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
)
691 # Compute an extra power in case the rank is equal to
692 # the dimension (otherwise, we would stop at x^(r-1)).
693 x_powers
= [ (L_x
**k
)*self
.one().to_vector()
694 for k
in range(n
+1) ]
695 A
= matrix
.column(F
, x_powers
[:n
])
696 AE
= A
.extended_echelon_form()
702 # The theory says that only the first "r" coefficients are
703 # nonzero, and they actually live in the original polynomial
704 # ring and not the fraction field. We negate them because
705 # in the actual characteristic polynomial, they get moved
706 # to the other side where x^r lives.
707 return -A_rref
.solve_right(E
*b
).change_ring(R
)[:r
]
712 Return the rank of this EJA.
714 This is a cached method because we know the rank a priori for
715 all of the algebras we can construct. Thus we can avoid the
716 expensive ``_charpoly_coefficients()`` call unless we truly
717 need to compute the whole characteristic polynomial.
721 sage: from mjo.eja.eja_algebra import (HadamardEJA,
723 ....: RealSymmetricEJA,
724 ....: ComplexHermitianEJA,
725 ....: QuaternionHermitianEJA,
730 The rank of the Jordan spin algebra is always two::
732 sage: JordanSpinEJA(2).rank()
734 sage: JordanSpinEJA(3).rank()
736 sage: JordanSpinEJA(4).rank()
739 The rank of the `n`-by-`n` Hermitian real, complex, or
740 quaternion matrices is `n`::
742 sage: RealSymmetricEJA(4).rank()
744 sage: ComplexHermitianEJA(3).rank()
746 sage: QuaternionHermitianEJA(2).rank()
751 Ensure that every EJA that we know how to construct has a
752 positive integer rank, unless the algebra is trivial in
753 which case its rank will be zero::
755 sage: set_random_seed()
756 sage: J = random_eja()
760 sage: r > 0 or (r == 0 and J.is_trivial())
763 Ensure that computing the rank actually works, since the ranks
764 of all simple algebras are known and will be cached by default::
766 sage: J = HadamardEJA(4)
767 sage: J.rank.clear_cache()
773 sage: J = JordanSpinEJA(4)
774 sage: J.rank.clear_cache()
780 sage: J = RealSymmetricEJA(3)
781 sage: J.rank.clear_cache()
787 sage: J = ComplexHermitianEJA(2)
788 sage: J.rank.clear_cache()
794 sage: J = QuaternionHermitianEJA(2)
795 sage: J.rank.clear_cache()
799 return len(self
._charpoly
_coefficients
())
802 def vector_space(self
):
804 Return the vector space that underlies this algebra.
808 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
812 sage: J = RealSymmetricEJA(2)
813 sage: J.vector_space()
814 Vector space of dimension 3 over...
817 return self
.zero().to_vector().parent().ambient_vector_space()
820 Element
= FiniteDimensionalEuclideanJordanAlgebraElement
823 class HadamardEJA(FiniteDimensionalEuclideanJordanAlgebra
):
825 Return the Euclidean Jordan Algebra corresponding to the set
826 `R^n` under the Hadamard product.
828 Note: this is nothing more than the Cartesian product of ``n``
829 copies of the spin algebra. Once Cartesian product algebras
830 are implemented, this can go.
834 sage: from mjo.eja.eja_algebra import HadamardEJA
838 This multiplication table can be verified by hand::
840 sage: J = HadamardEJA(3)
841 sage: e0,e1,e2 = J.gens()
857 We can change the generator prefix::
859 sage: HadamardEJA(3, prefix='r').gens()
863 def __init__(self
, n
, field
=AA
, **kwargs
):
864 V
= VectorSpace(field
, n
)
865 mult_table
= [ [ V
.gen(i
)*(i
== j
) for j
in range(n
) ]
868 fdeja
= super(HadamardEJA
, self
)
869 fdeja
.__init
__(field
, mult_table
, **kwargs
)
870 self
.rank
.set_cache(n
)
872 def inner_product(self
, x
, y
):
874 Faster to reimplement than to use natural representations.
878 sage: from mjo.eja.eja_algebra import HadamardEJA
882 Ensure that this is the usual inner product for the algebras
885 sage: set_random_seed()
886 sage: J = HadamardEJA.random_instance()
887 sage: x,y = J.random_elements(2)
888 sage: X = x.natural_representation()
889 sage: Y = y.natural_representation()
890 sage: x.inner_product(y) == J.natural_inner_product(X,Y)
894 return x
.to_vector().inner_product(y
.to_vector())
897 def random_eja(field
=AA
, nontrivial
=False):
899 Return a "random" finite-dimensional Euclidean Jordan Algebra.
903 sage: from mjo.eja.eja_algebra import random_eja
908 Euclidean Jordan algebra of dimension...
911 eja_classes
= [HadamardEJA
,
915 QuaternionHermitianEJA
]
917 eja_classes
.append(TrivialEJA
)
918 classname
= choice(eja_classes
)
919 return classname
.random_instance(field
=field
)
926 class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra
):
928 def _max_test_case_size():
929 # Play it safe, since this will be squared and the underlying
930 # field can have dimension 4 (quaternions) too.
933 def __init__(self
, field
, basis
, normalize_basis
=True, **kwargs
):
935 Compared to the superclass constructor, we take a basis instead of
936 a multiplication table because the latter can be computed in terms
937 of the former when the product is known (like it is here).
939 # Used in this class's fast _charpoly_coefficients() override.
940 self
._basis
_normalizers
= None
942 # We're going to loop through this a few times, so now's a good
943 # time to ensure that it isn't a generator expression.
946 if len(basis
) > 1 and normalize_basis
:
947 # We'll need sqrt(2) to normalize the basis, and this
948 # winds up in the multiplication table, so the whole
949 # algebra needs to be over the field extension.
950 R
= PolynomialRing(field
, 'z')
953 if p
.is_irreducible():
954 field
= field
.extension(p
, 'sqrt2', embedding
=RLF(2).sqrt())
955 basis
= tuple( s
.change_ring(field
) for s
in basis
)
956 self
._basis
_normalizers
= tuple(
957 ~
(self
.natural_inner_product(s
,s
).sqrt()) for s
in basis
)
958 basis
= tuple(s
*c
for (s
,c
) in zip(basis
,self
._basis
_normalizers
))
960 Qs
= self
.multiplication_table_from_matrix_basis(basis
)
962 fdeja
= super(MatrixEuclideanJordanAlgebra
, self
)
963 fdeja
.__init
__(field
, Qs
, natural_basis
=basis
, **kwargs
)
968 def _charpoly_coefficients(self
):
970 Override the parent method with something that tries to compute
971 over a faster (non-extension) field.
973 if self
._basis
_normalizers
is None:
974 # We didn't normalize, so assume that the basis we started
975 # with had entries in a nice field.
976 return super(MatrixEuclideanJordanAlgebra
, self
)._charpoly
_coefficients
()
978 basis
= ( (b
/n
) for (b
,n
) in zip(self
.natural_basis(),
979 self
._basis
_normalizers
) )
981 # Do this over the rationals and convert back at the end.
982 # Only works because we know the entries of the basis are
984 J
= MatrixEuclideanJordanAlgebra(QQ
,
986 normalize_basis
=False)
987 return J
._charpoly
_coefficients
()
991 def multiplication_table_from_matrix_basis(basis
):
993 At least three of the five simple Euclidean Jordan algebras have the
994 symmetric multiplication (A,B) |-> (AB + BA)/2, where the
995 multiplication on the right is matrix multiplication. Given a basis
996 for the underlying matrix space, this function returns a
997 multiplication table (obtained by looping through the basis
998 elements) for an algebra of those matrices.
1000 # In S^2, for example, we nominally have four coordinates even
1001 # though the space is of dimension three only. The vector space V
1002 # is supposed to hold the entire long vector, and the subspace W
1003 # of V will be spanned by the vectors that arise from symmetric
1004 # matrices. Thus for S^2, dim(V) == 4 and dim(W) == 3.
1005 field
= basis
[0].base_ring()
1006 dimension
= basis
[0].nrows()
1008 V
= VectorSpace(field
, dimension
**2)
1009 W
= V
.span_of_basis( _mat2vec(s
) for s
in basis
)
1011 mult_table
= [[W
.zero() for j
in range(n
)] for i
in range(n
)]
1014 mat_entry
= (basis
[i
]*basis
[j
] + basis
[j
]*basis
[i
])/2
1015 mult_table
[i
][j
] = W
.coordinate_vector(_mat2vec(mat_entry
))
1023 Embed the matrix ``M`` into a space of real matrices.
1025 The matrix ``M`` can have entries in any field at the moment:
1026 the real numbers, complex numbers, or quaternions. And although
1027 they are not a field, we can probably support octonions at some
1028 point, too. This function returns a real matrix that "acts like"
1029 the original with respect to matrix multiplication; i.e.
1031 real_embed(M*N) = real_embed(M)*real_embed(N)
1034 raise NotImplementedError
1038 def real_unembed(M
):
1040 The inverse of :meth:`real_embed`.
1042 raise NotImplementedError
1046 def natural_inner_product(cls
,X
,Y
):
1047 Xu
= cls
.real_unembed(X
)
1048 Yu
= cls
.real_unembed(Y
)
1049 tr
= (Xu
*Yu
).trace()
1052 # It's real already.
1055 # Otherwise, try the thing that works for complex numbers; and
1056 # if that doesn't work, the thing that works for quaternions.
1058 return tr
.vector()[0] # real part, imag part is index 1
1059 except AttributeError:
1060 # A quaternions doesn't have a vector() method, but does
1061 # have coefficient_tuple() method that returns the
1062 # coefficients of 1, i, j, and k -- in that order.
1063 return tr
.coefficient_tuple()[0]
1066 class RealMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra
):
1070 The identity function, for embedding real matrices into real
1076 def real_unembed(M
):
1078 The identity function, for unembedding real matrices from real
1084 class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra
):
1086 The rank-n simple EJA consisting of real symmetric n-by-n
1087 matrices, the usual symmetric Jordan product, and the trace inner
1088 product. It has dimension `(n^2 + n)/2` over the reals.
1092 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
1096 sage: J = RealSymmetricEJA(2)
1097 sage: e0, e1, e2 = J.gens()
1105 In theory, our "field" can be any subfield of the reals::
1107 sage: RealSymmetricEJA(2, RDF)
1108 Euclidean Jordan algebra of dimension 3 over Real Double Field
1109 sage: RealSymmetricEJA(2, RR)
1110 Euclidean Jordan algebra of dimension 3 over Real Field with
1111 53 bits of precision
1115 The dimension of this algebra is `(n^2 + n) / 2`::
1117 sage: set_random_seed()
1118 sage: n_max = RealSymmetricEJA._max_test_case_size()
1119 sage: n = ZZ.random_element(1, n_max)
1120 sage: J = RealSymmetricEJA(n)
1121 sage: J.dimension() == (n^2 + n)/2
1124 The Jordan multiplication is what we think it is::
1126 sage: set_random_seed()
1127 sage: J = RealSymmetricEJA.random_instance()
1128 sage: x,y = J.random_elements(2)
1129 sage: actual = (x*y).natural_representation()
1130 sage: X = x.natural_representation()
1131 sage: Y = y.natural_representation()
1132 sage: expected = (X*Y + Y*X)/2
1133 sage: actual == expected
1135 sage: J(expected) == x*y
1138 We can change the generator prefix::
1140 sage: RealSymmetricEJA(3, prefix='q').gens()
1141 (q0, q1, q2, q3, q4, q5)
1143 Our natural basis is normalized with respect to the natural inner
1144 product unless we specify otherwise::
1146 sage: set_random_seed()
1147 sage: J = RealSymmetricEJA.random_instance()
1148 sage: all( b.norm() == 1 for b in J.gens() )
1151 Since our natural basis is normalized with respect to the natural
1152 inner product, and since we know that this algebra is an EJA, any
1153 left-multiplication operator's matrix will be symmetric because
1154 natural->EJA basis representation is an isometry and within the EJA
1155 the operator is self-adjoint by the Jordan axiom::
1157 sage: set_random_seed()
1158 sage: x = RealSymmetricEJA.random_instance().random_element()
1159 sage: x.operator().matrix().is_symmetric()
1164 def _denormalized_basis(cls
, n
, field
):
1166 Return a basis for the space of real symmetric n-by-n matrices.
1170 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
1174 sage: set_random_seed()
1175 sage: n = ZZ.random_element(1,5)
1176 sage: B = RealSymmetricEJA._denormalized_basis(n,QQ)
1177 sage: all( M.is_symmetric() for M in B)
1181 # The basis of symmetric matrices, as matrices, in their R^(n-by-n)
1185 for j
in range(i
+1):
1186 Eij
= matrix(field
, n
, lambda k
,l
: k
==i
and l
==j
)
1190 Sij
= Eij
+ Eij
.transpose()
1196 def _max_test_case_size():
1197 return 4 # Dimension 10
1200 def __init__(self
, n
, field
=AA
, **kwargs
):
1201 basis
= self
._denormalized
_basis
(n
, field
)
1202 super(RealSymmetricEJA
, self
).__init
__(field
, basis
, **kwargs
)
1203 self
.rank
.set_cache(n
)
1206 class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra
):
1210 Embed the n-by-n complex matrix ``M`` into the space of real
1211 matrices of size 2n-by-2n via the map the sends each entry `z = a +
1212 bi` to the block matrix ``[[a,b],[-b,a]]``.
1216 sage: from mjo.eja.eja_algebra import \
1217 ....: ComplexMatrixEuclideanJordanAlgebra
1221 sage: F = QuadraticField(-1, 'I')
1222 sage: x1 = F(4 - 2*i)
1223 sage: x2 = F(1 + 2*i)
1226 sage: M = matrix(F,2,[[x1,x2],[x3,x4]])
1227 sage: ComplexMatrixEuclideanJordanAlgebra.real_embed(M)
1236 Embedding is a homomorphism (isomorphism, in fact)::
1238 sage: set_random_seed()
1239 sage: n_max = ComplexMatrixEuclideanJordanAlgebra._max_test_case_size()
1240 sage: n = ZZ.random_element(n_max)
1241 sage: F = QuadraticField(-1, 'I')
1242 sage: X = random_matrix(F, n)
1243 sage: Y = random_matrix(F, n)
1244 sage: Xe = ComplexMatrixEuclideanJordanAlgebra.real_embed(X)
1245 sage: Ye = ComplexMatrixEuclideanJordanAlgebra.real_embed(Y)
1246 sage: XYe = ComplexMatrixEuclideanJordanAlgebra.real_embed(X*Y)
1253 raise ValueError("the matrix 'M' must be square")
1255 # We don't need any adjoined elements...
1256 field
= M
.base_ring().base_ring()
1260 a
= z
.list()[0] # real part, I guess
1261 b
= z
.list()[1] # imag part, I guess
1262 blocks
.append(matrix(field
, 2, [[a
,b
],[-b
,a
]]))
1264 return matrix
.block(field
, n
, blocks
)
1268 def real_unembed(M
):
1270 The inverse of _embed_complex_matrix().
1274 sage: from mjo.eja.eja_algebra import \
1275 ....: ComplexMatrixEuclideanJordanAlgebra
1279 sage: A = matrix(QQ,[ [ 1, 2, 3, 4],
1280 ....: [-2, 1, -4, 3],
1281 ....: [ 9, 10, 11, 12],
1282 ....: [-10, 9, -12, 11] ])
1283 sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(A)
1285 [ 10*I + 9 12*I + 11]
1289 Unembedding is the inverse of embedding::
1291 sage: set_random_seed()
1292 sage: F = QuadraticField(-1, 'I')
1293 sage: M = random_matrix(F, 3)
1294 sage: Me = ComplexMatrixEuclideanJordanAlgebra.real_embed(M)
1295 sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(Me) == M
1301 raise ValueError("the matrix 'M' must be square")
1302 if not n
.mod(2).is_zero():
1303 raise ValueError("the matrix 'M' must be a complex embedding")
1305 # If "M" was normalized, its base ring might have roots
1306 # adjoined and they can stick around after unembedding.
1307 field
= M
.base_ring()
1308 R
= PolynomialRing(field
, 'z')
1311 # Sage doesn't know how to embed AA into QQbar, i.e. how
1312 # to adjoin sqrt(-1) to AA.
1315 F
= field
.extension(z
**2 + 1, 'I', embedding
=CLF(-1).sqrt())
1318 # Go top-left to bottom-right (reading order), converting every
1319 # 2-by-2 block we see to a single complex element.
1321 for k
in range(n
/2):
1322 for j
in range(n
/2):
1323 submat
= M
[2*k
:2*k
+2,2*j
:2*j
+2]
1324 if submat
[0,0] != submat
[1,1]:
1325 raise ValueError('bad on-diagonal submatrix')
1326 if submat
[0,1] != -submat
[1,0]:
1327 raise ValueError('bad off-diagonal submatrix')
1328 z
= submat
[0,0] + submat
[0,1]*i
1331 return matrix(F
, n
/2, elements
)
1335 def natural_inner_product(cls
,X
,Y
):
1337 Compute a natural inner product in this algebra directly from
1342 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
1346 This gives the same answer as the slow, default method implemented
1347 in :class:`MatrixEuclideanJordanAlgebra`::
1349 sage: set_random_seed()
1350 sage: J = ComplexHermitianEJA.random_instance()
1351 sage: x,y = J.random_elements(2)
1352 sage: Xe = x.natural_representation()
1353 sage: Ye = y.natural_representation()
1354 sage: X = ComplexHermitianEJA.real_unembed(Xe)
1355 sage: Y = ComplexHermitianEJA.real_unembed(Ye)
1356 sage: expected = (X*Y).trace().real()
1357 sage: actual = ComplexHermitianEJA.natural_inner_product(Xe,Ye)
1358 sage: actual == expected
1362 return RealMatrixEuclideanJordanAlgebra
.natural_inner_product(X
,Y
)/2
1365 class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra
):
1367 The rank-n simple EJA consisting of complex Hermitian n-by-n
1368 matrices over the real numbers, the usual symmetric Jordan product,
1369 and the real-part-of-trace inner product. It has dimension `n^2` over
1374 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
1378 In theory, our "field" can be any subfield of the reals::
1380 sage: ComplexHermitianEJA(2, RDF)
1381 Euclidean Jordan algebra of dimension 4 over Real Double Field
1382 sage: ComplexHermitianEJA(2, RR)
1383 Euclidean Jordan algebra of dimension 4 over Real Field with
1384 53 bits of precision
1388 The dimension of this algebra is `n^2`::
1390 sage: set_random_seed()
1391 sage: n_max = ComplexHermitianEJA._max_test_case_size()
1392 sage: n = ZZ.random_element(1, n_max)
1393 sage: J = ComplexHermitianEJA(n)
1394 sage: J.dimension() == n^2
1397 The Jordan multiplication is what we think it is::
1399 sage: set_random_seed()
1400 sage: J = ComplexHermitianEJA.random_instance()
1401 sage: x,y = J.random_elements(2)
1402 sage: actual = (x*y).natural_representation()
1403 sage: X = x.natural_representation()
1404 sage: Y = y.natural_representation()
1405 sage: expected = (X*Y + Y*X)/2
1406 sage: actual == expected
1408 sage: J(expected) == x*y
1411 We can change the generator prefix::
1413 sage: ComplexHermitianEJA(2, prefix='z').gens()
1416 Our natural basis is normalized with respect to the natural inner
1417 product unless we specify otherwise::
1419 sage: set_random_seed()
1420 sage: J = ComplexHermitianEJA.random_instance()
1421 sage: all( b.norm() == 1 for b in J.gens() )
1424 Since our natural basis is normalized with respect to the natural
1425 inner product, and since we know that this algebra is an EJA, any
1426 left-multiplication operator's matrix will be symmetric because
1427 natural->EJA basis representation is an isometry and within the EJA
1428 the operator is self-adjoint by the Jordan axiom::
1430 sage: set_random_seed()
1431 sage: x = ComplexHermitianEJA.random_instance().random_element()
1432 sage: x.operator().matrix().is_symmetric()
1438 def _denormalized_basis(cls
, n
, field
):
1440 Returns a basis for the space of complex Hermitian n-by-n matrices.
1442 Why do we embed these? Basically, because all of numerical linear
1443 algebra assumes that you're working with vectors consisting of `n`
1444 entries from a field and scalars from the same field. There's no way
1445 to tell SageMath that (for example) the vectors contain complex
1446 numbers, while the scalar field is real.
1450 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
1454 sage: set_random_seed()
1455 sage: n = ZZ.random_element(1,5)
1456 sage: field = QuadraticField(2, 'sqrt2')
1457 sage: B = ComplexHermitianEJA._denormalized_basis(n, field)
1458 sage: all( M.is_symmetric() for M in B)
1462 R
= PolynomialRing(field
, 'z')
1464 F
= field
.extension(z
**2 + 1, 'I')
1467 # This is like the symmetric case, but we need to be careful:
1469 # * We want conjugate-symmetry, not just symmetry.
1470 # * The diagonal will (as a result) be real.
1474 for j
in range(i
+1):
1475 Eij
= matrix(F
, n
, lambda k
,l
: k
==i
and l
==j
)
1477 Sij
= cls
.real_embed(Eij
)
1480 # The second one has a minus because it's conjugated.
1481 Sij_real
= cls
.real_embed(Eij
+ Eij
.transpose())
1483 Sij_imag
= cls
.real_embed(I
*Eij
- I
*Eij
.transpose())
1486 # Since we embedded these, we can drop back to the "field" that we
1487 # started with instead of the complex extension "F".
1488 return ( s
.change_ring(field
) for s
in S
)
1491 def __init__(self
, n
, field
=AA
, **kwargs
):
1492 basis
= self
._denormalized
_basis
(n
,field
)
1493 super(ComplexHermitianEJA
,self
).__init
__(field
, basis
, **kwargs
)
1494 self
.rank
.set_cache(n
)
1497 class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra
):
1501 Embed the n-by-n quaternion matrix ``M`` into the space of real
1502 matrices of size 4n-by-4n by first sending each quaternion entry `z
1503 = a + bi + cj + dk` to the block-complex matrix ``[[a + bi,
1504 c+di],[-c + di, a-bi]]`, and then embedding those into a real
1509 sage: from mjo.eja.eja_algebra import \
1510 ....: QuaternionMatrixEuclideanJordanAlgebra
1514 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1515 sage: i,j,k = Q.gens()
1516 sage: x = 1 + 2*i + 3*j + 4*k
1517 sage: M = matrix(Q, 1, [[x]])
1518 sage: QuaternionMatrixEuclideanJordanAlgebra.real_embed(M)
1524 Embedding is a homomorphism (isomorphism, in fact)::
1526 sage: set_random_seed()
1527 sage: n_max = QuaternionMatrixEuclideanJordanAlgebra._max_test_case_size()
1528 sage: n = ZZ.random_element(n_max)
1529 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1530 sage: X = random_matrix(Q, n)
1531 sage: Y = random_matrix(Q, n)
1532 sage: Xe = QuaternionMatrixEuclideanJordanAlgebra.real_embed(X)
1533 sage: Ye = QuaternionMatrixEuclideanJordanAlgebra.real_embed(Y)
1534 sage: XYe = QuaternionMatrixEuclideanJordanAlgebra.real_embed(X*Y)
1539 quaternions
= M
.base_ring()
1542 raise ValueError("the matrix 'M' must be square")
1544 F
= QuadraticField(-1, 'I')
1549 t
= z
.coefficient_tuple()
1554 cplxM
= matrix(F
, 2, [[ a
+ b
*i
, c
+ d
*i
],
1555 [-c
+ d
*i
, a
- b
*i
]])
1556 realM
= ComplexMatrixEuclideanJordanAlgebra
.real_embed(cplxM
)
1557 blocks
.append(realM
)
1559 # We should have real entries by now, so use the realest field
1560 # we've got for the return value.
1561 return matrix
.block(quaternions
.base_ring(), n
, blocks
)
1566 def real_unembed(M
):
1568 The inverse of _embed_quaternion_matrix().
1572 sage: from mjo.eja.eja_algebra import \
1573 ....: QuaternionMatrixEuclideanJordanAlgebra
1577 sage: M = matrix(QQ, [[ 1, 2, 3, 4],
1578 ....: [-2, 1, -4, 3],
1579 ....: [-3, 4, 1, -2],
1580 ....: [-4, -3, 2, 1]])
1581 sage: QuaternionMatrixEuclideanJordanAlgebra.real_unembed(M)
1582 [1 + 2*i + 3*j + 4*k]
1586 Unembedding is the inverse of embedding::
1588 sage: set_random_seed()
1589 sage: Q = QuaternionAlgebra(QQ, -1, -1)
1590 sage: M = random_matrix(Q, 3)
1591 sage: Me = QuaternionMatrixEuclideanJordanAlgebra.real_embed(M)
1592 sage: QuaternionMatrixEuclideanJordanAlgebra.real_unembed(Me) == M
1598 raise ValueError("the matrix 'M' must be square")
1599 if not n
.mod(4).is_zero():
1600 raise ValueError("the matrix 'M' must be a quaternion embedding")
1602 # Use the base ring of the matrix to ensure that its entries can be
1603 # multiplied by elements of the quaternion algebra.
1604 field
= M
.base_ring()
1605 Q
= QuaternionAlgebra(field
,-1,-1)
1608 # Go top-left to bottom-right (reading order), converting every
1609 # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1
1612 for l
in range(n
/4):
1613 for m
in range(n
/4):
1614 submat
= ComplexMatrixEuclideanJordanAlgebra
.real_unembed(
1615 M
[4*l
:4*l
+4,4*m
:4*m
+4] )
1616 if submat
[0,0] != submat
[1,1].conjugate():
1617 raise ValueError('bad on-diagonal submatrix')
1618 if submat
[0,1] != -submat
[1,0].conjugate():
1619 raise ValueError('bad off-diagonal submatrix')
1620 z
= submat
[0,0].real()
1621 z
+= submat
[0,0].imag()*i
1622 z
+= submat
[0,1].real()*j
1623 z
+= submat
[0,1].imag()*k
1626 return matrix(Q
, n
/4, elements
)
1630 def natural_inner_product(cls
,X
,Y
):
1632 Compute a natural inner product in this algebra directly from
1637 sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
1641 This gives the same answer as the slow, default method implemented
1642 in :class:`MatrixEuclideanJordanAlgebra`::
1644 sage: set_random_seed()
1645 sage: J = QuaternionHermitianEJA.random_instance()
1646 sage: x,y = J.random_elements(2)
1647 sage: Xe = x.natural_representation()
1648 sage: Ye = y.natural_representation()
1649 sage: X = QuaternionHermitianEJA.real_unembed(Xe)
1650 sage: Y = QuaternionHermitianEJA.real_unembed(Ye)
1651 sage: expected = (X*Y).trace().coefficient_tuple()[0]
1652 sage: actual = QuaternionHermitianEJA.natural_inner_product(Xe,Ye)
1653 sage: actual == expected
1657 return RealMatrixEuclideanJordanAlgebra
.natural_inner_product(X
,Y
)/4
1660 class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra
):
1662 The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
1663 matrices, the usual symmetric Jordan product, and the
1664 real-part-of-trace inner product. It has dimension `2n^2 - n` over
1669 sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
1673 In theory, our "field" can be any subfield of the reals::
1675 sage: QuaternionHermitianEJA(2, RDF)
1676 Euclidean Jordan algebra of dimension 6 over Real Double Field
1677 sage: QuaternionHermitianEJA(2, RR)
1678 Euclidean Jordan algebra of dimension 6 over Real Field with
1679 53 bits of precision
1683 The dimension of this algebra is `2*n^2 - n`::
1685 sage: set_random_seed()
1686 sage: n_max = QuaternionHermitianEJA._max_test_case_size()
1687 sage: n = ZZ.random_element(1, n_max)
1688 sage: J = QuaternionHermitianEJA(n)
1689 sage: J.dimension() == 2*(n^2) - n
1692 The Jordan multiplication is what we think it is::
1694 sage: set_random_seed()
1695 sage: J = QuaternionHermitianEJA.random_instance()
1696 sage: x,y = J.random_elements(2)
1697 sage: actual = (x*y).natural_representation()
1698 sage: X = x.natural_representation()
1699 sage: Y = y.natural_representation()
1700 sage: expected = (X*Y + Y*X)/2
1701 sage: actual == expected
1703 sage: J(expected) == x*y
1706 We can change the generator prefix::
1708 sage: QuaternionHermitianEJA(2, prefix='a').gens()
1709 (a0, a1, a2, a3, a4, a5)
1711 Our natural basis is normalized with respect to the natural inner
1712 product unless we specify otherwise::
1714 sage: set_random_seed()
1715 sage: J = QuaternionHermitianEJA.random_instance()
1716 sage: all( b.norm() == 1 for b in J.gens() )
1719 Since our natural basis is normalized with respect to the natural
1720 inner product, and since we know that this algebra is an EJA, any
1721 left-multiplication operator's matrix will be symmetric because
1722 natural->EJA basis representation is an isometry and within the EJA
1723 the operator is self-adjoint by the Jordan axiom::
1725 sage: set_random_seed()
1726 sage: x = QuaternionHermitianEJA.random_instance().random_element()
1727 sage: x.operator().matrix().is_symmetric()
1732 def _denormalized_basis(cls
, n
, field
):
1734 Returns a basis for the space of quaternion Hermitian n-by-n matrices.
1736 Why do we embed these? Basically, because all of numerical
1737 linear algebra assumes that you're working with vectors consisting
1738 of `n` entries from a field and scalars from the same field. There's
1739 no way to tell SageMath that (for example) the vectors contain
1740 complex numbers, while the scalar field is real.
1744 sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
1748 sage: set_random_seed()
1749 sage: n = ZZ.random_element(1,5)
1750 sage: B = QuaternionHermitianEJA._denormalized_basis(n,QQ)
1751 sage: all( M.is_symmetric() for M in B )
1755 Q
= QuaternionAlgebra(QQ
,-1,-1)
1758 # This is like the symmetric case, but we need to be careful:
1760 # * We want conjugate-symmetry, not just symmetry.
1761 # * The diagonal will (as a result) be real.
1765 for j
in range(i
+1):
1766 Eij
= matrix(Q
, n
, lambda k
,l
: k
==i
and l
==j
)
1768 Sij
= cls
.real_embed(Eij
)
1771 # The second, third, and fourth ones have a minus
1772 # because they're conjugated.
1773 Sij_real
= cls
.real_embed(Eij
+ Eij
.transpose())
1775 Sij_I
= cls
.real_embed(I
*Eij
- I
*Eij
.transpose())
1777 Sij_J
= cls
.real_embed(J
*Eij
- J
*Eij
.transpose())
1779 Sij_K
= cls
.real_embed(K
*Eij
- K
*Eij
.transpose())
1782 # Since we embedded these, we can drop back to the "field" that we
1783 # started with instead of the quaternion algebra "Q".
1784 return ( s
.change_ring(field
) for s
in S
)
1787 def __init__(self
, n
, field
=AA
, **kwargs
):
1788 basis
= self
._denormalized
_basis
(n
,field
)
1789 super(QuaternionHermitianEJA
,self
).__init
__(field
, basis
, **kwargs
)
1790 self
.rank
.set_cache(n
)
1793 class BilinearFormEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1795 The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
1796 with the half-trace inner product and jordan product ``x*y =
1797 (x0*y0 + <B*x_bar,y_bar>, x0*y_bar + y0*x_bar)`` where ``B`` is a
1798 symmetric positive-definite "bilinear form" matrix. It has
1799 dimension `n` over the reals, and reduces to the ``JordanSpinEJA``
1800 when ``B`` is the identity matrix of order ``n-1``.
1804 sage: from mjo.eja.eja_algebra import (BilinearFormEJA,
1805 ....: JordanSpinEJA)
1809 When no bilinear form is specified, the identity matrix is used,
1810 and the resulting algebra is the Jordan spin algebra::
1812 sage: J0 = BilinearFormEJA(3)
1813 sage: J1 = JordanSpinEJA(3)
1814 sage: J0.multiplication_table() == J0.multiplication_table()
1819 We can create a zero-dimensional algebra::
1821 sage: J = BilinearFormEJA(0)
1825 We can check the multiplication condition given in the Jordan, von
1826 Neumann, and Wigner paper (and also discussed on my "On the
1827 symmetry..." paper). Note that this relies heavily on the standard
1828 choice of basis, as does anything utilizing the bilinear form matrix::
1830 sage: set_random_seed()
1831 sage: n = ZZ.random_element(5)
1832 sage: M = matrix.random(QQ, max(0,n-1), algorithm='unimodular')
1833 sage: B = M.transpose()*M
1834 sage: J = BilinearFormEJA(n, B=B)
1835 sage: eis = VectorSpace(M.base_ring(), M.ncols()).basis()
1836 sage: V = J.vector_space()
1837 sage: sis = [ J.from_vector(V([0] + (M.inverse()*ei).list()))
1838 ....: for ei in eis ]
1839 sage: actual = [ sis[i]*sis[j]
1840 ....: for i in range(n-1)
1841 ....: for j in range(n-1) ]
1842 sage: expected = [ J.one() if i == j else J.zero()
1843 ....: for i in range(n-1)
1844 ....: for j in range(n-1) ]
1845 sage: actual == expected
1848 def __init__(self
, n
, field
=AA
, B
=None, **kwargs
):
1850 self
._B
= matrix
.identity(field
, max(0,n
-1))
1854 V
= VectorSpace(field
, n
)
1855 mult_table
= [[V
.zero() for j
in range(n
)] for i
in range(n
)]
1864 z0
= x0
*y0
+ (self
._B
*xbar
).inner_product(ybar
)
1865 zbar
= y0
*xbar
+ x0
*ybar
1866 z
= V([z0
] + zbar
.list())
1867 mult_table
[i
][j
] = z
1869 # The rank of this algebra is two, unless we're in a
1870 # one-dimensional ambient space (because the rank is bounded
1871 # by the ambient dimension).
1872 fdeja
= super(BilinearFormEJA
, self
)
1873 fdeja
.__init
__(field
, mult_table
, **kwargs
)
1874 self
.rank
.set_cache(min(n
,2))
1876 def inner_product(self
, x
, y
):
1878 Half of the trace inner product.
1880 This is defined so that the special case of the Jordan spin
1881 algebra gets the usual inner product.
1885 sage: from mjo.eja.eja_algebra import BilinearFormEJA
1889 Ensure that this is one-half of the trace inner-product when
1890 the algebra isn't just the reals (when ``n`` isn't one). This
1891 is in Faraut and Koranyi, and also my "On the symmetry..."
1894 sage: set_random_seed()
1895 sage: n = ZZ.random_element(2,5)
1896 sage: M = matrix.random(QQ, max(0,n-1), algorithm='unimodular')
1897 sage: B = M.transpose()*M
1898 sage: J = BilinearFormEJA(n, B=B)
1899 sage: x = J.random_element()
1900 sage: y = J.random_element()
1901 sage: x.inner_product(y) == (x*y).trace()/2
1905 xvec
= x
.to_vector()
1907 yvec
= y
.to_vector()
1909 return x
[0]*y
[0] + (self
._B
*xbar
).inner_product(ybar
)
1912 class JordanSpinEJA(BilinearFormEJA
):
1914 The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
1915 with the usual inner product and jordan product ``x*y =
1916 (<x,y>, x0*y_bar + y0*x_bar)``. It has dimension `n` over
1921 sage: from mjo.eja.eja_algebra import JordanSpinEJA
1925 This multiplication table can be verified by hand::
1927 sage: J = JordanSpinEJA(4)
1928 sage: e0,e1,e2,e3 = J.gens()
1944 We can change the generator prefix::
1946 sage: JordanSpinEJA(2, prefix='B').gens()
1951 Ensure that we have the usual inner product on `R^n`::
1953 sage: set_random_seed()
1954 sage: J = JordanSpinEJA.random_instance()
1955 sage: x,y = J.random_elements(2)
1956 sage: X = x.natural_representation()
1957 sage: Y = y.natural_representation()
1958 sage: x.inner_product(y) == J.natural_inner_product(X,Y)
1962 def __init__(self
, n
, field
=AA
, **kwargs
):
1963 # This is a special case of the BilinearFormEJA with the identity
1964 # matrix as its bilinear form.
1965 return super(JordanSpinEJA
, self
).__init
__(n
, field
, **kwargs
)
1968 class TrivialEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1970 The trivial Euclidean Jordan algebra consisting of only a zero element.
1974 sage: from mjo.eja.eja_algebra import TrivialEJA
1978 sage: J = TrivialEJA()
1985 sage: 7*J.one()*12*J.one()
1987 sage: J.one().inner_product(J.one())
1989 sage: J.one().norm()
1991 sage: J.one().subalgebra_generated_by()
1992 Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
1997 def __init__(self
, field
=AA
, **kwargs
):
1999 fdeja
= super(TrivialEJA
, self
)
2000 # The rank is zero using my definition, namely the dimension of the
2001 # largest subalgebra generated by any element.
2002 fdeja
.__init
__(field
, mult_table
, **kwargs
)
2003 self
.rank
.set_cache(0)