2 Euclidean Jordan Algebras. These are formally-real Jordan Algebras;
3 specifically those where u^2 + v^2 = 0 implies that u = v = 0. They
4 are used in optimization, and have some additional nice methods beyond
5 what can be supported in a general Jordan Algebra.
8 from itertools
import repeat
10 from sage
.algebras
.quatalg
.quaternion_algebra
import QuaternionAlgebra
11 from sage
.categories
.magmatic_algebras
import MagmaticAlgebras
12 from sage
.combinat
.free_module
import CombinatorialFreeModule
13 from sage
.matrix
.constructor
import matrix
14 from sage
.matrix
.matrix_space
import MatrixSpace
15 from sage
.misc
.cachefunc
import cached_method
16 from sage
.misc
.prandom
import choice
17 from sage
.misc
.table
import table
18 from sage
.modules
.free_module
import FreeModule
, VectorSpace
19 from sage
.rings
.integer_ring
import ZZ
20 from sage
.rings
.number_field
.number_field
import NumberField
, QuadraticField
21 from sage
.rings
.polynomial
.polynomial_ring_constructor
import PolynomialRing
22 from sage
.rings
.rational_field
import QQ
23 from sage
.rings
.real_lazy
import CLF
, RLF
24 from sage
.structure
.element
import is_Matrix
26 from mjo
.eja
.eja_element
import FiniteDimensionalEuclideanJordanAlgebraElement
27 from mjo
.eja
.eja_utils
import _mat2vec
29 class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule
):
30 # This is an ugly hack needed to prevent the category framework
31 # from implementing a coercion from our base ring (e.g. the
32 # rationals) into the algebra. First of all -- such a coercion is
33 # nonsense to begin with. But more importantly, it tries to do so
34 # in the category of rings, and since our algebras aren't
35 # associative they generally won't be rings.
36 _no_generic_basering_coercion
= True
48 sage: from mjo.eja.eja_algebra import random_eja
52 By definition, Jordan multiplication commutes::
54 sage: set_random_seed()
55 sage: J = random_eja()
56 sage: x,y = J.random_elements(2)
62 self
._natural
_basis
= natural_basis
64 # TODO: HACK for the charpoly.. needs redesign badly.
65 self
._basis
_normalizers
= None
68 category
= MagmaticAlgebras(field
).FiniteDimensional()
69 category
= category
.WithBasis().Unital()
71 fda
= super(FiniteDimensionalEuclideanJordanAlgebra
, self
)
73 range(len(mult_table
)),
76 self
.print_options(bracket
='')
78 # The multiplication table we're given is necessarily in terms
79 # of vectors, because we don't have an algebra yet for
80 # anything to be an element of. However, it's faster in the
81 # long run to have the multiplication table be in terms of
82 # algebra elements. We do this after calling the superclass
83 # constructor so that from_vector() knows what to do.
84 self
._multiplication
_table
= [ map(lambda x
: self
.from_vector(x
), ls
)
85 for ls
in mult_table
]
88 def _element_constructor_(self
, elt
):
90 Construct an element of this algebra from its natural
93 This gets called only after the parent element _call_ method
94 fails to find a coercion for the argument.
98 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
99 ....: RealCartesianProductEJA,
100 ....: RealSymmetricEJA)
104 The identity in `S^n` is converted to the identity in the EJA::
106 sage: J = RealSymmetricEJA(3)
107 sage: I = matrix.identity(QQ,3)
108 sage: J(I) == J.one()
111 This skew-symmetric matrix can't be represented in the EJA::
113 sage: J = RealSymmetricEJA(3)
114 sage: A = matrix(QQ,3, lambda i,j: i-j)
116 Traceback (most recent call last):
118 ArithmeticError: vector is not in free module
122 Ensure that we can convert any element of the two non-matrix
123 simple algebras (whose natural representations are their usual
124 vector representations) back and forth faithfully::
126 sage: set_random_seed()
127 sage: J = RealCartesianProductEJA.random_instance()
128 sage: x = J.random_element()
129 sage: J(x.to_vector().column()) == x
131 sage: J = JordanSpinEJA.random_instance()
132 sage: x = J.random_element()
133 sage: J(x.to_vector().column()) == x
138 # The superclass implementation of random_element()
139 # needs to be able to coerce "0" into the algebra.
142 natural_basis
= self
.natural_basis()
143 basis_space
= natural_basis
[0].matrix_space()
144 if elt
not in basis_space
:
145 raise ValueError("not a naturally-represented algebra element")
147 # Thanks for nothing! Matrix spaces aren't vector spaces in
148 # Sage, so we have to figure out its natural-basis coordinates
149 # ourselves. We use the basis space's ring instead of the
150 # element's ring because the basis space might be an algebraic
151 # closure whereas the base ring of the 3-by-3 identity matrix
152 # could be QQ instead of QQbar.
153 V
= VectorSpace(basis_space
.base_ring(), elt
.nrows()*elt
.ncols())
154 W
= V
.span_of_basis( _mat2vec(s
) for s
in natural_basis
)
155 coords
= W
.coordinate_vector(_mat2vec(elt
))
156 return self
.from_vector(coords
)
160 def _max_test_case_size():
162 Return an integer "size" that is an upper bound on the size of
163 this algebra when it is used in a random test
164 case. Unfortunately, the term "size" is quite vague -- when
165 dealing with `R^n` under either the Hadamard or Jordan spin
166 product, the "size" refers to the dimension `n`. When dealing
167 with a matrix algebra (real symmetric or complex/quaternion
168 Hermitian), it refers to the size of the matrix, which is
169 far less than the dimension of the underlying vector space.
171 We default to five in this class, which is safe in `R^n`. The
172 matrix algebra subclasses (or any class where the "size" is
173 interpreted to be far less than the dimension) should override
174 with a smaller number.
181 Return a string representation of ``self``.
185 sage: from mjo.eja.eja_algebra import JordanSpinEJA
189 Ensure that it says what we think it says::
191 sage: JordanSpinEJA(2, field=QQ)
192 Euclidean Jordan algebra of dimension 2 over Rational Field
193 sage: JordanSpinEJA(3, field=RDF)
194 Euclidean Jordan algebra of dimension 3 over Real Double Field
197 fmt
= "Euclidean Jordan algebra of dimension {} over {}"
198 return fmt
.format(self
.dimension(), self
.base_ring())
200 def product_on_basis(self
, i
, j
):
201 return self
._multiplication
_table
[i
][j
]
203 def _a_regular_element(self
):
205 Guess a regular element. Needed to compute the basis for our
206 characteristic polynomial coefficients.
210 sage: from mjo.eja.eja_algebra import random_eja
214 Ensure that this hacky method succeeds for every algebra that we
215 know how to construct::
217 sage: set_random_seed()
218 sage: J = random_eja()
219 sage: J._a_regular_element().is_regular()
224 z
= self
.sum( (i
+1)*gs
[i
] for i
in range(len(gs
)) )
225 if not z
.is_regular():
226 raise ValueError("don't know a regular element")
231 def _charpoly_basis_space(self
):
233 Return the vector space spanned by the basis used in our
234 characteristic polynomial coefficients. This is used not only to
235 compute those coefficients, but also any time we need to
236 evaluate the coefficients (like when we compute the trace or
239 z
= self
._a
_regular
_element
()
240 # Don't use the parent vector space directly here in case this
241 # happens to be a subalgebra. In that case, we would be e.g.
242 # two-dimensional but span_of_basis() would expect three
244 V
= VectorSpace(self
.base_ring(), self
.vector_space().dimension())
245 basis
= [ (z
**k
).to_vector() for k
in range(self
.rank()) ]
246 V1
= V
.span_of_basis( basis
)
247 b
= (V1
.basis() + V1
.complement().basis())
248 return V
.span_of_basis(b
)
253 def _charpoly_coeff(self
, i
):
255 Return the coefficient polynomial "a_{i}" of this algebra's
256 general characteristic polynomial.
258 Having this be a separate cached method lets us compute and
259 store the trace/determinant (a_{r-1} and a_{0} respectively)
260 separate from the entire characteristic polynomial.
262 if self
._basis
_normalizers
is not None:
263 # Must be a matrix class?
264 # WARNING/TODO: this whole mess is mis-designed.
265 n
= self
.natural_basis_space().nrows()
266 field
= self
.base_ring().base_ring() # yeeeeaaaahhh
267 J
= self
.__class
__(n
, field
, False)
268 (_
,x
,_
,_
) = J
._charpoly
_matrix
_system
()
269 p
= J
._charpoly
_coeff
(i
)
270 # p might be missing some vars, have to substitute "optionally"
271 pairs
= zip(x
.base_ring().gens(), self
._basis
_normalizers
)
272 substitutions
= { v: v*c for (v,c) in pairs }
273 return p
.subs(substitutions
)
275 (A_of_x
, x
, xr
, detA
) = self
._charpoly
_matrix
_system
()
276 R
= A_of_x
.base_ring()
278 # Guaranteed by theory
281 # Danger: the in-place modification is done for performance
282 # reasons (reconstructing a matrix with huge polynomial
283 # entries is slow), but I don't know how cached_method works,
284 # so it's highly possible that we're modifying some global
285 # list variable by reference, here. In other words, you
286 # probably shouldn't call this method twice on the same
287 # algebra, at the same time, in two threads
288 Ai_orig
= A_of_x
.column(i
)
289 A_of_x
.set_column(i
,xr
)
290 numerator
= A_of_x
.det()
291 A_of_x
.set_column(i
,Ai_orig
)
293 # We're relying on the theory here to ensure that each a_i is
294 # indeed back in R, and the added negative signs are to make
295 # the whole charpoly expression sum to zero.
296 return R(-numerator
/detA
)
300 def _charpoly_matrix_system(self
):
302 Compute the matrix whose entries A_ij are polynomials in
303 X1,...,XN, the vector ``x`` of variables X1,...,XN, the vector
304 corresponding to `x^r` and the determinent of the matrix A =
305 [A_ij]. In other words, all of the fixed (cachable) data needed
306 to compute the coefficients of the characteristic polynomial.
311 # Turn my vector space into a module so that "vectors" can
312 # have multivatiate polynomial entries.
313 names
= tuple('X' + str(i
) for i
in range(1,n
+1))
314 R
= PolynomialRing(self
.base_ring(), names
)
316 # Using change_ring() on the parent's vector space doesn't work
317 # here because, in a subalgebra, that vector space has a basis
318 # and change_ring() tries to bring the basis along with it. And
319 # that doesn't work unless the new ring is a PID, which it usually
323 # Now let x = (X1,X2,...,Xn) be the vector whose entries are
327 # And figure out the "left multiplication by x" matrix in
330 monomial_matrices
= [ self
.monomial(i
).operator().matrix()
331 for i
in range(n
) ] # don't recompute these!
333 ek
= self
.monomial(k
).to_vector()
335 sum( x
[i
]*(monomial_matrices
[i
]*ek
)
336 for i
in range(n
) ) )
337 Lx
= matrix
.column(R
, lmbx_cols
)
339 # Now we can compute powers of x "symbolically"
340 x_powers
= [self
.one().to_vector(), x
]
341 for d
in range(2, r
+1):
342 x_powers
.append( Lx
*(x_powers
[-1]) )
344 idmat
= matrix
.identity(R
, n
)
346 W
= self
._charpoly
_basis
_space
()
347 W
= W
.change_ring(R
.fraction_field())
349 # Starting with the standard coordinates x = (X1,X2,...,Xn)
350 # and then converting the entries to W-coordinates allows us
351 # to pass in the standard coordinates to the charpoly and get
352 # back the right answer. Specifically, with x = (X1,X2,...,Xn),
355 # W.coordinates(x^2) eval'd at (standard z-coords)
359 # W-coords of (standard coords of x^2 eval'd at std-coords of z)
361 # We want the middle equivalent thing in our matrix, but use
362 # the first equivalent thing instead so that we can pass in
363 # standard coordinates.
364 x_powers
= [ W
.coordinate_vector(xp
) for xp
in x_powers
]
365 l2
= [idmat
.column(k
-1) for k
in range(r
+1, n
+1)]
366 A_of_x
= matrix
.column(R
, n
, (x_powers
[:r
] + l2
))
367 return (A_of_x
, x
, x_powers
[r
], A_of_x
.det())
371 def characteristic_polynomial(self
):
373 Return a characteristic polynomial that works for all elements
376 The resulting polynomial has `n+1` variables, where `n` is the
377 dimension of this algebra. The first `n` variables correspond to
378 the coordinates of an algebra element: when evaluated at the
379 coordinates of an algebra element with respect to a certain
380 basis, the result is a univariate polynomial (in the one
381 remaining variable ``t``), namely the characteristic polynomial
386 sage: from mjo.eja.eja_algebra import JordanSpinEJA
390 The characteristic polynomial in the spin algebra is given in
391 Alizadeh, Example 11.11::
393 sage: J = JordanSpinEJA(3)
394 sage: p = J.characteristic_polynomial(); p
395 X1^2 - X2^2 - X3^2 + (-2*t)*X1 + t^2
396 sage: xvec = J.one().to_vector()
404 # The list of coefficient polynomials a_1, a_2, ..., a_n.
405 a
= [ self
._charpoly
_coeff
(i
) for i
in range(n
) ]
407 # We go to a bit of trouble here to reorder the
408 # indeterminates, so that it's easier to evaluate the
409 # characteristic polynomial at x's coordinates and get back
410 # something in terms of t, which is what we want.
412 S
= PolynomialRing(self
.base_ring(),'t')
414 S
= PolynomialRing(S
, R
.variable_names())
417 # Note: all entries past the rth should be zero. The
418 # coefficient of the highest power (x^r) is 1, but it doesn't
419 # appear in the solution vector which contains coefficients
420 # for the other powers (to make them sum to x^r).
422 a
[r
] = 1 # corresponds to x^r
424 # When the rank is equal to the dimension, trying to
425 # assign a[r] goes out-of-bounds.
426 a
.append(1) # corresponds to x^r
428 return sum( a
[k
]*(t
**k
) for k
in range(len(a
)) )
431 def inner_product(self
, x
, y
):
433 The inner product associated with this Euclidean Jordan algebra.
435 Defaults to the trace inner product, but can be overridden by
436 subclasses if they are sure that the necessary properties are
441 sage: from mjo.eja.eja_algebra import random_eja
445 Our inner product satisfies the Jordan axiom, which is also
446 referred to as "associativity" for a symmetric bilinear form::
448 sage: set_random_seed()
449 sage: J = random_eja()
450 sage: x,y,z = J.random_elements(3)
451 sage: (x*y).inner_product(z) == y.inner_product(x*z)
455 X
= x
.natural_representation()
456 Y
= y
.natural_representation()
457 return self
.natural_inner_product(X
,Y
)
460 def is_trivial(self
):
462 Return whether or not this algebra is trivial.
464 A trivial algebra contains only the zero element.
468 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
472 sage: J = ComplexHermitianEJA(3)
475 sage: A = J.zero().subalgebra_generated_by()
480 return self
.dimension() == 0
483 def multiplication_table(self
):
485 Return a visual representation of this algebra's multiplication
486 table (on basis elements).
490 sage: from mjo.eja.eja_algebra import JordanSpinEJA
494 sage: J = JordanSpinEJA(4)
495 sage: J.multiplication_table()
496 +----++----+----+----+----+
497 | * || e0 | e1 | e2 | e3 |
498 +====++====+====+====+====+
499 | e0 || e0 | e1 | e2 | e3 |
500 +----++----+----+----+----+
501 | e1 || e1 | e0 | 0 | 0 |
502 +----++----+----+----+----+
503 | e2 || e2 | 0 | e0 | 0 |
504 +----++----+----+----+----+
505 | e3 || e3 | 0 | 0 | e0 |
506 +----++----+----+----+----+
509 M
= list(self
._multiplication
_table
) # copy
510 for i
in range(len(M
)):
511 # M had better be "square"
512 M
[i
] = [self
.monomial(i
)] + M
[i
]
513 M
= [["*"] + list(self
.gens())] + M
514 return table(M
, header_row
=True, header_column
=True, frame
=True)
517 def natural_basis(self
):
519 Return a more-natural representation of this algebra's basis.
521 Every finite-dimensional Euclidean Jordan Algebra is a direct
522 sum of five simple algebras, four of which comprise Hermitian
523 matrices. This method returns the original "natural" basis
524 for our underlying vector space. (Typically, the natural basis
525 is used to construct the multiplication table in the first place.)
527 Note that this will always return a matrix. The standard basis
528 in `R^n` will be returned as `n`-by-`1` column matrices.
532 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
533 ....: RealSymmetricEJA)
537 sage: J = RealSymmetricEJA(2)
539 Finite family {0: e0, 1: e1, 2: e2}
540 sage: J.natural_basis()
542 [1 0] [ 0 1/2*sqrt2] [0 0]
543 [0 0], [1/2*sqrt2 0], [0 1]
548 sage: J = JordanSpinEJA(2)
550 Finite family {0: e0, 1: e1}
551 sage: J.natural_basis()
558 if self
._natural
_basis
is None:
559 M
= self
.natural_basis_space()
560 return tuple( M(b
.to_vector()) for b
in self
.basis() )
562 return self
._natural
_basis
565 def natural_basis_space(self
):
567 Return the matrix space in which this algebra's natural basis
570 if self
._natural
_basis
is None or len(self
._natural
_basis
) == 0:
571 return MatrixSpace(self
.base_ring(), self
.dimension(), 1)
573 return self
._natural
_basis
[0].matrix_space()
577 def natural_inner_product(X
,Y
):
579 Compute the inner product of two naturally-represented elements.
581 For example in the real symmetric matrix EJA, this will compute
582 the trace inner-product of two n-by-n symmetric matrices. The
583 default should work for the real cartesian product EJA, the
584 Jordan spin EJA, and the real symmetric matrices. The others
585 will have to be overridden.
587 return (X
.conjugate_transpose()*Y
).trace()
593 Return the unit element of this algebra.
597 sage: from mjo.eja.eja_algebra import (RealCartesianProductEJA,
602 sage: J = RealCartesianProductEJA(5)
604 e0 + e1 + e2 + e3 + e4
608 The identity element acts like the identity::
610 sage: set_random_seed()
611 sage: J = random_eja()
612 sage: x = J.random_element()
613 sage: J.one()*x == x and x*J.one() == x
616 The matrix of the unit element's operator is the identity::
618 sage: set_random_seed()
619 sage: J = random_eja()
620 sage: actual = J.one().operator().matrix()
621 sage: expected = matrix.identity(J.base_ring(), J.dimension())
622 sage: actual == expected
626 # We can brute-force compute the matrices of the operators
627 # that correspond to the basis elements of this algebra.
628 # If some linear combination of those basis elements is the
629 # algebra identity, then the same linear combination of
630 # their matrices has to be the identity matrix.
632 # Of course, matrices aren't vectors in sage, so we have to
633 # appeal to the "long vectors" isometry.
634 oper_vecs
= [ _mat2vec(g
.operator().matrix()) for g
in self
.gens() ]
636 # Now we use basis linear algebra to find the coefficients,
637 # of the matrices-as-vectors-linear-combination, which should
638 # work for the original algebra basis too.
639 A
= matrix
.column(self
.base_ring(), oper_vecs
)
641 # We used the isometry on the left-hand side already, but we
642 # still need to do it for the right-hand side. Recall that we
643 # wanted something that summed to the identity matrix.
644 b
= _mat2vec( matrix
.identity(self
.base_ring(), self
.dimension()) )
646 # Now if there's an identity element in the algebra, this should work.
647 coeffs
= A
.solve_right(b
)
648 return self
.linear_combination(zip(self
.gens(), coeffs
))
651 def random_element(self
):
652 # Temporary workaround for https://trac.sagemath.org/ticket/28327
653 if self
.is_trivial():
656 s
= super(FiniteDimensionalEuclideanJordanAlgebra
, self
)
657 return s
.random_element()
659 def random_elements(self
, count
):
661 Return ``count`` random elements as a tuple.
665 sage: from mjo.eja.eja_algebra import JordanSpinEJA
669 sage: J = JordanSpinEJA(3)
670 sage: x,y,z = J.random_elements(3)
671 sage: all( [ x in J, y in J, z in J ])
673 sage: len( J.random_elements(10) ) == 10
677 return tuple( self
.random_element() for idx
in xrange(count
) )
680 def random_instance(cls
, field
=QQ
, **kwargs
):
682 Return a random instance of this type of algebra.
684 In subclasses for algebras that we know how to construct, this
685 is a shortcut for constructing test cases and examples.
687 if cls
is FiniteDimensionalEuclideanJordanAlgebra
:
688 # Red flag! But in theory we could do this I guess. The
689 # only finite-dimensional exceptional EJA is the
690 # octononions. So, we could just create an EJA from an
691 # associative matrix algebra (generated by a subset of
692 # elements) with the symmetric product. Or, we could punt
693 # to random_eja() here, override it in our subclasses, and
694 # not worry about it.
695 raise NotImplementedError
697 n
= ZZ
.random_element(1, cls
._max
_test
_case
_size
())
698 return cls(n
, field
, **kwargs
)
703 Return the rank of this EJA.
707 The author knows of no algorithm to compute the rank of an EJA
708 where only the multiplication table is known. In lieu of one, we
709 require the rank to be specified when the algebra is created,
710 and simply pass along that number here.
714 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
715 ....: RealSymmetricEJA,
716 ....: ComplexHermitianEJA,
717 ....: QuaternionHermitianEJA,
722 The rank of the Jordan spin algebra is always two::
724 sage: JordanSpinEJA(2).rank()
726 sage: JordanSpinEJA(3).rank()
728 sage: JordanSpinEJA(4).rank()
731 The rank of the `n`-by-`n` Hermitian real, complex, or
732 quaternion matrices is `n`::
734 sage: RealSymmetricEJA(4).rank()
736 sage: ComplexHermitianEJA(3).rank()
738 sage: QuaternionHermitianEJA(2).rank()
743 Ensure that every EJA that we know how to construct has a
744 positive integer rank::
746 sage: set_random_seed()
747 sage: r = random_eja().rank()
748 sage: r in ZZ and r > 0
755 def vector_space(self
):
757 Return the vector space that underlies this algebra.
761 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
765 sage: J = RealSymmetricEJA(2)
766 sage: J.vector_space()
767 Vector space of dimension 3 over...
770 return self
.zero().to_vector().parent().ambient_vector_space()
773 Element
= FiniteDimensionalEuclideanJordanAlgebraElement
776 class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra
):
778 Return the Euclidean Jordan Algebra corresponding to the set
779 `R^n` under the Hadamard product.
781 Note: this is nothing more than the Cartesian product of ``n``
782 copies of the spin algebra. Once Cartesian product algebras
783 are implemented, this can go.
787 sage: from mjo.eja.eja_algebra import RealCartesianProductEJA
791 This multiplication table can be verified by hand::
793 sage: J = RealCartesianProductEJA(3)
794 sage: e0,e1,e2 = J.gens()
810 We can change the generator prefix::
812 sage: RealCartesianProductEJA(3, prefix='r').gens()
816 def __init__(self
, n
, field
=QQ
, **kwargs
):
817 V
= VectorSpace(field
, n
)
818 mult_table
= [ [ V
.gen(i
)*(i
== j
) for j
in range(n
) ]
821 fdeja
= super(RealCartesianProductEJA
, self
)
822 return fdeja
.__init
__(field
, mult_table
, rank
=n
, **kwargs
)
824 def inner_product(self
, x
, y
):
826 Faster to reimplement than to use natural representations.
830 sage: from mjo.eja.eja_algebra import RealCartesianProductEJA
834 Ensure that this is the usual inner product for the algebras
837 sage: set_random_seed()
838 sage: J = RealCartesianProductEJA.random_instance()
839 sage: x,y = J.random_elements(2)
840 sage: X = x.natural_representation()
841 sage: Y = y.natural_representation()
842 sage: x.inner_product(y) == J.natural_inner_product(X,Y)
846 return x
.to_vector().inner_product(y
.to_vector())
851 Return a "random" finite-dimensional Euclidean Jordan Algebra.
855 For now, we choose a random natural number ``n`` (greater than zero)
856 and then give you back one of the following:
858 * The cartesian product of the rational numbers ``n`` times; this is
859 ``QQ^n`` with the Hadamard product.
861 * The Jordan spin algebra on ``QQ^n``.
863 * The ``n``-by-``n`` rational symmetric matrices with the symmetric
866 * The ``n``-by-``n`` complex-rational Hermitian matrices embedded
867 in the space of ``2n``-by-``2n`` real symmetric matrices.
869 * The ``n``-by-``n`` quaternion-rational Hermitian matrices embedded
870 in the space of ``4n``-by-``4n`` real symmetric matrices.
872 Later this might be extended to return Cartesian products of the
877 sage: from mjo.eja.eja_algebra import random_eja
882 Euclidean Jordan algebra of dimension...
885 classname
= choice([RealCartesianProductEJA
,
889 QuaternionHermitianEJA
])
890 return classname
.random_instance()
897 class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra
):
899 def _max_test_case_size():
900 # Play it safe, since this will be squared and the underlying
901 # field can have dimension 4 (quaternions) too.
905 def _denormalized_basis(cls
, n
, field
):
906 raise NotImplementedError
908 def __init__(self
, n
, field
=QQ
, normalize_basis
=True, **kwargs
):
909 S
= self
._denormalized
_basis
(n
, field
)
911 if n
> 1 and normalize_basis
:
912 # We'll need sqrt(2) to normalize the basis, and this
913 # winds up in the multiplication table, so the whole
914 # algebra needs to be over the field extension.
915 R
= PolynomialRing(field
, 'z')
918 if p
.is_irreducible():
919 field
= NumberField(p
, 'sqrt2', embedding
=RLF(2).sqrt())
920 S
= [ s
.change_ring(field
) for s
in S
]
921 self
._basis
_normalizers
= tuple(
922 ~
(self
.natural_inner_product(s
,s
).sqrt()) for s
in S
)
923 S
= tuple( s
*c
for (s
,c
) in zip(S
,self
._basis
_normalizers
) )
925 Qs
= self
.multiplication_table_from_matrix_basis(S
)
927 fdeja
= super(MatrixEuclideanJordanAlgebra
, self
)
928 return fdeja
.__init
__(field
,
936 def multiplication_table_from_matrix_basis(basis
):
938 At least three of the five simple Euclidean Jordan algebras have the
939 symmetric multiplication (A,B) |-> (AB + BA)/2, where the
940 multiplication on the right is matrix multiplication. Given a basis
941 for the underlying matrix space, this function returns a
942 multiplication table (obtained by looping through the basis
943 elements) for an algebra of those matrices.
945 # In S^2, for example, we nominally have four coordinates even
946 # though the space is of dimension three only. The vector space V
947 # is supposed to hold the entire long vector, and the subspace W
948 # of V will be spanned by the vectors that arise from symmetric
949 # matrices. Thus for S^2, dim(V) == 4 and dim(W) == 3.
950 field
= basis
[0].base_ring()
951 dimension
= basis
[0].nrows()
953 V
= VectorSpace(field
, dimension
**2)
954 W
= V
.span_of_basis( _mat2vec(s
) for s
in basis
)
956 mult_table
= [[W
.zero() for j
in range(n
)] for i
in range(n
)]
959 mat_entry
= (basis
[i
]*basis
[j
] + basis
[j
]*basis
[i
])/2
960 mult_table
[i
][j
] = W
.coordinate_vector(_mat2vec(mat_entry
))
968 Embed the matrix ``M`` into a space of real matrices.
970 The matrix ``M`` can have entries in any field at the moment:
971 the real numbers, complex numbers, or quaternions. And although
972 they are not a field, we can probably support octonions at some
973 point, too. This function returns a real matrix that "acts like"
974 the original with respect to matrix multiplication; i.e.
976 real_embed(M*N) = real_embed(M)*real_embed(N)
979 raise NotImplementedError
985 The inverse of :meth:`real_embed`.
987 raise NotImplementedError
991 def natural_inner_product(cls
,X
,Y
):
992 Xu
= cls
.real_unembed(X
)
993 Yu
= cls
.real_unembed(Y
)
999 # Otherwise, try the thing that works for complex numbers; and
1000 # if that doesn't work, the thing that works for quaternions.
1002 return tr
.vector()[0] # real part, imag part is index 1
1003 except AttributeError:
1004 # A quaternions doesn't have a vector() method, but does
1005 # have coefficient_tuple() method that returns the
1006 # coefficients of 1, i, j, and k -- in that order.
1007 return tr
.coefficient_tuple()[0]
1010 class RealSymmetricEJA(MatrixEuclideanJordanAlgebra
):
1012 The rank-n simple EJA consisting of real symmetric n-by-n
1013 matrices, the usual symmetric Jordan product, and the trace inner
1014 product. It has dimension `(n^2 + n)/2` over the reals.
1018 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
1022 sage: J = RealSymmetricEJA(2)
1023 sage: e0, e1, e2 = J.gens()
1033 The dimension of this algebra is `(n^2 + n) / 2`::
1035 sage: set_random_seed()
1036 sage: n_max = RealSymmetricEJA._max_test_case_size()
1037 sage: n = ZZ.random_element(1, n_max)
1038 sage: J = RealSymmetricEJA(n)
1039 sage: J.dimension() == (n^2 + n)/2
1042 The Jordan multiplication is what we think it is::
1044 sage: set_random_seed()
1045 sage: J = RealSymmetricEJA.random_instance()
1046 sage: x,y = J.random_elements(2)
1047 sage: actual = (x*y).natural_representation()
1048 sage: X = x.natural_representation()
1049 sage: Y = y.natural_representation()
1050 sage: expected = (X*Y + Y*X)/2
1051 sage: actual == expected
1053 sage: J(expected) == x*y
1056 We can change the generator prefix::
1058 sage: RealSymmetricEJA(3, prefix='q').gens()
1059 (q0, q1, q2, q3, q4, q5)
1061 Our natural basis is normalized with respect to the natural inner
1062 product unless we specify otherwise::
1064 sage: set_random_seed()
1065 sage: J = RealSymmetricEJA.random_instance()
1066 sage: all( b.norm() == 1 for b in J.gens() )
1069 Since our natural basis is normalized with respect to the natural
1070 inner product, and since we know that this algebra is an EJA, any
1071 left-multiplication operator's matrix will be symmetric because
1072 natural->EJA basis representation is an isometry and within the EJA
1073 the operator is self-adjoint by the Jordan axiom::
1075 sage: set_random_seed()
1076 sage: x = RealSymmetricEJA.random_instance().random_element()
1077 sage: x.operator().matrix().is_symmetric()
1082 def _denormalized_basis(cls
, n
, field
):
1084 Return a basis for the space of real symmetric n-by-n matrices.
1088 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
1092 sage: set_random_seed()
1093 sage: n = ZZ.random_element(1,5)
1094 sage: B = RealSymmetricEJA._denormalized_basis(n,QQ)
1095 sage: all( M.is_symmetric() for M in B)
1099 # The basis of symmetric matrices, as matrices, in their R^(n-by-n)
1103 for j
in xrange(i
+1):
1104 Eij
= matrix(field
, n
, lambda k
,l
: k
==i
and l
==j
)
1108 Sij
= Eij
+ Eij
.transpose()
1114 def _max_test_case_size():
1115 return 5 # Dimension 10
1120 Embed the matrix ``M`` into a space of real matrices.
1122 The matrix ``M`` can have entries in any field at the moment:
1123 the real numbers, complex numbers, or quaternions. And although
1124 they are not a field, we can probably support octonions at some
1125 point, too. This function returns a real matrix that "acts like"
1126 the original with respect to matrix multiplication; i.e.
1128 real_embed(M*N) = real_embed(M)*real_embed(N)
1135 def real_unembed(M
):
1137 The inverse of :meth:`real_embed`.
1143 class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra
):
1147 Embed the n-by-n complex matrix ``M`` into the space of real
1148 matrices of size 2n-by-2n via the map the sends each entry `z = a +
1149 bi` to the block matrix ``[[a,b],[-b,a]]``.
1153 sage: from mjo.eja.eja_algebra import \
1154 ....: ComplexMatrixEuclideanJordanAlgebra
1158 sage: F = QuadraticField(-1, 'i')
1159 sage: x1 = F(4 - 2*i)
1160 sage: x2 = F(1 + 2*i)
1163 sage: M = matrix(F,2,[[x1,x2],[x3,x4]])
1164 sage: ComplexMatrixEuclideanJordanAlgebra.real_embed(M)
1173 Embedding is a homomorphism (isomorphism, in fact)::
1175 sage: set_random_seed()
1176 sage: n_max = ComplexMatrixEuclideanJordanAlgebra._max_test_case_size()
1177 sage: n = ZZ.random_element(n_max)
1178 sage: F = QuadraticField(-1, 'i')
1179 sage: X = random_matrix(F, n)
1180 sage: Y = random_matrix(F, n)
1181 sage: Xe = ComplexMatrixEuclideanJordanAlgebra.real_embed(X)
1182 sage: Ye = ComplexMatrixEuclideanJordanAlgebra.real_embed(Y)
1183 sage: XYe = ComplexMatrixEuclideanJordanAlgebra.real_embed(X*Y)
1190 raise ValueError("the matrix 'M' must be square")
1191 field
= M
.base_ring()
1194 a
= z
.vector()[0] # real part, I guess
1195 b
= z
.vector()[1] # imag part, I guess
1196 blocks
.append(matrix(field
, 2, [[a
,b
],[-b
,a
]]))
1198 # We can drop the imaginaries here.
1199 return matrix
.block(field
.base_ring(), n
, blocks
)
1203 def real_unembed(M
):
1205 The inverse of _embed_complex_matrix().
1209 sage: from mjo.eja.eja_algebra import \
1210 ....: ComplexMatrixEuclideanJordanAlgebra
1214 sage: A = matrix(QQ,[ [ 1, 2, 3, 4],
1215 ....: [-2, 1, -4, 3],
1216 ....: [ 9, 10, 11, 12],
1217 ....: [-10, 9, -12, 11] ])
1218 sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(A)
1220 [ 10*i + 9 12*i + 11]
1224 Unembedding is the inverse of embedding::
1226 sage: set_random_seed()
1227 sage: F = QuadraticField(-1, 'i')
1228 sage: M = random_matrix(F, 3)
1229 sage: Me = ComplexMatrixEuclideanJordanAlgebra.real_embed(M)
1230 sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(Me) == M
1236 raise ValueError("the matrix 'M' must be square")
1237 if not n
.mod(2).is_zero():
1238 raise ValueError("the matrix 'M' must be a complex embedding")
1240 field
= M
.base_ring() # This should already have sqrt2
1241 R
= PolynomialRing(field
, 'z')
1243 F
= NumberField(z
**2 + 1,'i', embedding
=CLF(-1).sqrt())
1246 # Go top-left to bottom-right (reading order), converting every
1247 # 2-by-2 block we see to a single complex element.
1249 for k
in xrange(n
/2):
1250 for j
in xrange(n
/2):
1251 submat
= M
[2*k
:2*k
+2,2*j
:2*j
+2]
1252 if submat
[0,0] != submat
[1,1]:
1253 raise ValueError('bad on-diagonal submatrix')
1254 if submat
[0,1] != -submat
[1,0]:
1255 raise ValueError('bad off-diagonal submatrix')
1256 z
= submat
[0,0] + submat
[0,1]*i
1259 return matrix(F
, n
/2, elements
)
1262 class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra
):
1264 The rank-n simple EJA consisting of complex Hermitian n-by-n
1265 matrices over the real numbers, the usual symmetric Jordan product,
1266 and the real-part-of-trace inner product. It has dimension `n^2` over
1271 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
1275 The dimension of this algebra is `n^2`::
1277 sage: set_random_seed()
1278 sage: n_max = ComplexHermitianEJA._max_test_case_size()
1279 sage: n = ZZ.random_element(1, n_max)
1280 sage: J = ComplexHermitianEJA(n)
1281 sage: J.dimension() == n^2
1284 The Jordan multiplication is what we think it is::
1286 sage: set_random_seed()
1287 sage: J = ComplexHermitianEJA.random_instance()
1288 sage: x,y = J.random_elements(2)
1289 sage: actual = (x*y).natural_representation()
1290 sage: X = x.natural_representation()
1291 sage: Y = y.natural_representation()
1292 sage: expected = (X*Y + Y*X)/2
1293 sage: actual == expected
1295 sage: J(expected) == x*y
1298 We can change the generator prefix::
1300 sage: ComplexHermitianEJA(2, prefix='z').gens()
1303 Our natural basis is normalized with respect to the natural inner
1304 product unless we specify otherwise::
1306 sage: set_random_seed()
1307 sage: J = ComplexHermitianEJA.random_instance()
1308 sage: all( b.norm() == 1 for b in J.gens() )
1311 Since our natural basis is normalized with respect to the natural
1312 inner product, and since we know that this algebra is an EJA, any
1313 left-multiplication operator's matrix will be symmetric because
1314 natural->EJA basis representation is an isometry and within the EJA
1315 the operator is self-adjoint by the Jordan axiom::
1317 sage: set_random_seed()
1318 sage: x = ComplexHermitianEJA.random_instance().random_element()
1319 sage: x.operator().matrix().is_symmetric()
1324 def _denormalized_basis(cls
, n
, field
):
1326 Returns a basis for the space of complex Hermitian n-by-n matrices.
1328 Why do we embed these? Basically, because all of numerical linear
1329 algebra assumes that you're working with vectors consisting of `n`
1330 entries from a field and scalars from the same field. There's no way
1331 to tell SageMath that (for example) the vectors contain complex
1332 numbers, while the scalar field is real.
1336 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
1340 sage: set_random_seed()
1341 sage: n = ZZ.random_element(1,5)
1342 sage: field = QuadraticField(2, 'sqrt2')
1343 sage: B = ComplexHermitianEJA._denormalized_basis(n, field)
1344 sage: all( M.is_symmetric() for M in B)
1348 R
= PolynomialRing(field
, 'z')
1350 F
= NumberField(z
**2 + 1, 'I', embedding
=CLF(-1).sqrt())
1353 # This is like the symmetric case, but we need to be careful:
1355 # * We want conjugate-symmetry, not just symmetry.
1356 # * The diagonal will (as a result) be real.
1360 for j
in xrange(i
+1):
1361 Eij
= matrix(F
, n
, lambda k
,l
: k
==i
and l
==j
)
1363 Sij
= cls
.real_embed(Eij
)
1366 # The second one has a minus because it's conjugated.
1367 Sij_real
= cls
.real_embed(Eij
+ Eij
.transpose())
1369 Sij_imag
= cls
.real_embed(I
*Eij
- I
*Eij
.transpose())
1372 # Since we embedded these, we can drop back to the "field" that we
1373 # started with instead of the complex extension "F".
1374 return tuple( s
.change_ring(field
) for s
in S
)
1378 class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra
):
1382 Embed the n-by-n quaternion matrix ``M`` into the space of real
1383 matrices of size 4n-by-4n by first sending each quaternion entry `z
1384 = a + bi + cj + dk` to the block-complex matrix ``[[a + bi,
1385 c+di],[-c + di, a-bi]]`, and then embedding those into a real
1390 sage: from mjo.eja.eja_algebra import \
1391 ....: QuaternionMatrixEuclideanJordanAlgebra
1395 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1396 sage: i,j,k = Q.gens()
1397 sage: x = 1 + 2*i + 3*j + 4*k
1398 sage: M = matrix(Q, 1, [[x]])
1399 sage: QuaternionMatrixEuclideanJordanAlgebra.real_embed(M)
1405 Embedding is a homomorphism (isomorphism, in fact)::
1407 sage: set_random_seed()
1408 sage: n_max = QuaternionMatrixEuclideanJordanAlgebra._max_test_case_size()
1409 sage: n = ZZ.random_element(n_max)
1410 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1411 sage: X = random_matrix(Q, n)
1412 sage: Y = random_matrix(Q, n)
1413 sage: Xe = QuaternionMatrixEuclideanJordanAlgebra.real_embed(X)
1414 sage: Ye = QuaternionMatrixEuclideanJordanAlgebra.real_embed(Y)
1415 sage: XYe = QuaternionMatrixEuclideanJordanAlgebra.real_embed(X*Y)
1420 quaternions
= M
.base_ring()
1423 raise ValueError("the matrix 'M' must be square")
1425 F
= QuadraticField(-1, 'i')
1430 t
= z
.coefficient_tuple()
1435 cplxM
= matrix(F
, 2, [[ a
+ b
*i
, c
+ d
*i
],
1436 [-c
+ d
*i
, a
- b
*i
]])
1437 realM
= ComplexMatrixEuclideanJordanAlgebra
.real_embed(cplxM
)
1438 blocks
.append(realM
)
1440 # We should have real entries by now, so use the realest field
1441 # we've got for the return value.
1442 return matrix
.block(quaternions
.base_ring(), n
, blocks
)
1447 def real_unembed(M
):
1449 The inverse of _embed_quaternion_matrix().
1453 sage: from mjo.eja.eja_algebra import \
1454 ....: QuaternionMatrixEuclideanJordanAlgebra
1458 sage: M = matrix(QQ, [[ 1, 2, 3, 4],
1459 ....: [-2, 1, -4, 3],
1460 ....: [-3, 4, 1, -2],
1461 ....: [-4, -3, 2, 1]])
1462 sage: QuaternionMatrixEuclideanJordanAlgebra.real_unembed(M)
1463 [1 + 2*i + 3*j + 4*k]
1467 Unembedding is the inverse of embedding::
1469 sage: set_random_seed()
1470 sage: Q = QuaternionAlgebra(QQ, -1, -1)
1471 sage: M = random_matrix(Q, 3)
1472 sage: Me = QuaternionMatrixEuclideanJordanAlgebra.real_embed(M)
1473 sage: QuaternionMatrixEuclideanJordanAlgebra.real_unembed(Me) == M
1479 raise ValueError("the matrix 'M' must be square")
1480 if not n
.mod(4).is_zero():
1481 raise ValueError("the matrix 'M' must be a complex embedding")
1483 # Use the base ring of the matrix to ensure that its entries can be
1484 # multiplied by elements of the quaternion algebra.
1485 field
= M
.base_ring()
1486 Q
= QuaternionAlgebra(field
,-1,-1)
1489 # Go top-left to bottom-right (reading order), converting every
1490 # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1
1493 for l
in xrange(n
/4):
1494 for m
in xrange(n
/4):
1495 submat
= ComplexMatrixEuclideanJordanAlgebra
.real_unembed(
1496 M
[4*l
:4*l
+4,4*m
:4*m
+4] )
1497 if submat
[0,0] != submat
[1,1].conjugate():
1498 raise ValueError('bad on-diagonal submatrix')
1499 if submat
[0,1] != -submat
[1,0].conjugate():
1500 raise ValueError('bad off-diagonal submatrix')
1501 z
= submat
[0,0].vector()[0] # real part
1502 z
+= submat
[0,0].vector()[1]*i
# imag part
1503 z
+= submat
[0,1].vector()[0]*j
# real part
1504 z
+= submat
[0,1].vector()[1]*k
# imag part
1507 return matrix(Q
, n
/4, elements
)
1511 class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra
):
1513 The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
1514 matrices, the usual symmetric Jordan product, and the
1515 real-part-of-trace inner product. It has dimension `2n^2 - n` over
1520 sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
1524 The dimension of this algebra is `2*n^2 - n`::
1526 sage: set_random_seed()
1527 sage: n_max = QuaternionHermitianEJA._max_test_case_size()
1528 sage: n = ZZ.random_element(1, n_max)
1529 sage: J = QuaternionHermitianEJA(n)
1530 sage: J.dimension() == 2*(n^2) - n
1533 The Jordan multiplication is what we think it is::
1535 sage: set_random_seed()
1536 sage: J = QuaternionHermitianEJA.random_instance()
1537 sage: x,y = J.random_elements(2)
1538 sage: actual = (x*y).natural_representation()
1539 sage: X = x.natural_representation()
1540 sage: Y = y.natural_representation()
1541 sage: expected = (X*Y + Y*X)/2
1542 sage: actual == expected
1544 sage: J(expected) == x*y
1547 We can change the generator prefix::
1549 sage: QuaternionHermitianEJA(2, prefix='a').gens()
1550 (a0, a1, a2, a3, a4, a5)
1552 Our natural basis is normalized with respect to the natural inner
1553 product unless we specify otherwise::
1555 sage: set_random_seed()
1556 sage: J = QuaternionHermitianEJA.random_instance()
1557 sage: all( b.norm() == 1 for b in J.gens() )
1560 Since our natural basis is normalized with respect to the natural
1561 inner product, and since we know that this algebra is an EJA, any
1562 left-multiplication operator's matrix will be symmetric because
1563 natural->EJA basis representation is an isometry and within the EJA
1564 the operator is self-adjoint by the Jordan axiom::
1566 sage: set_random_seed()
1567 sage: x = QuaternionHermitianEJA.random_instance().random_element()
1568 sage: x.operator().matrix().is_symmetric()
1573 def _denormalized_basis(cls
, n
, field
):
1575 Returns a basis for the space of quaternion Hermitian n-by-n matrices.
1577 Why do we embed these? Basically, because all of numerical
1578 linear algebra assumes that you're working with vectors consisting
1579 of `n` entries from a field and scalars from the same field. There's
1580 no way to tell SageMath that (for example) the vectors contain
1581 complex numbers, while the scalar field is real.
1585 sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
1589 sage: set_random_seed()
1590 sage: n = ZZ.random_element(1,5)
1591 sage: B = QuaternionHermitianEJA._denormalized_basis(n,QQ)
1592 sage: all( M.is_symmetric() for M in B )
1596 Q
= QuaternionAlgebra(QQ
,-1,-1)
1599 # This is like the symmetric case, but we need to be careful:
1601 # * We want conjugate-symmetry, not just symmetry.
1602 # * The diagonal will (as a result) be real.
1606 for j
in xrange(i
+1):
1607 Eij
= matrix(Q
, n
, lambda k
,l
: k
==i
and l
==j
)
1609 Sij
= cls
.real_embed(Eij
)
1612 # The second, third, and fourth ones have a minus
1613 # because they're conjugated.
1614 Sij_real
= cls
.real_embed(Eij
+ Eij
.transpose())
1616 Sij_I
= cls
.real_embed(I
*Eij
- I
*Eij
.transpose())
1618 Sij_J
= cls
.real_embed(J
*Eij
- J
*Eij
.transpose())
1620 Sij_K
= cls
.real_embed(K
*Eij
- K
*Eij
.transpose())
1626 class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1628 The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
1629 with the usual inner product and jordan product ``x*y =
1630 (<x_bar,y_bar>, x0*y_bar + y0*x_bar)``. It has dimension `n` over
1635 sage: from mjo.eja.eja_algebra import JordanSpinEJA
1639 This multiplication table can be verified by hand::
1641 sage: J = JordanSpinEJA(4)
1642 sage: e0,e1,e2,e3 = J.gens()
1658 We can change the generator prefix::
1660 sage: JordanSpinEJA(2, prefix='B').gens()
1664 def __init__(self
, n
, field
=QQ
, **kwargs
):
1665 V
= VectorSpace(field
, n
)
1666 mult_table
= [[V
.zero() for j
in range(n
)] for i
in range(n
)]
1676 z0
= x
.inner_product(y
)
1677 zbar
= y0
*xbar
+ x0
*ybar
1678 z
= V([z0
] + zbar
.list())
1679 mult_table
[i
][j
] = z
1681 # The rank of the spin algebra is two, unless we're in a
1682 # one-dimensional ambient space (because the rank is bounded by
1683 # the ambient dimension).
1684 fdeja
= super(JordanSpinEJA
, self
)
1685 return fdeja
.__init
__(field
, mult_table
, rank
=min(n
,2), **kwargs
)
1687 def inner_product(self
, x
, y
):
1689 Faster to reimplement than to use natural representations.
1693 sage: from mjo.eja.eja_algebra import JordanSpinEJA
1697 Ensure that this is the usual inner product for the algebras
1700 sage: set_random_seed()
1701 sage: J = JordanSpinEJA.random_instance()
1702 sage: x,y = J.random_elements(2)
1703 sage: X = x.natural_representation()
1704 sage: Y = y.natural_representation()
1705 sage: x.inner_product(y) == J.natural_inner_product(X,Y)
1709 return x
.to_vector().inner_product(y
.to_vector())