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
.lazy_import
import lazy_import
17 from sage
.misc
.prandom
import choice
18 from sage
.misc
.table
import table
19 from sage
.modules
.free_module
import FreeModule
, VectorSpace
20 from sage
.rings
.all
import (ZZ
, QQ
, AA
, QQbar
, RR
, RLF
, CLF
,
23 from mjo
.eja
.eja_element
import FiniteDimensionalEuclideanJordanAlgebraElement
24 lazy_import('mjo.eja.eja_subalgebra',
25 'FiniteDimensionalEuclideanJordanSubalgebra')
26 from mjo
.eja
.eja_utils
import _mat2vec
28 class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule
):
30 def _coerce_map_from_base_ring(self
):
32 Disable the map from the base ring into the algebra.
34 Performing a nonsense conversion like this automatically
35 is counterpedagogical. The fallback is to try the usual
36 element constructor, which should also fail.
40 sage: from mjo.eja.eja_algebra import random_eja
44 sage: set_random_seed()
45 sage: J = random_eja()
47 Traceback (most recent call last):
49 ValueError: not a naturally-represented algebra element
64 sage: from mjo.eja.eja_algebra import (JordanSpinEJA, random_eja)
68 By definition, Jordan multiplication commutes::
70 sage: set_random_seed()
71 sage: J = random_eja()
72 sage: x,y = J.random_elements(2)
78 The ``field`` we're given must be real::
80 sage: JordanSpinEJA(2,QQbar)
81 Traceback (most recent call last):
83 ValueError: field is not real
87 if not field
.is_subring(RR
):
88 # Note: this does return true for the real algebraic
89 # field, and any quadratic field where we've specified
91 raise ValueError('field is not real')
93 self
._natural
_basis
= natural_basis
96 category
= MagmaticAlgebras(field
).FiniteDimensional()
97 category
= category
.WithBasis().Unital()
99 fda
= super(FiniteDimensionalEuclideanJordanAlgebra
, self
)
101 range(len(mult_table
)),
104 self
.print_options(bracket
='')
106 # The multiplication table we're given is necessarily in terms
107 # of vectors, because we don't have an algebra yet for
108 # anything to be an element of. However, it's faster in the
109 # long run to have the multiplication table be in terms of
110 # algebra elements. We do this after calling the superclass
111 # constructor so that from_vector() knows what to do.
112 self
._multiplication
_table
= [
113 list(map(lambda x
: self
.from_vector(x
), ls
))
118 def _element_constructor_(self
, elt
):
120 Construct an element of this algebra from its natural
123 This gets called only after the parent element _call_ method
124 fails to find a coercion for the argument.
128 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
130 ....: RealSymmetricEJA)
134 The identity in `S^n` is converted to the identity in the EJA::
136 sage: J = RealSymmetricEJA(3)
137 sage: I = matrix.identity(QQ,3)
138 sage: J(I) == J.one()
141 This skew-symmetric matrix can't be represented in the EJA::
143 sage: J = RealSymmetricEJA(3)
144 sage: A = matrix(QQ,3, lambda i,j: i-j)
146 Traceback (most recent call last):
148 ArithmeticError: vector is not in free module
152 Ensure that we can convert any element of the two non-matrix
153 simple algebras (whose natural representations are their usual
154 vector representations) back and forth faithfully::
156 sage: set_random_seed()
157 sage: J = HadamardEJA.random_instance()
158 sage: x = J.random_element()
159 sage: J(x.to_vector().column()) == x
161 sage: J = JordanSpinEJA.random_instance()
162 sage: x = J.random_element()
163 sage: J(x.to_vector().column()) == x
167 msg
= "not a naturally-represented algebra element"
169 # The superclass implementation of random_element()
170 # needs to be able to coerce "0" into the algebra.
172 elif elt
in self
.base_ring():
173 # Ensure that no base ring -> algebra coercion is performed
174 # by this method. There's some stupidity in sage that would
175 # otherwise propagate to this method; for example, sage thinks
176 # that the integer 3 belongs to the space of 2-by-2 matrices.
177 raise ValueError(msg
)
179 natural_basis
= self
.natural_basis()
180 basis_space
= natural_basis
[0].matrix_space()
181 if elt
not in basis_space
:
182 raise ValueError(msg
)
184 # Thanks for nothing! Matrix spaces aren't vector spaces in
185 # Sage, so we have to figure out its natural-basis coordinates
186 # ourselves. We use the basis space's ring instead of the
187 # element's ring because the basis space might be an algebraic
188 # closure whereas the base ring of the 3-by-3 identity matrix
189 # could be QQ instead of QQbar.
190 V
= VectorSpace(basis_space
.base_ring(), elt
.nrows()*elt
.ncols())
191 W
= V
.span_of_basis( _mat2vec(s
) for s
in natural_basis
)
192 coords
= W
.coordinate_vector(_mat2vec(elt
))
193 return self
.from_vector(coords
)
196 def _max_test_case_size():
198 Return an integer "size" that is an upper bound on the size of
199 this algebra when it is used in a random test
200 case. Unfortunately, the term "size" is quite vague -- when
201 dealing with `R^n` under either the Hadamard or Jordan spin
202 product, the "size" refers to the dimension `n`. When dealing
203 with a matrix algebra (real symmetric or complex/quaternion
204 Hermitian), it refers to the size of the matrix, which is
205 far less than the dimension of the underlying vector space.
207 We default to five in this class, which is safe in `R^n`. The
208 matrix algebra subclasses (or any class where the "size" is
209 interpreted to be far less than the dimension) should override
210 with a smaller number.
216 Return a string representation of ``self``.
220 sage: from mjo.eja.eja_algebra import JordanSpinEJA
224 Ensure that it says what we think it says::
226 sage: JordanSpinEJA(2, field=AA)
227 Euclidean Jordan algebra of dimension 2 over Algebraic Real Field
228 sage: JordanSpinEJA(3, field=RDF)
229 Euclidean Jordan algebra of dimension 3 over Real Double Field
232 fmt
= "Euclidean Jordan algebra of dimension {} over {}"
233 return fmt
.format(self
.dimension(), self
.base_ring())
235 def product_on_basis(self
, i
, j
):
236 return self
._multiplication
_table
[i
][j
]
239 def characteristic_polynomial(self
):
241 Return a characteristic polynomial that works for all elements
244 The resulting polynomial has `n+1` variables, where `n` is the
245 dimension of this algebra. The first `n` variables correspond to
246 the coordinates of an algebra element: when evaluated at the
247 coordinates of an algebra element with respect to a certain
248 basis, the result is a univariate polynomial (in the one
249 remaining variable ``t``), namely the characteristic polynomial
254 sage: from mjo.eja.eja_algebra import JordanSpinEJA, TrivialEJA
258 The characteristic polynomial in the spin algebra is given in
259 Alizadeh, Example 11.11::
261 sage: J = JordanSpinEJA(3)
262 sage: p = J.characteristic_polynomial(); p
263 X1^2 - X2^2 - X3^2 + (-2*t)*X1 + t^2
264 sage: xvec = J.one().to_vector()
268 By definition, the characteristic polynomial is a monic
269 degree-zero polynomial in a rank-zero algebra. Note that
270 Cayley-Hamilton is indeed satisfied since the polynomial
271 ``1`` evaluates to the identity element of the algebra on
274 sage: J = TrivialEJA()
275 sage: J.characteristic_polynomial()
282 # The list of coefficient polynomials a_0, a_1, a_2, ..., a_(r-1).
283 a
= self
._charpoly
_coefficients
()
285 # We go to a bit of trouble here to reorder the
286 # indeterminates, so that it's easier to evaluate the
287 # characteristic polynomial at x's coordinates and get back
288 # something in terms of t, which is what we want.
289 S
= PolynomialRing(self
.base_ring(),'t')
293 S
= PolynomialRing(S
, R
.variable_names())
296 return (t
**r
+ sum( a
[k
]*(t
**k
) for k
in range(r
) ))
299 def inner_product(self
, x
, y
):
301 The inner product associated with this Euclidean Jordan algebra.
303 Defaults to the trace inner product, but can be overridden by
304 subclasses if they are sure that the necessary properties are
309 sage: from mjo.eja.eja_algebra import random_eja
313 Our inner product is "associative," which means the following for
314 a symmetric bilinear form::
316 sage: set_random_seed()
317 sage: J = random_eja()
318 sage: x,y,z = J.random_elements(3)
319 sage: (x*y).inner_product(z) == y.inner_product(x*z)
323 X
= x
.natural_representation()
324 Y
= y
.natural_representation()
325 return self
.natural_inner_product(X
,Y
)
328 def is_trivial(self
):
330 Return whether or not this algebra is trivial.
332 A trivial algebra contains only the zero element.
336 sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
341 sage: J = ComplexHermitianEJA(3)
347 sage: J = TrivialEJA()
352 return self
.dimension() == 0
355 def multiplication_table(self
):
357 Return a visual representation of this algebra's multiplication
358 table (on basis elements).
362 sage: from mjo.eja.eja_algebra import JordanSpinEJA
366 sage: J = JordanSpinEJA(4)
367 sage: J.multiplication_table()
368 +----++----+----+----+----+
369 | * || e0 | e1 | e2 | e3 |
370 +====++====+====+====+====+
371 | e0 || e0 | e1 | e2 | e3 |
372 +----++----+----+----+----+
373 | e1 || e1 | e0 | 0 | 0 |
374 +----++----+----+----+----+
375 | e2 || e2 | 0 | e0 | 0 |
376 +----++----+----+----+----+
377 | e3 || e3 | 0 | 0 | e0 |
378 +----++----+----+----+----+
381 M
= list(self
._multiplication
_table
) # copy
382 for i
in range(len(M
)):
383 # M had better be "square"
384 M
[i
] = [self
.monomial(i
)] + M
[i
]
385 M
= [["*"] + list(self
.gens())] + M
386 return table(M
, header_row
=True, header_column
=True, frame
=True)
389 def natural_basis(self
):
391 Return a more-natural representation of this algebra's basis.
393 Every finite-dimensional Euclidean Jordan Algebra is a direct
394 sum of five simple algebras, four of which comprise Hermitian
395 matrices. This method returns the original "natural" basis
396 for our underlying vector space. (Typically, the natural basis
397 is used to construct the multiplication table in the first place.)
399 Note that this will always return a matrix. The standard basis
400 in `R^n` will be returned as `n`-by-`1` column matrices.
404 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
405 ....: RealSymmetricEJA)
409 sage: J = RealSymmetricEJA(2)
411 Finite family {0: e0, 1: e1, 2: e2}
412 sage: J.natural_basis()
414 [1 0] [ 0 0.7071067811865475?] [0 0]
415 [0 0], [0.7071067811865475? 0], [0 1]
420 sage: J = JordanSpinEJA(2)
422 Finite family {0: e0, 1: e1}
423 sage: J.natural_basis()
430 if self
._natural
_basis
is None:
431 M
= self
.natural_basis_space()
432 return tuple( M(b
.to_vector()) for b
in self
.basis() )
434 return self
._natural
_basis
437 def natural_basis_space(self
):
439 Return the matrix space in which this algebra's natural basis
442 if self
._natural
_basis
is None or len(self
._natural
_basis
) == 0:
443 return MatrixSpace(self
.base_ring(), self
.dimension(), 1)
445 return self
._natural
_basis
[0].matrix_space()
449 def natural_inner_product(X
,Y
):
451 Compute the inner product of two naturally-represented elements.
453 For example in the real symmetric matrix EJA, this will compute
454 the trace inner-product of two n-by-n symmetric matrices. The
455 default should work for the real cartesian product EJA, the
456 Jordan spin EJA, and the real symmetric matrices. The others
457 will have to be overridden.
459 return (X
.conjugate_transpose()*Y
).trace()
465 Return the unit element of this algebra.
469 sage: from mjo.eja.eja_algebra import (HadamardEJA,
474 sage: J = HadamardEJA(5)
476 e0 + e1 + e2 + e3 + e4
480 The identity element acts like the identity::
482 sage: set_random_seed()
483 sage: J = random_eja()
484 sage: x = J.random_element()
485 sage: J.one()*x == x and x*J.one() == x
488 The matrix of the unit element's operator is the identity::
490 sage: set_random_seed()
491 sage: J = random_eja()
492 sage: actual = J.one().operator().matrix()
493 sage: expected = matrix.identity(J.base_ring(), J.dimension())
494 sage: actual == expected
498 # We can brute-force compute the matrices of the operators
499 # that correspond to the basis elements of this algebra.
500 # If some linear combination of those basis elements is the
501 # algebra identity, then the same linear combination of
502 # their matrices has to be the identity matrix.
504 # Of course, matrices aren't vectors in sage, so we have to
505 # appeal to the "long vectors" isometry.
506 oper_vecs
= [ _mat2vec(g
.operator().matrix()) for g
in self
.gens() ]
508 # Now we use basis linear algebra to find the coefficients,
509 # of the matrices-as-vectors-linear-combination, which should
510 # work for the original algebra basis too.
511 A
= matrix
.column(self
.base_ring(), oper_vecs
)
513 # We used the isometry on the left-hand side already, but we
514 # still need to do it for the right-hand side. Recall that we
515 # wanted something that summed to the identity matrix.
516 b
= _mat2vec( matrix
.identity(self
.base_ring(), self
.dimension()) )
518 # Now if there's an identity element in the algebra, this should work.
519 coeffs
= A
.solve_right(b
)
520 return self
.linear_combination(zip(self
.gens(), coeffs
))
523 def peirce_decomposition(self
, c
):
525 The Peirce decomposition of this algebra relative to the
528 In the future, this can be extended to a complete system of
529 orthogonal idempotents.
533 - ``c`` -- an idempotent of this algebra.
537 A triple (J0, J5, J1) containing two subalgebras and one subspace
540 - ``J0`` -- the algebra on the eigenspace of ``c.operator()``
541 corresponding to the eigenvalue zero.
543 - ``J5`` -- the eigenspace (NOT a subalgebra) of ``c.operator()``
544 corresponding to the eigenvalue one-half.
546 - ``J1`` -- the algebra on the eigenspace of ``c.operator()``
547 corresponding to the eigenvalue one.
549 These are the only possible eigenspaces for that operator, and this
550 algebra is a direct sum of them. The spaces ``J0`` and ``J1`` are
551 orthogonal, and are subalgebras of this algebra with the appropriate
556 sage: from mjo.eja.eja_algebra import random_eja, RealSymmetricEJA
560 The canonical example comes from the symmetric matrices, which
561 decompose into diagonal and off-diagonal parts::
563 sage: J = RealSymmetricEJA(3)
564 sage: C = matrix(QQ, [ [1,0,0],
568 sage: J0,J5,J1 = J.peirce_decomposition(c)
570 Euclidean Jordan algebra of dimension 1...
572 Vector space of degree 6 and dimension 2...
574 Euclidean Jordan algebra of dimension 3...
578 Every algebra decomposes trivially with respect to its identity
581 sage: set_random_seed()
582 sage: J = random_eja()
583 sage: J0,J5,J1 = J.peirce_decomposition(J.one())
584 sage: J0.dimension() == 0 and J5.dimension() == 0
586 sage: J1.superalgebra() == J and J1.dimension() == J.dimension()
589 The identity elements in the two subalgebras are the
590 projections onto their respective subspaces of the
591 superalgebra's identity element::
593 sage: set_random_seed()
594 sage: J = random_eja()
595 sage: x = J.random_element()
596 sage: if not J.is_trivial():
597 ....: while x.is_nilpotent():
598 ....: x = J.random_element()
599 sage: c = x.subalgebra_idempotent()
600 sage: J0,J5,J1 = J.peirce_decomposition(c)
601 sage: J1(c) == J1.one()
603 sage: J0(J.one() - c) == J0.one()
607 if not c
.is_idempotent():
608 raise ValueError("element is not idempotent: %s" % c
)
610 # Default these to what they should be if they turn out to be
611 # trivial, because eigenspaces_left() won't return eigenvalues
612 # corresponding to trivial spaces (e.g. it returns only the
613 # eigenspace corresponding to lambda=1 if you take the
614 # decomposition relative to the identity element).
615 trivial
= FiniteDimensionalEuclideanJordanSubalgebra(self
, ())
616 J0
= trivial
# eigenvalue zero
617 J5
= VectorSpace(self
.base_ring(), 0) # eigenvalue one-half
618 J1
= trivial
# eigenvalue one
620 for (eigval
, eigspace
) in c
.operator().matrix().right_eigenspaces():
621 if eigval
== ~
(self
.base_ring()(2)):
624 gens
= tuple( self
.from_vector(b
) for b
in eigspace
.basis() )
625 subalg
= FiniteDimensionalEuclideanJordanSubalgebra(self
, gens
)
631 raise ValueError("unexpected eigenvalue: %s" % eigval
)
636 def random_elements(self
, count
):
638 Return ``count`` random elements as a tuple.
642 sage: from mjo.eja.eja_algebra import JordanSpinEJA
646 sage: J = JordanSpinEJA(3)
647 sage: x,y,z = J.random_elements(3)
648 sage: all( [ x in J, y in J, z in J ])
650 sage: len( J.random_elements(10) ) == 10
654 return tuple( self
.random_element() for idx
in range(count
) )
657 def random_instance(cls
, field
=AA
, **kwargs
):
659 Return a random instance of this type of algebra.
661 Beware, this will crash for "most instances" because the
662 constructor below looks wrong.
664 if cls
is TrivialEJA
:
665 # The TrivialEJA class doesn't take an "n" argument because
669 n
= ZZ
.random_element(cls
._max
_test
_case
_size
()) + 1
670 return cls(n
, field
, **kwargs
)
673 def _charpoly_coefficients(self
):
675 The `r` polynomial coefficients of the "characteristic polynomial
679 var_names
= [ "X" + str(z
) for z
in range(1,n
+1) ]
680 R
= PolynomialRing(self
.base_ring(), var_names
)
682 F
= R
.fraction_field()
685 # From a result in my book, these are the entries of the
686 # basis representation of L_x.
687 return sum( vars[k
]*self
.monomial(k
).operator().matrix()[i
,j
]
690 L_x
= matrix(F
, n
, n
, L_x_i_j
)
693 if self
.rank
.is_in_cache():
695 # There's no need to pad the system with redundant
696 # columns if we *know* they'll be redundant.
699 # Compute an extra power in case the rank is equal to
700 # the dimension (otherwise, we would stop at x^(r-1)).
701 x_powers
= [ (L_x
**k
)*self
.one().to_vector()
702 for k
in range(n
+1) ]
703 A
= matrix
.column(F
, x_powers
[:n
])
704 AE
= A
.extended_echelon_form()
711 # The theory says that only the first "r" coefficients are
712 # nonzero, and they actually live in the original polynomial
713 # ring and not the fraction field. We negate them because
714 # in the actual characteristic polynomial, they get moved
715 # to the other side where x^r lives.
716 return -A_rref
.solve_right(E
*b
).change_ring(R
)[:r
]
721 Return the rank of this EJA.
723 This is a cached method because we know the rank a priori for
724 all of the algebras we can construct. Thus we can avoid the
725 expensive ``_charpoly_coefficients()`` call unless we truly
726 need to compute the whole characteristic polynomial.
730 sage: from mjo.eja.eja_algebra import (HadamardEJA,
732 ....: RealSymmetricEJA,
733 ....: ComplexHermitianEJA,
734 ....: QuaternionHermitianEJA,
739 The rank of the Jordan spin algebra is always two::
741 sage: JordanSpinEJA(2).rank()
743 sage: JordanSpinEJA(3).rank()
745 sage: JordanSpinEJA(4).rank()
748 The rank of the `n`-by-`n` Hermitian real, complex, or
749 quaternion matrices is `n`::
751 sage: RealSymmetricEJA(4).rank()
753 sage: ComplexHermitianEJA(3).rank()
755 sage: QuaternionHermitianEJA(2).rank()
760 Ensure that every EJA that we know how to construct has a
761 positive integer rank, unless the algebra is trivial in
762 which case its rank will be zero::
764 sage: set_random_seed()
765 sage: J = random_eja()
769 sage: r > 0 or (r == 0 and J.is_trivial())
772 Ensure that computing the rank actually works, since the ranks
773 of all simple algebras are known and will be cached by default::
775 sage: J = HadamardEJA(4)
776 sage: J.rank.clear_cache()
782 sage: J = JordanSpinEJA(4)
783 sage: J.rank.clear_cache()
789 sage: J = RealSymmetricEJA(3)
790 sage: J.rank.clear_cache()
796 sage: J = ComplexHermitianEJA(2)
797 sage: J.rank.clear_cache()
803 sage: J = QuaternionHermitianEJA(2)
804 sage: J.rank.clear_cache()
808 return len(self
._charpoly
_coefficients
())
811 def vector_space(self
):
813 Return the vector space that underlies this algebra.
817 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
821 sage: J = RealSymmetricEJA(2)
822 sage: J.vector_space()
823 Vector space of dimension 3 over...
826 return self
.zero().to_vector().parent().ambient_vector_space()
829 Element
= FiniteDimensionalEuclideanJordanAlgebraElement
832 class HadamardEJA(FiniteDimensionalEuclideanJordanAlgebra
):
834 Return the Euclidean Jordan Algebra corresponding to the set
835 `R^n` under the Hadamard product.
837 Note: this is nothing more than the Cartesian product of ``n``
838 copies of the spin algebra. Once Cartesian product algebras
839 are implemented, this can go.
843 sage: from mjo.eja.eja_algebra import HadamardEJA
847 This multiplication table can be verified by hand::
849 sage: J = HadamardEJA(3)
850 sage: e0,e1,e2 = J.gens()
866 We can change the generator prefix::
868 sage: HadamardEJA(3, prefix='r').gens()
872 def __init__(self
, n
, field
=AA
, **kwargs
):
873 V
= VectorSpace(field
, n
)
874 mult_table
= [ [ V
.gen(i
)*(i
== j
) for j
in range(n
) ]
877 fdeja
= super(HadamardEJA
, self
)
878 fdeja
.__init
__(field
, mult_table
, **kwargs
)
879 self
.rank
.set_cache(n
)
881 def inner_product(self
, x
, y
):
883 Faster to reimplement than to use natural representations.
887 sage: from mjo.eja.eja_algebra import HadamardEJA
891 Ensure that this is the usual inner product for the algebras
894 sage: set_random_seed()
895 sage: J = HadamardEJA.random_instance()
896 sage: x,y = J.random_elements(2)
897 sage: X = x.natural_representation()
898 sage: Y = y.natural_representation()
899 sage: x.inner_product(y) == J.natural_inner_product(X,Y)
903 return x
.to_vector().inner_product(y
.to_vector())
906 def random_eja(field
=AA
, nontrivial
=False):
908 Return a "random" finite-dimensional Euclidean Jordan Algebra.
912 sage: from mjo.eja.eja_algebra import random_eja
917 Euclidean Jordan algebra of dimension...
920 eja_classes
= [HadamardEJA
,
924 QuaternionHermitianEJA
]
926 eja_classes
.append(TrivialEJA
)
927 classname
= choice(eja_classes
)
928 return classname
.random_instance(field
=field
)
935 class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra
):
937 def _max_test_case_size():
938 # Play it safe, since this will be squared and the underlying
939 # field can have dimension 4 (quaternions) too.
942 def __init__(self
, field
, basis
, normalize_basis
=True, **kwargs
):
944 Compared to the superclass constructor, we take a basis instead of
945 a multiplication table because the latter can be computed in terms
946 of the former when the product is known (like it is here).
948 # Used in this class's fast _charpoly_coefficients() override.
949 self
._basis
_normalizers
= None
951 # We're going to loop through this a few times, so now's a good
952 # time to ensure that it isn't a generator expression.
955 if len(basis
) > 1 and normalize_basis
:
956 # We'll need sqrt(2) to normalize the basis, and this
957 # winds up in the multiplication table, so the whole
958 # algebra needs to be over the field extension.
959 R
= PolynomialRing(field
, 'z')
962 if p
.is_irreducible():
963 field
= field
.extension(p
, 'sqrt2', embedding
=RLF(2).sqrt())
964 basis
= tuple( s
.change_ring(field
) for s
in basis
)
965 self
._basis
_normalizers
= tuple(
966 ~
(self
.natural_inner_product(s
,s
).sqrt()) for s
in basis
)
967 basis
= tuple(s
*c
for (s
,c
) in zip(basis
,self
._basis
_normalizers
))
969 Qs
= self
.multiplication_table_from_matrix_basis(basis
)
971 fdeja
= super(MatrixEuclideanJordanAlgebra
, self
)
972 fdeja
.__init
__(field
, Qs
, natural_basis
=basis
, **kwargs
)
977 def _charpoly_coefficients(self
):
979 Override the parent method with something that tries to compute
980 over a faster (non-extension) field.
982 if self
._basis
_normalizers
is None:
983 # We didn't normalize, so assume that the basis we started
984 # with had entries in a nice field.
985 return super(MatrixEuclideanJordanAlgebra
, self
)._charpoly
_coefficients
()
987 basis
= ( (b
/n
) for (b
,n
) in zip(self
.natural_basis(),
988 self
._basis
_normalizers
) )
990 # Do this over the rationals and convert back at the end.
991 # Only works because we know the entries of the basis are
993 J
= MatrixEuclideanJordanAlgebra(QQ
,
995 normalize_basis
=False)
996 return J
._charpoly
_coefficients
()
1000 def multiplication_table_from_matrix_basis(basis
):
1002 At least three of the five simple Euclidean Jordan algebras have the
1003 symmetric multiplication (A,B) |-> (AB + BA)/2, where the
1004 multiplication on the right is matrix multiplication. Given a basis
1005 for the underlying matrix space, this function returns a
1006 multiplication table (obtained by looping through the basis
1007 elements) for an algebra of those matrices.
1009 # In S^2, for example, we nominally have four coordinates even
1010 # though the space is of dimension three only. The vector space V
1011 # is supposed to hold the entire long vector, and the subspace W
1012 # of V will be spanned by the vectors that arise from symmetric
1013 # matrices. Thus for S^2, dim(V) == 4 and dim(W) == 3.
1014 field
= basis
[0].base_ring()
1015 dimension
= basis
[0].nrows()
1017 V
= VectorSpace(field
, dimension
**2)
1018 W
= V
.span_of_basis( _mat2vec(s
) for s
in basis
)
1020 mult_table
= [[W
.zero() for j
in range(n
)] for i
in range(n
)]
1023 mat_entry
= (basis
[i
]*basis
[j
] + basis
[j
]*basis
[i
])/2
1024 mult_table
[i
][j
] = W
.coordinate_vector(_mat2vec(mat_entry
))
1032 Embed the matrix ``M`` into a space of real matrices.
1034 The matrix ``M`` can have entries in any field at the moment:
1035 the real numbers, complex numbers, or quaternions. And although
1036 they are not a field, we can probably support octonions at some
1037 point, too. This function returns a real matrix that "acts like"
1038 the original with respect to matrix multiplication; i.e.
1040 real_embed(M*N) = real_embed(M)*real_embed(N)
1043 raise NotImplementedError
1047 def real_unembed(M
):
1049 The inverse of :meth:`real_embed`.
1051 raise NotImplementedError
1055 def natural_inner_product(cls
,X
,Y
):
1056 Xu
= cls
.real_unembed(X
)
1057 Yu
= cls
.real_unembed(Y
)
1058 tr
= (Xu
*Yu
).trace()
1061 # It's real already.
1064 # Otherwise, try the thing that works for complex numbers; and
1065 # if that doesn't work, the thing that works for quaternions.
1067 return tr
.vector()[0] # real part, imag part is index 1
1068 except AttributeError:
1069 # A quaternions doesn't have a vector() method, but does
1070 # have coefficient_tuple() method that returns the
1071 # coefficients of 1, i, j, and k -- in that order.
1072 return tr
.coefficient_tuple()[0]
1075 class RealMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra
):
1079 The identity function, for embedding real matrices into real
1085 def real_unembed(M
):
1087 The identity function, for unembedding real matrices from real
1093 class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra
):
1095 The rank-n simple EJA consisting of real symmetric n-by-n
1096 matrices, the usual symmetric Jordan product, and the trace inner
1097 product. It has dimension `(n^2 + n)/2` over the reals.
1101 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
1105 sage: J = RealSymmetricEJA(2)
1106 sage: e0, e1, e2 = J.gens()
1114 In theory, our "field" can be any subfield of the reals::
1116 sage: RealSymmetricEJA(2, RDF)
1117 Euclidean Jordan algebra of dimension 3 over Real Double Field
1118 sage: RealSymmetricEJA(2, RR)
1119 Euclidean Jordan algebra of dimension 3 over Real Field with
1120 53 bits of precision
1124 The dimension of this algebra is `(n^2 + n) / 2`::
1126 sage: set_random_seed()
1127 sage: n_max = RealSymmetricEJA._max_test_case_size()
1128 sage: n = ZZ.random_element(1, n_max)
1129 sage: J = RealSymmetricEJA(n)
1130 sage: J.dimension() == (n^2 + n)/2
1133 The Jordan multiplication is what we think it is::
1135 sage: set_random_seed()
1136 sage: J = RealSymmetricEJA.random_instance()
1137 sage: x,y = J.random_elements(2)
1138 sage: actual = (x*y).natural_representation()
1139 sage: X = x.natural_representation()
1140 sage: Y = y.natural_representation()
1141 sage: expected = (X*Y + Y*X)/2
1142 sage: actual == expected
1144 sage: J(expected) == x*y
1147 We can change the generator prefix::
1149 sage: RealSymmetricEJA(3, prefix='q').gens()
1150 (q0, q1, q2, q3, q4, q5)
1152 Our natural basis is normalized with respect to the natural inner
1153 product unless we specify otherwise::
1155 sage: set_random_seed()
1156 sage: J = RealSymmetricEJA.random_instance()
1157 sage: all( b.norm() == 1 for b in J.gens() )
1160 Since our natural basis is normalized with respect to the natural
1161 inner product, and since we know that this algebra is an EJA, any
1162 left-multiplication operator's matrix will be symmetric because
1163 natural->EJA basis representation is an isometry and within the EJA
1164 the operator is self-adjoint by the Jordan axiom::
1166 sage: set_random_seed()
1167 sage: x = RealSymmetricEJA.random_instance().random_element()
1168 sage: x.operator().matrix().is_symmetric()
1173 def _denormalized_basis(cls
, n
, field
):
1175 Return a basis for the space of real symmetric n-by-n matrices.
1179 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
1183 sage: set_random_seed()
1184 sage: n = ZZ.random_element(1,5)
1185 sage: B = RealSymmetricEJA._denormalized_basis(n,QQ)
1186 sage: all( M.is_symmetric() for M in B)
1190 # The basis of symmetric matrices, as matrices, in their R^(n-by-n)
1194 for j
in range(i
+1):
1195 Eij
= matrix(field
, n
, lambda k
,l
: k
==i
and l
==j
)
1199 Sij
= Eij
+ Eij
.transpose()
1205 def _max_test_case_size():
1206 return 4 # Dimension 10
1209 def __init__(self
, n
, field
=AA
, **kwargs
):
1210 basis
= self
._denormalized
_basis
(n
, field
)
1211 super(RealSymmetricEJA
, self
).__init
__(field
, basis
, **kwargs
)
1212 self
.rank
.set_cache(n
)
1215 class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra
):
1219 Embed the n-by-n complex matrix ``M`` into the space of real
1220 matrices of size 2n-by-2n via the map the sends each entry `z = a +
1221 bi` to the block matrix ``[[a,b],[-b,a]]``.
1225 sage: from mjo.eja.eja_algebra import \
1226 ....: ComplexMatrixEuclideanJordanAlgebra
1230 sage: F = QuadraticField(-1, 'I')
1231 sage: x1 = F(4 - 2*i)
1232 sage: x2 = F(1 + 2*i)
1235 sage: M = matrix(F,2,[[x1,x2],[x3,x4]])
1236 sage: ComplexMatrixEuclideanJordanAlgebra.real_embed(M)
1245 Embedding is a homomorphism (isomorphism, in fact)::
1247 sage: set_random_seed()
1248 sage: n_max = ComplexMatrixEuclideanJordanAlgebra._max_test_case_size()
1249 sage: n = ZZ.random_element(n_max)
1250 sage: F = QuadraticField(-1, 'I')
1251 sage: X = random_matrix(F, n)
1252 sage: Y = random_matrix(F, n)
1253 sage: Xe = ComplexMatrixEuclideanJordanAlgebra.real_embed(X)
1254 sage: Ye = ComplexMatrixEuclideanJordanAlgebra.real_embed(Y)
1255 sage: XYe = ComplexMatrixEuclideanJordanAlgebra.real_embed(X*Y)
1262 raise ValueError("the matrix 'M' must be square")
1264 # We don't need any adjoined elements...
1265 field
= M
.base_ring().base_ring()
1269 a
= z
.list()[0] # real part, I guess
1270 b
= z
.list()[1] # imag part, I guess
1271 blocks
.append(matrix(field
, 2, [[a
,b
],[-b
,a
]]))
1273 return matrix
.block(field
, n
, blocks
)
1277 def real_unembed(M
):
1279 The inverse of _embed_complex_matrix().
1283 sage: from mjo.eja.eja_algebra import \
1284 ....: ComplexMatrixEuclideanJordanAlgebra
1288 sage: A = matrix(QQ,[ [ 1, 2, 3, 4],
1289 ....: [-2, 1, -4, 3],
1290 ....: [ 9, 10, 11, 12],
1291 ....: [-10, 9, -12, 11] ])
1292 sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(A)
1294 [ 10*I + 9 12*I + 11]
1298 Unembedding is the inverse of embedding::
1300 sage: set_random_seed()
1301 sage: F = QuadraticField(-1, 'I')
1302 sage: M = random_matrix(F, 3)
1303 sage: Me = ComplexMatrixEuclideanJordanAlgebra.real_embed(M)
1304 sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(Me) == M
1310 raise ValueError("the matrix 'M' must be square")
1311 if not n
.mod(2).is_zero():
1312 raise ValueError("the matrix 'M' must be a complex embedding")
1314 # If "M" was normalized, its base ring might have roots
1315 # adjoined and they can stick around after unembedding.
1316 field
= M
.base_ring()
1317 R
= PolynomialRing(field
, 'z')
1320 # Sage doesn't know how to embed AA into QQbar, i.e. how
1321 # to adjoin sqrt(-1) to AA.
1324 F
= field
.extension(z
**2 + 1, 'I', embedding
=CLF(-1).sqrt())
1327 # Go top-left to bottom-right (reading order), converting every
1328 # 2-by-2 block we see to a single complex element.
1330 for k
in range(n
/2):
1331 for j
in range(n
/2):
1332 submat
= M
[2*k
:2*k
+2,2*j
:2*j
+2]
1333 if submat
[0,0] != submat
[1,1]:
1334 raise ValueError('bad on-diagonal submatrix')
1335 if submat
[0,1] != -submat
[1,0]:
1336 raise ValueError('bad off-diagonal submatrix')
1337 z
= submat
[0,0] + submat
[0,1]*i
1340 return matrix(F
, n
/2, elements
)
1344 def natural_inner_product(cls
,X
,Y
):
1346 Compute a natural inner product in this algebra directly from
1351 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
1355 This gives the same answer as the slow, default method implemented
1356 in :class:`MatrixEuclideanJordanAlgebra`::
1358 sage: set_random_seed()
1359 sage: J = ComplexHermitianEJA.random_instance()
1360 sage: x,y = J.random_elements(2)
1361 sage: Xe = x.natural_representation()
1362 sage: Ye = y.natural_representation()
1363 sage: X = ComplexHermitianEJA.real_unembed(Xe)
1364 sage: Y = ComplexHermitianEJA.real_unembed(Ye)
1365 sage: expected = (X*Y).trace().real()
1366 sage: actual = ComplexHermitianEJA.natural_inner_product(Xe,Ye)
1367 sage: actual == expected
1371 return RealMatrixEuclideanJordanAlgebra
.natural_inner_product(X
,Y
)/2
1374 class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra
):
1376 The rank-n simple EJA consisting of complex Hermitian n-by-n
1377 matrices over the real numbers, the usual symmetric Jordan product,
1378 and the real-part-of-trace inner product. It has dimension `n^2` over
1383 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
1387 In theory, our "field" can be any subfield of the reals::
1389 sage: ComplexHermitianEJA(2, RDF)
1390 Euclidean Jordan algebra of dimension 4 over Real Double Field
1391 sage: ComplexHermitianEJA(2, RR)
1392 Euclidean Jordan algebra of dimension 4 over Real Field with
1393 53 bits of precision
1397 The dimension of this algebra is `n^2`::
1399 sage: set_random_seed()
1400 sage: n_max = ComplexHermitianEJA._max_test_case_size()
1401 sage: n = ZZ.random_element(1, n_max)
1402 sage: J = ComplexHermitianEJA(n)
1403 sage: J.dimension() == n^2
1406 The Jordan multiplication is what we think it is::
1408 sage: set_random_seed()
1409 sage: J = ComplexHermitianEJA.random_instance()
1410 sage: x,y = J.random_elements(2)
1411 sage: actual = (x*y).natural_representation()
1412 sage: X = x.natural_representation()
1413 sage: Y = y.natural_representation()
1414 sage: expected = (X*Y + Y*X)/2
1415 sage: actual == expected
1417 sage: J(expected) == x*y
1420 We can change the generator prefix::
1422 sage: ComplexHermitianEJA(2, prefix='z').gens()
1425 Our natural basis is normalized with respect to the natural inner
1426 product unless we specify otherwise::
1428 sage: set_random_seed()
1429 sage: J = ComplexHermitianEJA.random_instance()
1430 sage: all( b.norm() == 1 for b in J.gens() )
1433 Since our natural basis is normalized with respect to the natural
1434 inner product, and since we know that this algebra is an EJA, any
1435 left-multiplication operator's matrix will be symmetric because
1436 natural->EJA basis representation is an isometry and within the EJA
1437 the operator is self-adjoint by the Jordan axiom::
1439 sage: set_random_seed()
1440 sage: x = ComplexHermitianEJA.random_instance().random_element()
1441 sage: x.operator().matrix().is_symmetric()
1447 def _denormalized_basis(cls
, n
, field
):
1449 Returns a basis for the space of complex Hermitian n-by-n matrices.
1451 Why do we embed these? Basically, because all of numerical linear
1452 algebra assumes that you're working with vectors consisting of `n`
1453 entries from a field and scalars from the same field. There's no way
1454 to tell SageMath that (for example) the vectors contain complex
1455 numbers, while the scalar field is real.
1459 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
1463 sage: set_random_seed()
1464 sage: n = ZZ.random_element(1,5)
1465 sage: field = QuadraticField(2, 'sqrt2')
1466 sage: B = ComplexHermitianEJA._denormalized_basis(n, field)
1467 sage: all( M.is_symmetric() for M in B)
1471 R
= PolynomialRing(field
, 'z')
1473 F
= field
.extension(z
**2 + 1, 'I')
1476 # This is like the symmetric case, but we need to be careful:
1478 # * We want conjugate-symmetry, not just symmetry.
1479 # * The diagonal will (as a result) be real.
1483 for j
in range(i
+1):
1484 Eij
= matrix(F
, n
, lambda k
,l
: k
==i
and l
==j
)
1486 Sij
= cls
.real_embed(Eij
)
1489 # The second one has a minus because it's conjugated.
1490 Sij_real
= cls
.real_embed(Eij
+ Eij
.transpose())
1492 Sij_imag
= cls
.real_embed(I
*Eij
- I
*Eij
.transpose())
1495 # Since we embedded these, we can drop back to the "field" that we
1496 # started with instead of the complex extension "F".
1497 return ( s
.change_ring(field
) for s
in S
)
1500 def __init__(self
, n
, field
=AA
, **kwargs
):
1501 basis
= self
._denormalized
_basis
(n
,field
)
1502 super(ComplexHermitianEJA
,self
).__init
__(field
, basis
, **kwargs
)
1503 self
.rank
.set_cache(n
)
1506 class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra
):
1510 Embed the n-by-n quaternion matrix ``M`` into the space of real
1511 matrices of size 4n-by-4n by first sending each quaternion entry `z
1512 = a + bi + cj + dk` to the block-complex matrix ``[[a + bi,
1513 c+di],[-c + di, a-bi]]`, and then embedding those into a real
1518 sage: from mjo.eja.eja_algebra import \
1519 ....: QuaternionMatrixEuclideanJordanAlgebra
1523 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1524 sage: i,j,k = Q.gens()
1525 sage: x = 1 + 2*i + 3*j + 4*k
1526 sage: M = matrix(Q, 1, [[x]])
1527 sage: QuaternionMatrixEuclideanJordanAlgebra.real_embed(M)
1533 Embedding is a homomorphism (isomorphism, in fact)::
1535 sage: set_random_seed()
1536 sage: n_max = QuaternionMatrixEuclideanJordanAlgebra._max_test_case_size()
1537 sage: n = ZZ.random_element(n_max)
1538 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1539 sage: X = random_matrix(Q, n)
1540 sage: Y = random_matrix(Q, n)
1541 sage: Xe = QuaternionMatrixEuclideanJordanAlgebra.real_embed(X)
1542 sage: Ye = QuaternionMatrixEuclideanJordanAlgebra.real_embed(Y)
1543 sage: XYe = QuaternionMatrixEuclideanJordanAlgebra.real_embed(X*Y)
1548 quaternions
= M
.base_ring()
1551 raise ValueError("the matrix 'M' must be square")
1553 F
= QuadraticField(-1, 'I')
1558 t
= z
.coefficient_tuple()
1563 cplxM
= matrix(F
, 2, [[ a
+ b
*i
, c
+ d
*i
],
1564 [-c
+ d
*i
, a
- b
*i
]])
1565 realM
= ComplexMatrixEuclideanJordanAlgebra
.real_embed(cplxM
)
1566 blocks
.append(realM
)
1568 # We should have real entries by now, so use the realest field
1569 # we've got for the return value.
1570 return matrix
.block(quaternions
.base_ring(), n
, blocks
)
1575 def real_unembed(M
):
1577 The inverse of _embed_quaternion_matrix().
1581 sage: from mjo.eja.eja_algebra import \
1582 ....: QuaternionMatrixEuclideanJordanAlgebra
1586 sage: M = matrix(QQ, [[ 1, 2, 3, 4],
1587 ....: [-2, 1, -4, 3],
1588 ....: [-3, 4, 1, -2],
1589 ....: [-4, -3, 2, 1]])
1590 sage: QuaternionMatrixEuclideanJordanAlgebra.real_unembed(M)
1591 [1 + 2*i + 3*j + 4*k]
1595 Unembedding is the inverse of embedding::
1597 sage: set_random_seed()
1598 sage: Q = QuaternionAlgebra(QQ, -1, -1)
1599 sage: M = random_matrix(Q, 3)
1600 sage: Me = QuaternionMatrixEuclideanJordanAlgebra.real_embed(M)
1601 sage: QuaternionMatrixEuclideanJordanAlgebra.real_unembed(Me) == M
1607 raise ValueError("the matrix 'M' must be square")
1608 if not n
.mod(4).is_zero():
1609 raise ValueError("the matrix 'M' must be a quaternion embedding")
1611 # Use the base ring of the matrix to ensure that its entries can be
1612 # multiplied by elements of the quaternion algebra.
1613 field
= M
.base_ring()
1614 Q
= QuaternionAlgebra(field
,-1,-1)
1617 # Go top-left to bottom-right (reading order), converting every
1618 # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1
1621 for l
in range(n
/4):
1622 for m
in range(n
/4):
1623 submat
= ComplexMatrixEuclideanJordanAlgebra
.real_unembed(
1624 M
[4*l
:4*l
+4,4*m
:4*m
+4] )
1625 if submat
[0,0] != submat
[1,1].conjugate():
1626 raise ValueError('bad on-diagonal submatrix')
1627 if submat
[0,1] != -submat
[1,0].conjugate():
1628 raise ValueError('bad off-diagonal submatrix')
1629 z
= submat
[0,0].real()
1630 z
+= submat
[0,0].imag()*i
1631 z
+= submat
[0,1].real()*j
1632 z
+= submat
[0,1].imag()*k
1635 return matrix(Q
, n
/4, elements
)
1639 def natural_inner_product(cls
,X
,Y
):
1641 Compute a natural inner product in this algebra directly from
1646 sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
1650 This gives the same answer as the slow, default method implemented
1651 in :class:`MatrixEuclideanJordanAlgebra`::
1653 sage: set_random_seed()
1654 sage: J = QuaternionHermitianEJA.random_instance()
1655 sage: x,y = J.random_elements(2)
1656 sage: Xe = x.natural_representation()
1657 sage: Ye = y.natural_representation()
1658 sage: X = QuaternionHermitianEJA.real_unembed(Xe)
1659 sage: Y = QuaternionHermitianEJA.real_unembed(Ye)
1660 sage: expected = (X*Y).trace().coefficient_tuple()[0]
1661 sage: actual = QuaternionHermitianEJA.natural_inner_product(Xe,Ye)
1662 sage: actual == expected
1666 return RealMatrixEuclideanJordanAlgebra
.natural_inner_product(X
,Y
)/4
1669 class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra
):
1671 The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
1672 matrices, the usual symmetric Jordan product, and the
1673 real-part-of-trace inner product. It has dimension `2n^2 - n` over
1678 sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
1682 In theory, our "field" can be any subfield of the reals::
1684 sage: QuaternionHermitianEJA(2, RDF)
1685 Euclidean Jordan algebra of dimension 6 over Real Double Field
1686 sage: QuaternionHermitianEJA(2, RR)
1687 Euclidean Jordan algebra of dimension 6 over Real Field with
1688 53 bits of precision
1692 The dimension of this algebra is `2*n^2 - n`::
1694 sage: set_random_seed()
1695 sage: n_max = QuaternionHermitianEJA._max_test_case_size()
1696 sage: n = ZZ.random_element(1, n_max)
1697 sage: J = QuaternionHermitianEJA(n)
1698 sage: J.dimension() == 2*(n^2) - n
1701 The Jordan multiplication is what we think it is::
1703 sage: set_random_seed()
1704 sage: J = QuaternionHermitianEJA.random_instance()
1705 sage: x,y = J.random_elements(2)
1706 sage: actual = (x*y).natural_representation()
1707 sage: X = x.natural_representation()
1708 sage: Y = y.natural_representation()
1709 sage: expected = (X*Y + Y*X)/2
1710 sage: actual == expected
1712 sage: J(expected) == x*y
1715 We can change the generator prefix::
1717 sage: QuaternionHermitianEJA(2, prefix='a').gens()
1718 (a0, a1, a2, a3, a4, a5)
1720 Our natural basis is normalized with respect to the natural inner
1721 product unless we specify otherwise::
1723 sage: set_random_seed()
1724 sage: J = QuaternionHermitianEJA.random_instance()
1725 sage: all( b.norm() == 1 for b in J.gens() )
1728 Since our natural basis is normalized with respect to the natural
1729 inner product, and since we know that this algebra is an EJA, any
1730 left-multiplication operator's matrix will be symmetric because
1731 natural->EJA basis representation is an isometry and within the EJA
1732 the operator is self-adjoint by the Jordan axiom::
1734 sage: set_random_seed()
1735 sage: x = QuaternionHermitianEJA.random_instance().random_element()
1736 sage: x.operator().matrix().is_symmetric()
1741 def _denormalized_basis(cls
, n
, field
):
1743 Returns a basis for the space of quaternion Hermitian n-by-n matrices.
1745 Why do we embed these? Basically, because all of numerical
1746 linear algebra assumes that you're working with vectors consisting
1747 of `n` entries from a field and scalars from the same field. There's
1748 no way to tell SageMath that (for example) the vectors contain
1749 complex numbers, while the scalar field is real.
1753 sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
1757 sage: set_random_seed()
1758 sage: n = ZZ.random_element(1,5)
1759 sage: B = QuaternionHermitianEJA._denormalized_basis(n,QQ)
1760 sage: all( M.is_symmetric() for M in B )
1764 Q
= QuaternionAlgebra(QQ
,-1,-1)
1767 # This is like the symmetric case, but we need to be careful:
1769 # * We want conjugate-symmetry, not just symmetry.
1770 # * The diagonal will (as a result) be real.
1774 for j
in range(i
+1):
1775 Eij
= matrix(Q
, n
, lambda k
,l
: k
==i
and l
==j
)
1777 Sij
= cls
.real_embed(Eij
)
1780 # The second, third, and fourth ones have a minus
1781 # because they're conjugated.
1782 Sij_real
= cls
.real_embed(Eij
+ Eij
.transpose())
1784 Sij_I
= cls
.real_embed(I
*Eij
- I
*Eij
.transpose())
1786 Sij_J
= cls
.real_embed(J
*Eij
- J
*Eij
.transpose())
1788 Sij_K
= cls
.real_embed(K
*Eij
- K
*Eij
.transpose())
1791 # Since we embedded these, we can drop back to the "field" that we
1792 # started with instead of the quaternion algebra "Q".
1793 return ( s
.change_ring(field
) for s
in S
)
1796 def __init__(self
, n
, field
=AA
, **kwargs
):
1797 basis
= self
._denormalized
_basis
(n
,field
)
1798 super(QuaternionHermitianEJA
,self
).__init
__(field
, basis
, **kwargs
)
1799 self
.rank
.set_cache(n
)
1802 class BilinearFormEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1804 The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
1805 with the half-trace inner product and jordan product ``x*y =
1806 (x0*y0 + <B*x_bar,y_bar>, x0*y_bar + y0*x_bar)`` where ``B`` is a
1807 symmetric positive-definite "bilinear form" matrix. It has
1808 dimension `n` over the reals, and reduces to the ``JordanSpinEJA``
1809 when ``B`` is the identity matrix of order ``n-1``.
1813 sage: from mjo.eja.eja_algebra import (BilinearFormEJA,
1814 ....: JordanSpinEJA)
1818 When no bilinear form is specified, the identity matrix is used,
1819 and the resulting algebra is the Jordan spin algebra::
1821 sage: J0 = BilinearFormEJA(3)
1822 sage: J1 = JordanSpinEJA(3)
1823 sage: J0.multiplication_table() == J0.multiplication_table()
1828 We can create a zero-dimensional algebra::
1830 sage: J = BilinearFormEJA(0)
1834 We can check the multiplication condition given in the Jordan, von
1835 Neumann, and Wigner paper (and also discussed on my "On the
1836 symmetry..." paper). Note that this relies heavily on the standard
1837 choice of basis, as does anything utilizing the bilinear form matrix::
1839 sage: set_random_seed()
1840 sage: n = ZZ.random_element(5)
1841 sage: M = matrix.random(QQ, max(0,n-1), algorithm='unimodular')
1842 sage: B = M.transpose()*M
1843 sage: J = BilinearFormEJA(n, B=B)
1844 sage: eis = VectorSpace(M.base_ring(), M.ncols()).basis()
1845 sage: V = J.vector_space()
1846 sage: sis = [ J.from_vector(V([0] + (M.inverse()*ei).list()))
1847 ....: for ei in eis ]
1848 sage: actual = [ sis[i]*sis[j]
1849 ....: for i in range(n-1)
1850 ....: for j in range(n-1) ]
1851 sage: expected = [ J.one() if i == j else J.zero()
1852 ....: for i in range(n-1)
1853 ....: for j in range(n-1) ]
1854 sage: actual == expected
1857 def __init__(self
, n
, field
=AA
, B
=None, **kwargs
):
1859 self
._B
= matrix
.identity(field
, max(0,n
-1))
1863 V
= VectorSpace(field
, n
)
1864 mult_table
= [[V
.zero() for j
in range(n
)] for i
in range(n
)]
1873 z0
= x0
*y0
+ (self
._B
*xbar
).inner_product(ybar
)
1874 zbar
= y0
*xbar
+ x0
*ybar
1875 z
= V([z0
] + zbar
.list())
1876 mult_table
[i
][j
] = z
1878 # The rank of this algebra is two, unless we're in a
1879 # one-dimensional ambient space (because the rank is bounded
1880 # by the ambient dimension).
1881 fdeja
= super(BilinearFormEJA
, self
)
1882 fdeja
.__init
__(field
, mult_table
, **kwargs
)
1883 self
.rank
.set_cache(min(n
,2))
1885 def inner_product(self
, x
, y
):
1887 Half of the trace inner product.
1889 This is defined so that the special case of the Jordan spin
1890 algebra gets the usual inner product.
1894 sage: from mjo.eja.eja_algebra import BilinearFormEJA
1898 Ensure that this is one-half of the trace inner-product when
1899 the algebra isn't just the reals (when ``n`` isn't one). This
1900 is in Faraut and Koranyi, and also my "On the symmetry..."
1903 sage: set_random_seed()
1904 sage: n = ZZ.random_element(2,5)
1905 sage: M = matrix.random(QQ, max(0,n-1), algorithm='unimodular')
1906 sage: B = M.transpose()*M
1907 sage: J = BilinearFormEJA(n, B=B)
1908 sage: x = J.random_element()
1909 sage: y = J.random_element()
1910 sage: x.inner_product(y) == (x*y).trace()/2
1914 xvec
= x
.to_vector()
1916 yvec
= y
.to_vector()
1918 return x
[0]*y
[0] + (self
._B
*xbar
).inner_product(ybar
)
1921 class JordanSpinEJA(BilinearFormEJA
):
1923 The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
1924 with the usual inner product and jordan product ``x*y =
1925 (<x,y>, x0*y_bar + y0*x_bar)``. It has dimension `n` over
1930 sage: from mjo.eja.eja_algebra import JordanSpinEJA
1934 This multiplication table can be verified by hand::
1936 sage: J = JordanSpinEJA(4)
1937 sage: e0,e1,e2,e3 = J.gens()
1953 We can change the generator prefix::
1955 sage: JordanSpinEJA(2, prefix='B').gens()
1960 Ensure that we have the usual inner product on `R^n`::
1962 sage: set_random_seed()
1963 sage: J = JordanSpinEJA.random_instance()
1964 sage: x,y = J.random_elements(2)
1965 sage: X = x.natural_representation()
1966 sage: Y = y.natural_representation()
1967 sage: x.inner_product(y) == J.natural_inner_product(X,Y)
1971 def __init__(self
, n
, field
=AA
, **kwargs
):
1972 # This is a special case of the BilinearFormEJA with the identity
1973 # matrix as its bilinear form.
1974 return super(JordanSpinEJA
, self
).__init
__(n
, field
, **kwargs
)
1977 class TrivialEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1979 The trivial Euclidean Jordan algebra consisting of only a zero element.
1983 sage: from mjo.eja.eja_algebra import TrivialEJA
1987 sage: J = TrivialEJA()
1994 sage: 7*J.one()*12*J.one()
1996 sage: J.one().inner_product(J.one())
1998 sage: J.one().norm()
2000 sage: J.one().subalgebra_generated_by()
2001 Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
2006 def __init__(self
, field
=AA
, **kwargs
):
2008 fdeja
= super(TrivialEJA
, self
)
2009 # The rank is zero using my definition, namely the dimension of the
2010 # largest subalgebra generated by any element.
2011 fdeja
.__init
__(field
, mult_table
, **kwargs
)
2012 self
.rank
.set_cache(0)