanSubalgebra(self,
+ gens,
+ check_axioms=False)
if eigval == 0:
J0 = subalg
elif eigval == 1:
return (J0, J5, J1)
- def random_elements(self, count):
+ def random_element(self, thorough=False):
+ r"""
+ Return a random element of this algebra.
+
+ Our algebra superclass method only returns a linear
+ combination of at most two basis elements. We instead
+ want the vector space "random element" method that
+ returns a more diverse selection.
+
+ INPUT:
+
+ - ``thorough`` -- (boolean; default False) whether or not we
+ should generate irrational coefficients for the random
+ element when our base ring is irrational; this slows the
+ algebra operations to a crawl, but any truly random method
+ should include them
+
+ """
+ # 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()
+
+ return self.from_vector(V.coordinate_vector(v))
+
+ def random_elements(self, count, thorough=False):
"""
Return ``count`` random elements as a tuple.
+ INPUT:
+
+ - ``thorough`` -- (boolean; default False) whether or not we
+ should generate irrational coefficients for the random
+ elements when our base ring is irrational; this slows the
+ algebra operations to a crawl, but any truly random method
+ should include them
+
SETUP::
sage: from mjo.eja.eja_algebra import JordanSpinEJA
True
"""
- return tuple( self.random_element() for idx in xrange(count) )
+ return tuple( self.random_element(thorough)
+ for idx in range(count) )
- def rank(self):
+ @cached_method
+ def _charpoly_coefficients(self):
+ r"""
+ The `r` polynomial coefficients of the "characteristic polynomial
+ of" function.
"""
- Return the rank of this EJA.
+ n = self.dimension()
+ var_names = [ "X" + str(z) for z in range(1,n+1) ]
+ R = PolynomialRing(self.base_ring(), var_names)
+ 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)
+
+ r = None
+ if self.rank.is_in_cache():
+ r = self.rank()
+ # There's no need to pad the system with redundant
+ # columns if we *know* they'll be redundant.
+ n = r
+
+ # Compute an extra power in case the rank is equal to
+ # the dimension (otherwise, we would stop at x^(r-1)).
+ x_powers = [ (L_x**k)*self.one().to_vector()
+ for k in range(n+1) ]
+ A = matrix.column(F, x_powers[:n])
+ AE = A.extended_echelon_form()
+ E = AE[:,n:]
+ A_rref = AE[:,:n]
+ if r is None:
+ r = A_rref.rank()
+ b = x_powers[r]
+
+ # 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]
- ALGORITHM:
+ @cached_method
+ def rank(self):
+ r"""
+ Return the rank of this EJA.
- The author knows of no algorithm to compute the rank of an EJA
- where only the multiplication table is known. In lieu of one, we
- require the rank to be specified when the algebra is created,
- and simply pass along that number here.
+ This is a cached method because we know the rank a priori for
+ all of the algebras we can construct. Thus we can avoid the
+ expensive ``_charpoly_coefficients()`` call unless we truly
+ need to compute the whole characteristic polynomial.
SETUP::
- sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
+ sage: from mjo.eja.eja_algebra import (HadamardEJA,
+ ....: JordanSpinEJA,
....: RealSymmetricEJA,
....: ComplexHermitianEJA,
....: QuaternionHermitianEJA,
sage: r > 0 or (r == 0 and J.is_trivial())
True
+ Ensure that computing the rank actually works, since the ranks
+ of all simple algebras are known and will be cached by default::
+
+ sage: J = HadamardEJA(4)
+ sage: J.rank.clear_cache()
+ sage: J.rank()
+ 4
+
+ ::
+
+ sage: J = JordanSpinEJA(4)
+ sage: J.rank.clear_cache()
+ sage: J.rank()
+ 2
+
+ ::
+
+ sage: J = RealSymmetricEJA(3)
+ sage: J.rank.clear_cache()
+ sage: J.rank()
+ 3
+
+ ::
+
+ sage: J = ComplexHermitianEJA(2)
+ sage: J.rank.clear_cache()
+ sage: J.rank()
+ 2
+
+ ::
+
+ sage: J = QuaternionHermitianEJA(2)
+ sage: J.rank.clear_cache()
+ sage: J.rank()
+ 2
"""
- return self._rank
+ return len(self._charpoly_coefficients())
def vector_space(self):
Element = FiniteDimensionalEuclideanJordanAlgebraElement
-class KnownRankEJA(object):
- """
- A class for algebras that we actually know we can construct. The
- main issue is that, for most of our methods to make sense, we need
- to know the rank of our algebra. Thus we can't simply generate a
- "random" algebra, or even check that a given basis and product
- satisfy the axioms; because even if everything looks OK, we wouldn't
- know the rank we need to actuallty build the thing.
-
- Not really a subclass of FDEJA because doing that causes method
- resolution errors, e.g.
-
- TypeError: Error when calling the metaclass bases
- Cannot create a consistent method resolution
- order (MRO) for bases FiniteDimensionalEuclideanJordanAlgebra,
- KnownRankEJA
+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.
"""
- @staticmethod
- def _max_test_case_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 quite vague -- 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.
-
- We default to five in this class, which is safe in `R^n`. The
- matrix algebra subclasses (or any class where the "size" is
- interpreted to be far less than the dimension) should override
- with a smaller number.
- """
- return 5
-
- @classmethod
- def random_instance(cls, field=QQ, **kwargs):
- """
- Return a random instance of this type of algebra.
+ @cached_method
+ def _charpoly_coefficients(self):
+ r"""
+ Override the parent method with something that tries to compute
+ over a faster (non-extension) field.
- Beware, this will crash for "most instances" because the
- constructor below looks wrong.
- """
- if cls is TrivialEJA:
- # The TrivialEJA class doesn't take an "n" argument because
- # there's only one.
- return cls(field)
+ SETUP::
- n = ZZ.random_element(cls._max_test_case_size()) + 1
- return cls(n, field, **kwargs)
+ sage: from mjo.eja.eja_algebra import JordanSpinEJA
+ EXAMPLES:
-class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra,
- KnownRankEJA):
- """
- Return the Euclidean Jordan Algebra corresponding to the set
- `R^n` under the Hadamard product.
+ The base ring of the resulting polynomial coefficients is what
+ it should be, and not the rationals (unless the algebra was
+ already over the rationals)::
- 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.
+ sage: J = JordanSpinEJA(3)
+ sage: J._charpoly_coefficients()
+ (X1^2 - X2^2 - X3^2, -2*X1)
+ sage: a0 = J._charpoly_coefficients()[0]
+ sage: J.base_ring()
+ Algebraic Real Field
+ sage: a0.base_ring()
+ 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(
+ map(lambda x: x.to_vector(), ls)
+ for ls in self._multiplication_table
+ )
+
+ # 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))
+
+
+class ConcreteEuclideanJordanAlgebra:
+ r"""
+ A class for the Euclidean Jordan algebras that we know by name.
+
+ These are the Jordan algebras whose basis, multiplication table,
+ rank, and so on are known a priori. More to the point, they are
+ the Euclidean Jordan algebras for which we are able to conjure up
+ a "random instance."
SETUP::
- sage: from mjo.eja.eja_algebra import RealCartesianProductEJA
-
- EXAMPLES:
+ sage: from mjo.eja.eja_algebra import ConcreteEuclideanJordanAlgebra
- This multiplication table can be verified by hand::
+ TESTS:
- sage: J = RealCartesianProductEJA(3)
- sage: e0,e1,e2 = J.gens()
- sage: e0*e0
- e0
- sage: e0*e1
- 0
- sage: e0*e2
- 0
- sage: e1*e1
- e1
- sage: e1*e2
- 0
- sage: e2*e2
- e2
+ Our natural basis is normalized with respect to the natural inner
+ product unless we specify otherwise::
- TESTS:
+ sage: set_random_seed()
+ sage: J = ConcreteEuclideanJordanAlgebra.random_instance()
+ sage: all( b.norm() == 1 for b in J.gens() )
+ True
- We can change the generator prefix::
+ Since our natural basis is normalized with respect to the natural
+ 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::
- sage: RealCartesianProductEJA(3, prefix='r').gens()
- (r0, r1, r2)
+ sage: set_random_seed()
+ sage: J = ConcreteEuclideanJordanAlgebra.random_instance()
+ sage: x = J.random_element()
+ sage: x.operator().matrix().is_symmetric()
+ True
"""
- def __init__(self, n, field=QQ, **kwargs):
- V = VectorSpace(field, n)
- mult_table = [ [ V.gen(i)*(i == j) for j in xrange(n) ]
- for i in xrange(n) ]
- fdeja = super(RealCartesianProductEJA, self)
- return fdeja.__init__(field, mult_table, rank=n, **kwargs)
-
- def inner_product(self, x, y):
+ @staticmethod
+ def _max_random_instance_size():
"""
- Faster to reimplement than to use natural representations.
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import RealCartesianProductEJA
-
- TESTS:
-
- Ensure that this is the usual inner product for the algebras
- over `R^n`::
-
- sage: set_random_seed()
- sage: J = RealCartesianProductEJA.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 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.
"""
- return x.to_vector().inner_product(y.to_vector())
-
-
-def random_eja(field=QQ, nontrivial=False):
- """
- Return a "random" finite-dimensional Euclidean Jordan Algebra.
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import random_eja
-
- TESTS::
-
- sage: random_eja()
- Euclidean Jordan algebra of dimension...
-
- """
- eja_classes = KnownRankEJA.__subclasses__()
- if nontrivial:
- eja_classes.remove(TrivialEJA)
- classname = choice(eja_classes)
- return classname.random_instance(field=field)
-
-
+ raise NotImplementedError
+ @classmethod
+ def random_instance(cls, field=AA, **kwargs):
+ """
+ Return a random instance of this type of algebra.
+ 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):
- @staticmethod
- def _max_test_case_size():
- # Play it safe, since this will be squared and the underlying
- # field can have dimension 4 (quaternions) too.
- return 2
- def __init__(self, field, basis, rank, normalize_basis=True, **kwargs):
+ 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_coeff() override.
+ # 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)
- if rank > 1 and normalize_basis:
+ algebra_dim = len(basis)
+ 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.
Qs = self.multiplication_table_from_matrix_basis(basis)
- fdeja = super(MatrixEuclideanJordanAlgebra, self)
- return fdeja.__init__(field,
- Qs,
- rank=rank,
- natural_basis=basis,
- **kwargs)
+ super(MatrixEuclideanJordanAlgebra, self).__init__(field,
+ Qs,
+ natural_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)))
@cached_method
- def _charpoly_coeff(self, i):
- """
+ def _charpoly_coefficients(self):
+ r"""
Override the parent method with something that tries to compute
over a faster (non-extension) field.
"""
- if self._basis_normalizers is None:
- # We didn't normalize, so assume that the basis we started
- # with had entries in a nice field.
- return super(MatrixEuclideanJordanAlgebra, self)._charpoly_coeff(i)
- else:
- basis = ( (b/n) for (b,n) in zip(self.natural_basis(),
- self._basis_normalizers) )
-
- # Do this over the rationals and convert back at the end.
- J = MatrixEuclideanJordanAlgebra(QQ,
- basis,
- self.rank(),
- normalize_basis=False)
- (_,x,_,_) = J._charpoly_matrix_system()
- p = J._charpoly_coeff(i)
- # p might be missing some vars, have to substitute "optionally"
- pairs = zip(x.base_ring().gens(), self._basis_normalizers)
- substitutions = { v: v*c for (v,c) in pairs }
- result = p.subs(substitutions)
-
- # The result of "subs" can be either a coefficient-ring
- # element or a polynomial. Gotta handle both cases.
- if result in QQ:
- return self.base_ring()(result)
- else:
- return result.change_ring(self.base_ring())
+ 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.natural_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 )
@staticmethod
# 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 xrange(n)] for i in xrange(n)]
- for i in xrange(n):
- for j in xrange(n):
+ 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))
Yu = cls.real_unembed(Y)
tr = (Xu*Yu).trace()
- if tr in RLF:
- # It's real already.
- return tr
-
- # Otherwise, try the thing that works for complex numbers; and
- # if that doesn't work, the thing that works for quaternions.
try:
- return tr.vector()[0] # real part, imag part is index 1
+ # Works in QQ, AA, RDF, et cetera.
+ return tr.real()
except AttributeError:
- # A quaternions doesn't have a vector() method, but does
+ # 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]
return M
-class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra, KnownRankEJA):
+class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra,
+ ConcreteEuclideanJordanAlgebra):
"""
The rank-n simple EJA consisting of real symmetric n-by-n
matrices, the usual symmetric Jordan product, and the trace inner
In theory, our "field" can be any subfield of the reals::
- sage: RealSymmetricEJA(2, AA)
- Euclidean Jordan algebra of dimension 3 over Algebraic Real Field
+ sage: RealSymmetricEJA(2, RDF)
+ Euclidean Jordan algebra of dimension 3 over Real Double Field
sage: RealSymmetricEJA(2, RR)
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_test_case_size()
+ sage: n_max = RealSymmetricEJA._max_random_instance_size()
sage: n = ZZ.random_element(1, n_max)
sage: J = RealSymmetricEJA(n)
sage: J.dimension() == (n^2 + n)/2
sage: RealSymmetricEJA(3, prefix='q').gens()
(q0, q1, q2, q3, q4, q5)
- Our natural basis is normalized with respect to the natural inner
- product unless we specify otherwise::
-
- sage: set_random_seed()
- sage: J = RealSymmetricEJA.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
- 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::
+ We can construct the (trivial) algebra of rank zero::
- sage: set_random_seed()
- sage: x = RealSymmetricEJA.random_instance().random_element()
- sage: x.operator().matrix().is_symmetric()
- True
+ sage: RealSymmetricEJA(0)
+ Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
"""
@classmethod
# The basis of symmetric matrices, as matrices, in their R^(n-by-n)
# coordinates.
S = []
- for i in xrange(n):
- for j in xrange(i+1):
+ 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
@staticmethod
- def _max_test_case_size():
+ def _max_random_instance_size():
return 4 # Dimension 10
+ @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)
- def __init__(self, n, field=QQ, **kwargs):
+ def __init__(self, n, field=AA, **kwargs):
basis = self._denormalized_basis(n, field)
- super(RealSymmetricEJA, self).__init__(field, basis, n, **kwargs)
+ super(RealSymmetricEJA, self).__init__(field,
+ basis,
+ check_axioms=False,
+ **kwargs)
+ self.rank.set_cache(n)
class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra):
EXAMPLES::
- sage: F = QuadraticField(-1, 'i')
+ sage: F = QuadraticField(-1, 'I')
sage: x1 = F(4 - 2*i)
sage: x2 = F(1 + 2*i)
sage: x3 = F(-i)
Embedding is a homomorphism (isomorphism, in fact)::
sage: set_random_seed()
- sage: n_max = ComplexMatrixEuclideanJordanAlgebra._max_test_case_size()
- sage: n = ZZ.random_element(n_max)
- sage: F = QuadraticField(-1, 'i')
+ 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)
....: [ 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]
+ [ 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: F = QuadraticField(-1, 'I')
sage: M = random_matrix(F, 3)
sage: Me = ComplexMatrixEuclideanJordanAlgebra.real_embed(M)
sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(Me) == M
field = M.base_ring()
R = PolynomialRing(field, 'z')
z = R.gen()
- F = field.extension(z**2 + 1, 'i', embedding=CLF(-1).sqrt())
+ 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 xrange(n/2):
- for j in xrange(n/2):
+ 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')
sage: Ye = y.natural_representation()
sage: X = ComplexHermitianEJA.real_unembed(Xe)
sage: Y = ComplexHermitianEJA.real_unembed(Ye)
- sage: expected = (X*Y).trace().vector()[0]
+ sage: expected = (X*Y).trace().real()
sage: actual = ComplexHermitianEJA.natural_inner_product(Xe,Ye)
sage: actual == expected
True
return RealMatrixEuclideanJordanAlgebra.natural_inner_product(X,Y)/2
-class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra, KnownRankEJA):
+class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra,
+ ConcreteEuclideanJordanAlgebra):
"""
The rank-n simple EJA consisting of complex Hermitian n-by-n
matrices over the real numbers, the usual symmetric Jordan product,
In theory, our "field" can be any subfield of the reals::
- sage: ComplexHermitianEJA(2, AA)
- Euclidean Jordan algebra of dimension 4 over Algebraic Real Field
+ 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
The dimension of this algebra is `n^2`::
sage: set_random_seed()
- sage: n_max = ComplexHermitianEJA._max_test_case_size()
+ sage: n_max = ComplexHermitianEJA._max_random_instance_size()
sage: n = ZZ.random_element(1, n_max)
sage: J = ComplexHermitianEJA(n)
sage: J.dimension() == n^2
sage: ComplexHermitianEJA(2, prefix='z').gens()
(z0, z1, z2, z3)
- Our natural basis is normalized with respect to the natural inner
- product unless we specify otherwise::
+ We can construct the (trivial) algebra of rank zero::
- sage: set_random_seed()
- sage: J = ComplexHermitianEJA.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
- 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::
-
- sage: set_random_seed()
- sage: x = ComplexHermitianEJA.random_instance().random_element()
- sage: x.operator().matrix().is_symmetric()
- True
+ sage: ComplexHermitianEJA(0)
+ Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
"""
# * The diagonal will (as a result) be real.
#
S = []
- for i in xrange(n):
- for j in xrange(i+1):
+ 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)
return ( s.change_ring(field) for s in S )
- def __init__(self, n, field=QQ, **kwargs):
+ def __init__(self, n, field=AA, **kwargs):
basis = self._denormalized_basis(n,field)
- super(ComplexHermitianEJA,self).__init__(field, basis, n, **kwargs)
+ 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
Embedding is a homomorphism (isomorphism, in fact)::
sage: set_random_seed()
- sage: n_max = QuaternionMatrixEuclideanJordanAlgebra._max_test_case_size()
- sage: n = ZZ.random_element(n_max)
+ sage: n = ZZ.random_element(2)
sage: Q = QuaternionAlgebra(QQ,-1,-1)
sage: X = random_matrix(Q, n)
sage: Y = random_matrix(Q, n)
if M.ncols() != n:
raise ValueError("the matrix 'M' must be square")
- F = QuadraticField(-1, 'i')
+ F = QuadraticField(-1, 'I')
i = F.gen()
blocks = []
# 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1
# quaternion block.
elements = []
- for l in xrange(n/4):
- for m in xrange(n/4):
+ 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].vector()[0] # real part
- z += submat[0,0].vector()[1]*i # imag part
- z += submat[0,1].vector()[0]*j # real part
- z += submat[0,1].vector()[1]*k # imag part
+ 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)
class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra,
- KnownRankEJA):
- """
+ ConcreteEuclideanJordanAlgebra):
+ r"""
The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
matrices, the usual symmetric Jordan product, and the
real-part-of-trace inner product. It has dimension `2n^2 - n` over
In theory, our "field" can be any subfield of the reals::
- sage: QuaternionHermitianEJA(2, AA)
- Euclidean Jordan algebra of dimension 6 over Algebraic Real Field
+ sage: QuaternionHermitianEJA(2, RDF)
+ Euclidean Jordan algebra of dimension 6 over Real Double Field
sage: QuaternionHermitianEJA(2, RR)
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_test_case_size()
+ sage: n_max = QuaternionHermitianEJA._max_random_instance_size()
sage: n = ZZ.random_element(1, n_max)
sage: J = QuaternionHermitianEJA(n)
sage: J.dimension() == 2*(n^2) - n
sage: QuaternionHermitianEJA(2, prefix='a').gens()
(a0, a1, a2, a3, a4, a5)
- Our natural basis is normalized with respect to the natural inner
- product unless we specify otherwise::
-
- sage: set_random_seed()
- sage: J = QuaternionHermitianEJA.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
- 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::
+ We can construct the (trivial) algebra of rank zero::
- sage: set_random_seed()
- sage: x = QuaternionHermitianEJA.random_instance().random_element()
- sage: x.operator().matrix().is_symmetric()
- True
+ sage: QuaternionHermitianEJA(0)
+ Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
"""
@classmethod
# * The diagonal will (as a result) be real.
#
S = []
- for i in xrange(n):
- for j in xrange(i+1):
+ 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)
return ( s.change_ring(field) for s in S )
- def __init__(self, n, field=QQ, **kwargs):
+ def __init__(self, n, field=AA, **kwargs):
basis = self._denormalized_basis(n,field)
- super(QuaternionHermitianEJA,self).__init__(field, basis, n, **kwargs)
+ super(QuaternionHermitianEJA,self).__init__(field,
+ basis,
+ check_axioms=False,
+ **kwargs)
+ self.rank.set_cache(n)
+ @staticmethod
+ def _max_random_instance_size():
+ r"""
+ The maximum rank of a random QuaternionHermitianEJA.
+ """
+ return 2 # Dimension 6
-class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA):
+ @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 HadamardEJA(RationalBasisEuclideanJordanAlgebra,
+ ConcreteEuclideanJordanAlgebra):
"""
- The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
- with the usual inner product and jordan product ``x*y =
- (<x_bar,y_bar>, x0*y_bar + y0*x_bar)``. It has dimension `n` over
- the reals.
+ Return the Euclidean Jordan Algebra corresponding to the set
+ `R^n` under the Hadamard 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.
SETUP::
- sage: from mjo.eja.eja_algebra import JordanSpinEJA
+ sage: from mjo.eja.eja_algebra import HadamardEJA
EXAMPLES:
This multiplication table can be verified by hand::
- sage: J = JordanSpinEJA(4)
- sage: e0,e1,e2,e3 = J.gens()
+ sage: J = HadamardEJA(3)
+ sage: e0,e1,e2 = J.gens()
sage: e0*e0
e0
sage: e0*e1
- e1
- sage: e0*e2
- e2
- sage: e0*e3
- e3
- sage: e1*e2
0
- sage: e1*e3
+ sage: e0*e2
0
- sage: e2*e3
+ sage: e1*e1
+ e1
+ sage: e1*e2
0
+ sage: e2*e2
+ e2
+
+ TESTS:
We can change the generator prefix::
- sage: JordanSpinEJA(2, prefix='B').gens()
- (B0, B1)
+ sage: HadamardEJA(3, prefix='r').gens()
+ (r0, r1, r2)
+
+ """
+ def __init__(self, n, field=AA, **kwargs):
+ V = VectorSpace(field, n)
+ mult_table = [ [ V.gen(i)*(i == j) for j in range(n) ]
+ for i in range(n) ]
+
+ super(HadamardEJA, self).__init__(field,
+ mult_table,
+ check_axioms=False,
+ **kwargs)
+ self.rank.set_cache(n)
+
+ if n == 0:
+ self.one.set_cache( self.zero() )
+ else:
+ self.one.set_cache( sum(self.gens()) )
+
+ @staticmethod
+ def _max_random_instance_size():
+ r"""
+ The maximum dimension of a random HadamardEJA.
+ """
+ return 5
+
+ @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)
+
+
+ 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"""
+ The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
+ with the half-trace inner product and jordan product ``x*y =
+ (<Bx,y>,y_bar>, x0*y_bar + y0*x_bar)`` where `B = 1 \times B22` is
+ a symmetric positive-definite "bilinear form" matrix. Its
+ dimension is the size of `B`, and it has rank two in dimensions
+ larger than two. It reduces to the ``JordanSpinEJA`` when `B` is
+ the identity matrix of order ``n``.
+
+ We insist that the one-by-one upper-left identity block of `B` be
+ passed in as well so that we can be passed a matrix of size zero
+ to construct a trivial algebra.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (BilinearFormEJA,
+ ....: JordanSpinEJA)
+
+ EXAMPLES:
+
+ When no bilinear form is specified, the identity matrix is used,
+ and the resulting algebra is the Jordan spin algebra::
+
+ sage: B = matrix.identity(AA,3)
+ sage: J0 = BilinearFormEJA(B)
+ sage: J1 = JordanSpinEJA(3)
+ sage: J0.multiplication_table() == J0.multiplication_table()
+ True
+
+ An error is raised if the matrix `B` does not correspond to a
+ positive-definite bilinear form::
+
+ sage: B = matrix.random(QQ,2,3)
+ sage: J = BilinearFormEJA(B)
+ Traceback (most recent call last):
+ ...
+ ValueError: bilinear form is not positive-definite
+ sage: B = matrix.zero(QQ,3)
+ sage: J = BilinearFormEJA(B)
+ Traceback (most recent call last):
+ ...
+ ValueError: bilinear form is not positive-definite
+
+ TESTS:
+
+ We can create a zero-dimensional algebra::
+
+ sage: B = matrix.identity(AA,0)
+ sage: J = BilinearFormEJA(B)
+ sage: J.basis()
+ Finite family {}
+
+ 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::
+
+ 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: eis = VectorSpace(M.base_ring(), M.ncols()).basis()
+ sage: V = J.vector_space()
+ sage: sis = [ J.from_vector(V([0] + (M.inverse()*ei).list()))
+ ....: for ei in eis ]
+ sage: actual = [ sis[i]*sis[j]
+ ....: for i in range(n-1)
+ ....: for j in range(n-1) ]
+ sage: expected = [ J.one() if i == j else J.zero()
+ ....: for i in range(n-1)
+ ....: for j in range(n-1) ]
+ sage: actual == expected
+ True
"""
- def __init__(self, n, field=QQ, **kwargs):
+ def __init__(self, B, field=AA, **kwargs):
+ self._B = B
+ n = B.nrows()
+
+ if not B.is_positive_definite():
+ raise ValueError("bilinear form is not positive-definite")
+
V = VectorSpace(field, n)
- mult_table = [[V.zero() for j in xrange(n)] for i in xrange(n)]
- for i in xrange(n):
- for j in xrange(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:]
- # z = x*y
- z0 = x.inner_product(y)
+ z0 = (B*x).inner_product(y)
zbar = y0*xbar + x0*ybar
z = V([z0] + zbar.list())
mult_table[i][j] = z
- # The rank of the spin algebra is two, unless we're in a
- # one-dimensional ambient space (because the rank is bounded by
- # the ambient dimension).
- fdeja = super(JordanSpinEJA, self)
- return fdeja.__init__(field, mult_table, rank=min(n,2), **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).
+ super(BilinearFormEJA, self).__init__(field,
+ mult_table,
+ check_axioms=False,
+ **kwargs)
+ self.rank.set_cache(min(n,2))
+
+ if n == 0:
+ self.one.set_cache( self.zero() )
+ else:
+ self.one.set_cache( self.monomial(0) )
- def inner_product(self, x, y):
+ @staticmethod
+ def _max_random_instance_size():
+ r"""
+ The maximum dimension of a random BilinearFormEJA.
"""
- Faster to reimplement than to use natural representations.
+ return 5
+
+ @classmethod
+ def random_instance(cls, field=AA, **kwargs):
+ """
+ Return a random instance of this algebra.
+ """
+ n = ZZ.random_element(cls._max_random_instance_size() + 1)
+ if n.is_zero():
+ B = matrix.identity(field, n)
+ return cls(B, field, **kwargs)
+
+ B11 = matrix.identity(field,1)
+ M = matrix.random(field, n-1)
+ I = matrix.identity(field, n-1)
+ alpha = field.zero()
+ while alpha.is_zero():
+ alpha = field.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)
+
+ 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 JordanSpinEJA
+ sage: from mjo.eja.eja_algebra import BilinearFormEJA
TESTS:
- Ensure that this is the usual inner product for the algebras
- over `R^n`::
+ 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):
+ """
+ The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
+ with the usual inner product and jordan product ``x*y =
+ (<x,y>, x0*y_bar + y0*x_bar)``. It has dimension `n` over
+ the reals.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import JordanSpinEJA
+
+ EXAMPLES:
+
+ 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
+ 0
+ sage: e1*e3
+ 0
+ sage: e2*e3
+ 0
+
+ We can change the generator prefix::
+
+ sage: JordanSpinEJA(2, prefix='B').gens()
+ (B0, B1)
+
+ TESTS:
+
+ Ensure that we have the usual inner product on `R^n`::
sage: set_random_seed()
sage: J = JordanSpinEJA.random_instance()
sage: x.inner_product(y) == J.natural_inner_product(X,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)
+
+ @staticmethod
+ def _max_random_instance_size():
+ r"""
+ The maximum dimension of a random JordanSpinEJA.
"""
- return x.to_vector().inner_product(y.to_vector())
+ return 5
+ @classmethod
+ def random_instance(cls, field=AA, **kwargs):
+ """
+ Return a random instance of this type of algebra.
-class TrivialEJA(FiniteDimensionalEuclideanJordanAlgebra, KnownRankEJA):
+ Needed here to override the implementation for ``BilinearFormEJA``.
+ """
+ n = ZZ.random_element(cls._max_random_instance_size() + 1)
+ return cls(n, field, **kwargs)
+
+
+class TrivialEJA(FiniteDimensionalEuclideanJordanAlgebra,
+ ConcreteEuclideanJordanAlgebra):
"""
The trivial Euclidean Jordan algebra consisting of only a zero element.
sage: J.one().norm()
0
sage: J.one().subalgebra_generated_by()
- Euclidean Jordan algebra of dimension 0 over Rational Field
+ Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
sage: J.rank()
0
"""
- def __init__(self, field=QQ, **kwargs):
+ def __init__(self, field=AA, **kwargs):
mult_table = []
- fdeja = super(TrivialEJA, self)
+ super(TrivialEJA, self).__init__(field,
+ mult_table,
+ check_axioms=False,
+ **kwargs)
# The rank is zero using my definition, namely the dimension of the
# largest subalgebra generated by any element.
- return fdeja.__init__(field, mult_table, rank=0, **kwargs)
+ self.rank.set_cache(0)
+ self.one.set_cache( self.zero() )
+
+ @classmethod
+ def random_instance(cls, field=AA, **kwargs):
+ # We don't take a "size" argument so the superclass method is
+ # inappropriate for us.
+ return cls(field, **kwargs)
+
+class DirectSumEJA(FiniteDimensionalEuclideanJordanAlgebra):
+ 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.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (random_eja,
+ ....: HadamardEJA,
+ ....: RealSymmetricEJA,
+ ....: DirectSumEJA)
+
+ EXAMPLES::
+
+ sage: J1 = HadamardEJA(2)
+ sage: J2 = RealSymmetricEJA(3)
+ sage: J = DirectSumEJA(J1,J2)
+ sage: J.dimension()
+ 8
+ sage: J.rank()
+ 5
+
+ TESTS:
+
+ The external direct sum construction is only valid when the two factors
+ have the same base ring; an error is raised otherwise::
+
+ sage: set_random_seed()
+ sage: J1 = random_eja(AA)
+ sage: J2 = random_eja(QQ)
+ sage: J = DirectSumEJA(J1,J2)
+ Traceback (most recent call last):
+ ...
+ ValueError: algebras must share the same base field
+
+ """
+ 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):
+ r"""
+ Return the pair of this algebra's factors.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (HadamardEJA,
+ ....: JordanSpinEJA,
+ ....: DirectSumEJA)
+
+ EXAMPLES::
+
+ 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)
+
+ """
+ return self._factors
+
+ def projections(self):
+ r"""
+ Return a pair of projections onto this algebra's factors.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
+ ....: ComplexHermitianEJA,
+ ....: DirectSumEJA)
+
+ EXAMPLES::
+
+ 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: pi_left(J.one()).to_vector()
+ (1, 0)
+ sage: pi_right(J.one()).to_vector()
+ (1, 0, 0, 1)
+
+ """
+ (J1,J2) = self.factors()
+ n = J1.dimension()
+ V_basis = self.vector_space().basis()
+ P1 = matrix(self.base_ring(), V_basis[:n])
+ P2 = matrix(self.base_ring(), V_basis[n:])
+ 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.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (random_eja,
+ ....: JordanSpinEJA,
+ ....: RealSymmetricEJA,
+ ....: DirectSumEJA)
+
+ EXAMPLES::
+
+ sage: J1 = JordanSpinEJA(3)
+ sage: J2 = RealSymmetricEJA(2)
+ sage: J = DirectSumEJA(J1,J2)
+ sage: (iota_left, iota_right) = J.inclusions()
+ sage: iota_left(J1.zero()) == J.zero()
+ True
+ sage: iota_right(J2.zero()) == J.zero()
+ True
+ sage: J1.one().to_vector()
+ (1, 0, 0)
+ sage: iota_left(J1.one()).to_vector()
+ (1, 0, 0, 0, 0, 0)
+ sage: J2.one().to_vector()
+ (1, 0, 1)
+ sage: iota_right(J2.one()).to_vector()
+ (0, 0, 0, 1, 0, 1)
+ sage: J.one().to_vector()
+ (1, 0, 0, 1, 0, 1)
+
+ TESTS:
+
+ 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: pi_left*iota_left == J1.one().operator()
+ True
+ sage: pi_right*iota_right == J2.one().operator()
+ True
+ sage: (pi_left*iota_right).is_zero()
+ True
+ sage: (pi_right*iota_left).is_zero()
+ True
+
+ """
+ (J1,J2) = self.factors()
+ n = J1.dimension()
+ V_basis = self.vector_space().basis()
+ I1 = matrix.column(self.base_ring(), V_basis[:n])
+ I2 = matrix.column(self.base_ring(), V_basis[n:])
+ iota_left = FiniteDimensionalEuclideanJordanAlgebraOperator(J1,self,I1)
+ iota_right = FiniteDimensionalEuclideanJordanAlgebraOperator(J2,self,I2)
+ return (iota_left, iota_right)
+
+ def inner_product(self, x, y):
+ r"""
+ The standard Cartesian inner-product.
+
+ We project ``x`` and ``y`` onto our factors, and add up the
+ inner-products from the subalgebras.
+
+ 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
+
+ """
+ (pi_left, pi_right) = self.projections()
+ x1 = pi_left(x)
+ x2 = pi_right(x)
+ y1 = pi_left(y)
+ y2 = pi_right(y)
+
+ return (x1.inner_product(y1) + x2.inner_product(y2))
+
+
+
+random_eja = ConcreteEuclideanJordanAlgebra.random_instance