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
65 sage: from mjo.eja.eja_algebra import (JordanSpinEJA, random_eja)
69 By definition, Jordan multiplication commutes::
71 sage: set_random_seed()
72 sage: J = random_eja()
73 sage: x,y = J.random_elements(2)
79 The ``field`` we're given must be real::
81 sage: JordanSpinEJA(2,QQbar)
82 Traceback (most recent call last):
84 ValueError: field is not real
88 if not field
.is_subring(RR
):
89 # Note: this does return true for the real algebraic
90 # field, and any quadratic field where we've specified
92 raise ValueError('field is not real')
95 self
._natural
_basis
= natural_basis
98 category
= MagmaticAlgebras(field
).FiniteDimensional()
99 category
= category
.WithBasis().Unital()
101 fda
= super(FiniteDimensionalEuclideanJordanAlgebra
, self
)
103 range(len(mult_table
)),
106 self
.print_options(bracket
='')
108 # The multiplication table we're given is necessarily in terms
109 # of vectors, because we don't have an algebra yet for
110 # anything to be an element of. However, it's faster in the
111 # long run to have the multiplication table be in terms of
112 # algebra elements. We do this after calling the superclass
113 # constructor so that from_vector() knows what to do.
114 self
._multiplication
_table
= [
115 list(map(lambda x
: self
.from_vector(x
), ls
))
120 def _element_constructor_(self
, elt
):
122 Construct an element of this algebra from its natural
125 This gets called only after the parent element _call_ method
126 fails to find a coercion for the argument.
130 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
132 ....: RealSymmetricEJA)
136 The identity in `S^n` is converted to the identity in the EJA::
138 sage: J = RealSymmetricEJA(3)
139 sage: I = matrix.identity(QQ,3)
140 sage: J(I) == J.one()
143 This skew-symmetric matrix can't be represented in the EJA::
145 sage: J = RealSymmetricEJA(3)
146 sage: A = matrix(QQ,3, lambda i,j: i-j)
148 Traceback (most recent call last):
150 ArithmeticError: vector is not in free module
154 Ensure that we can convert any element of the two non-matrix
155 simple algebras (whose natural representations are their usual
156 vector representations) back and forth faithfully::
158 sage: set_random_seed()
159 sage: J = HadamardEJA.random_instance()
160 sage: x = J.random_element()
161 sage: J(x.to_vector().column()) == x
163 sage: J = JordanSpinEJA.random_instance()
164 sage: x = J.random_element()
165 sage: J(x.to_vector().column()) == x
169 msg
= "not a naturally-represented algebra element"
171 # The superclass implementation of random_element()
172 # needs to be able to coerce "0" into the algebra.
174 elif elt
in self
.base_ring():
175 # Ensure that no base ring -> algebra coercion is performed
176 # by this method. There's some stupidity in sage that would
177 # otherwise propagate to this method; for example, sage thinks
178 # that the integer 3 belongs to the space of 2-by-2 matrices.
179 raise ValueError(msg
)
181 natural_basis
= self
.natural_basis()
182 basis_space
= natural_basis
[0].matrix_space()
183 if elt
not in basis_space
:
184 raise ValueError(msg
)
186 # Thanks for nothing! Matrix spaces aren't vector spaces in
187 # Sage, so we have to figure out its natural-basis coordinates
188 # ourselves. We use the basis space's ring instead of the
189 # element's ring because the basis space might be an algebraic
190 # closure whereas the base ring of the 3-by-3 identity matrix
191 # could be QQ instead of QQbar.
192 V
= VectorSpace(basis_space
.base_ring(), elt
.nrows()*elt
.ncols())
193 W
= V
.span_of_basis( _mat2vec(s
) for s
in natural_basis
)
194 coords
= W
.coordinate_vector(_mat2vec(elt
))
195 return self
.from_vector(coords
)
200 Return a string representation of ``self``.
204 sage: from mjo.eja.eja_algebra import JordanSpinEJA
208 Ensure that it says what we think it says::
210 sage: JordanSpinEJA(2, field=AA)
211 Euclidean Jordan algebra of dimension 2 over Algebraic Real Field
212 sage: JordanSpinEJA(3, field=RDF)
213 Euclidean Jordan algebra of dimension 3 over Real Double Field
216 fmt
= "Euclidean Jordan algebra of dimension {} over {}"
217 return fmt
.format(self
.dimension(), self
.base_ring())
219 def product_on_basis(self
, i
, j
):
220 return self
._multiplication
_table
[i
][j
]
222 def _a_regular_element(self
):
224 Guess a regular element. Needed to compute the basis for our
225 characteristic polynomial coefficients.
229 sage: from mjo.eja.eja_algebra import random_eja
233 Ensure that this hacky method succeeds for every algebra that we
234 know how to construct::
236 sage: set_random_seed()
237 sage: J = random_eja()
238 sage: J._a_regular_element().is_regular()
243 z
= self
.sum( (i
+1)*gs
[i
] for i
in range(len(gs
)) )
244 if not z
.is_regular():
245 raise ValueError("don't know a regular element")
250 def _charpoly_basis_space(self
):
252 Return the vector space spanned by the basis used in our
253 characteristic polynomial coefficients. This is used not only to
254 compute those coefficients, but also any time we need to
255 evaluate the coefficients (like when we compute the trace or
258 z
= self
._a
_regular
_element
()
259 # Don't use the parent vector space directly here in case this
260 # happens to be a subalgebra. In that case, we would be e.g.
261 # two-dimensional but span_of_basis() would expect three
263 V
= VectorSpace(self
.base_ring(), self
.vector_space().dimension())
264 basis
= [ (z
**k
).to_vector() for k
in range(self
.rank()) ]
265 V1
= V
.span_of_basis( basis
)
266 b
= (V1
.basis() + V1
.complement().basis())
267 return V
.span_of_basis(b
)
272 def _charpoly_coeff(self
, i
):
274 Return the coefficient polynomial "a_{i}" of this algebra's
275 general characteristic polynomial.
277 Having this be a separate cached method lets us compute and
278 store the trace/determinant (a_{r-1} and a_{0} respectively)
279 separate from the entire characteristic polynomial.
281 (A_of_x
, x
, xr
, detA
) = self
._charpoly
_matrix
_system
()
282 R
= A_of_x
.base_ring()
287 # Guaranteed by theory
290 # Danger: the in-place modification is done for performance
291 # reasons (reconstructing a matrix with huge polynomial
292 # entries is slow), but I don't know how cached_method works,
293 # so it's highly possible that we're modifying some global
294 # list variable by reference, here. In other words, you
295 # probably shouldn't call this method twice on the same
296 # algebra, at the same time, in two threads
297 Ai_orig
= A_of_x
.column(i
)
298 A_of_x
.set_column(i
,xr
)
299 numerator
= A_of_x
.det()
300 A_of_x
.set_column(i
,Ai_orig
)
302 # We're relying on the theory here to ensure that each a_i is
303 # indeed back in R, and the added negative signs are to make
304 # the whole charpoly expression sum to zero.
305 return R(-numerator
/detA
)
309 def _charpoly_matrix_system(self
):
311 Compute the matrix whose entries A_ij are polynomials in
312 X1,...,XN, the vector ``x`` of variables X1,...,XN, the vector
313 corresponding to `x^r` and the determinent of the matrix A =
314 [A_ij]. In other words, all of the fixed (cachable) data needed
315 to compute the coefficients of the characteristic polynomial.
320 # Turn my vector space into a module so that "vectors" can
321 # have multivatiate polynomial entries.
322 names
= tuple('X' + str(i
) for i
in range(1,n
+1))
323 R
= PolynomialRing(self
.base_ring(), names
)
325 # Using change_ring() on the parent's vector space doesn't work
326 # here because, in a subalgebra, that vector space has a basis
327 # and change_ring() tries to bring the basis along with it. And
328 # that doesn't work unless the new ring is a PID, which it usually
332 # Now let x = (X1,X2,...,Xn) be the vector whose entries are
336 # And figure out the "left multiplication by x" matrix in
339 monomial_matrices
= [ self
.monomial(i
).operator().matrix()
340 for i
in range(n
) ] # don't recompute these!
342 ek
= self
.monomial(k
).to_vector()
344 sum( x
[i
]*(monomial_matrices
[i
]*ek
)
345 for i
in range(n
) ) )
346 Lx
= matrix
.column(R
, lmbx_cols
)
348 # Now we can compute powers of x "symbolically"
349 x_powers
= [self
.one().to_vector(), x
]
350 for d
in range(2, r
+1):
351 x_powers
.append( Lx
*(x_powers
[-1]) )
353 idmat
= matrix
.identity(R
, n
)
355 W
= self
._charpoly
_basis
_space
()
356 W
= W
.change_ring(R
.fraction_field())
358 # Starting with the standard coordinates x = (X1,X2,...,Xn)
359 # and then converting the entries to W-coordinates allows us
360 # to pass in the standard coordinates to the charpoly and get
361 # back the right answer. Specifically, with x = (X1,X2,...,Xn),
364 # W.coordinates(x^2) eval'd at (standard z-coords)
368 # W-coords of (standard coords of x^2 eval'd at std-coords of z)
370 # We want the middle equivalent thing in our matrix, but use
371 # the first equivalent thing instead so that we can pass in
372 # standard coordinates.
373 x_powers
= [ W
.coordinate_vector(xp
) for xp
in x_powers
]
374 l2
= [idmat
.column(k
-1) for k
in range(r
+1, n
+1)]
375 A_of_x
= matrix
.column(R
, n
, (x_powers
[:r
] + l2
))
376 return (A_of_x
, x
, x_powers
[r
], A_of_x
.det())
380 def characteristic_polynomial(self
):
382 Return a characteristic polynomial that works for all elements
385 The resulting polynomial has `n+1` variables, where `n` is the
386 dimension of this algebra. The first `n` variables correspond to
387 the coordinates of an algebra element: when evaluated at the
388 coordinates of an algebra element with respect to a certain
389 basis, the result is a univariate polynomial (in the one
390 remaining variable ``t``), namely the characteristic polynomial
395 sage: from mjo.eja.eja_algebra import JordanSpinEJA, TrivialEJA
399 The characteristic polynomial in the spin algebra is given in
400 Alizadeh, Example 11.11::
402 sage: J = JordanSpinEJA(3)
403 sage: p = J.characteristic_polynomial(); p
404 X1^2 - X2^2 - X3^2 + (-2*t)*X1 + t^2
405 sage: xvec = J.one().to_vector()
409 By definition, the characteristic polynomial is a monic
410 degree-zero polynomial in a rank-zero algebra. Note that
411 Cayley-Hamilton is indeed satisfied since the polynomial
412 ``1`` evaluates to the identity element of the algebra on
415 sage: J = TrivialEJA()
416 sage: J.characteristic_polynomial()
423 # The list of coefficient polynomials a_0, a_1, a_2, ..., a_n.
424 a
= [ self
._charpoly
_coeff
(i
) for i
in range(r
+1) ]
426 # We go to a bit of trouble here to reorder the
427 # indeterminates, so that it's easier to evaluate the
428 # characteristic polynomial at x's coordinates and get back
429 # something in terms of t, which is what we want.
431 S
= PolynomialRing(self
.base_ring(),'t')
433 S
= PolynomialRing(S
, R
.variable_names())
436 return sum( a
[k
]*(t
**k
) for k
in range(len(a
)) )
439 def inner_product(self
, x
, y
):
441 The inner product associated with this Euclidean Jordan algebra.
443 Defaults to the trace inner product, but can be overridden by
444 subclasses if they are sure that the necessary properties are
449 sage: from mjo.eja.eja_algebra import random_eja
453 Our inner product is "associative," which means the following for
454 a symmetric bilinear form::
456 sage: set_random_seed()
457 sage: J = random_eja()
458 sage: x,y,z = J.random_elements(3)
459 sage: (x*y).inner_product(z) == y.inner_product(x*z)
463 X
= x
.natural_representation()
464 Y
= y
.natural_representation()
465 return self
.natural_inner_product(X
,Y
)
468 def is_trivial(self
):
470 Return whether or not this algebra is trivial.
472 A trivial algebra contains only the zero element.
476 sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
481 sage: J = ComplexHermitianEJA(3)
487 sage: J = TrivialEJA()
492 return self
.dimension() == 0
495 def multiplication_table(self
):
497 Return a visual representation of this algebra's multiplication
498 table (on basis elements).
502 sage: from mjo.eja.eja_algebra import JordanSpinEJA
506 sage: J = JordanSpinEJA(4)
507 sage: J.multiplication_table()
508 +----++----+----+----+----+
509 | * || e0 | e1 | e2 | e3 |
510 +====++====+====+====+====+
511 | e0 || e0 | e1 | e2 | e3 |
512 +----++----+----+----+----+
513 | e1 || e1 | e0 | 0 | 0 |
514 +----++----+----+----+----+
515 | e2 || e2 | 0 | e0 | 0 |
516 +----++----+----+----+----+
517 | e3 || e3 | 0 | 0 | e0 |
518 +----++----+----+----+----+
521 M
= list(self
._multiplication
_table
) # copy
522 for i
in range(len(M
)):
523 # M had better be "square"
524 M
[i
] = [self
.monomial(i
)] + M
[i
]
525 M
= [["*"] + list(self
.gens())] + M
526 return table(M
, header_row
=True, header_column
=True, frame
=True)
529 def natural_basis(self
):
531 Return a more-natural representation of this algebra's basis.
533 Every finite-dimensional Euclidean Jordan Algebra is a direct
534 sum of five simple algebras, four of which comprise Hermitian
535 matrices. This method returns the original "natural" basis
536 for our underlying vector space. (Typically, the natural basis
537 is used to construct the multiplication table in the first place.)
539 Note that this will always return a matrix. The standard basis
540 in `R^n` will be returned as `n`-by-`1` column matrices.
544 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
545 ....: RealSymmetricEJA)
549 sage: J = RealSymmetricEJA(2)
551 Finite family {0: e0, 1: e1, 2: e2}
552 sage: J.natural_basis()
554 [1 0] [ 0 0.7071067811865475?] [0 0]
555 [0 0], [0.7071067811865475? 0], [0 1]
560 sage: J = JordanSpinEJA(2)
562 Finite family {0: e0, 1: e1}
563 sage: J.natural_basis()
570 if self
._natural
_basis
is None:
571 M
= self
.natural_basis_space()
572 return tuple( M(b
.to_vector()) for b
in self
.basis() )
574 return self
._natural
_basis
577 def natural_basis_space(self
):
579 Return the matrix space in which this algebra's natural basis
582 if self
._natural
_basis
is None or len(self
._natural
_basis
) == 0:
583 return MatrixSpace(self
.base_ring(), self
.dimension(), 1)
585 return self
._natural
_basis
[0].matrix_space()
589 def natural_inner_product(X
,Y
):
591 Compute the inner product of two naturally-represented elements.
593 For example in the real symmetric matrix EJA, this will compute
594 the trace inner-product of two n-by-n symmetric matrices. The
595 default should work for the real cartesian product EJA, the
596 Jordan spin EJA, and the real symmetric matrices. The others
597 will have to be overridden.
599 return (X
.conjugate_transpose()*Y
).trace()
605 Return the unit element of this algebra.
609 sage: from mjo.eja.eja_algebra import (HadamardEJA,
614 sage: J = HadamardEJA(5)
616 e0 + e1 + e2 + e3 + e4
620 The identity element acts like the identity::
622 sage: set_random_seed()
623 sage: J = random_eja()
624 sage: x = J.random_element()
625 sage: J.one()*x == x and x*J.one() == x
628 The matrix of the unit element's operator is the identity::
630 sage: set_random_seed()
631 sage: J = random_eja()
632 sage: actual = J.one().operator().matrix()
633 sage: expected = matrix.identity(J.base_ring(), J.dimension())
634 sage: actual == expected
638 # We can brute-force compute the matrices of the operators
639 # that correspond to the basis elements of this algebra.
640 # If some linear combination of those basis elements is the
641 # algebra identity, then the same linear combination of
642 # their matrices has to be the identity matrix.
644 # Of course, matrices aren't vectors in sage, so we have to
645 # appeal to the "long vectors" isometry.
646 oper_vecs
= [ _mat2vec(g
.operator().matrix()) for g
in self
.gens() ]
648 # Now we use basis linear algebra to find the coefficients,
649 # of the matrices-as-vectors-linear-combination, which should
650 # work for the original algebra basis too.
651 A
= matrix
.column(self
.base_ring(), oper_vecs
)
653 # We used the isometry on the left-hand side already, but we
654 # still need to do it for the right-hand side. Recall that we
655 # wanted something that summed to the identity matrix.
656 b
= _mat2vec( matrix
.identity(self
.base_ring(), self
.dimension()) )
658 # Now if there's an identity element in the algebra, this should work.
659 coeffs
= A
.solve_right(b
)
660 return self
.linear_combination(zip(self
.gens(), coeffs
))
663 def peirce_decomposition(self
, c
):
665 The Peirce decomposition of this algebra relative to the
668 In the future, this can be extended to a complete system of
669 orthogonal idempotents.
673 - ``c`` -- an idempotent of this algebra.
677 A triple (J0, J5, J1) containing two subalgebras and one subspace
680 - ``J0`` -- the algebra on the eigenspace of ``c.operator()``
681 corresponding to the eigenvalue zero.
683 - ``J5`` -- the eigenspace (NOT a subalgebra) of ``c.operator()``
684 corresponding to the eigenvalue one-half.
686 - ``J1`` -- the algebra on the eigenspace of ``c.operator()``
687 corresponding to the eigenvalue one.
689 These are the only possible eigenspaces for that operator, and this
690 algebra is a direct sum of them. The spaces ``J0`` and ``J1`` are
691 orthogonal, and are subalgebras of this algebra with the appropriate
696 sage: from mjo.eja.eja_algebra import random_eja, RealSymmetricEJA
700 The canonical example comes from the symmetric matrices, which
701 decompose into diagonal and off-diagonal parts::
703 sage: J = RealSymmetricEJA(3)
704 sage: C = matrix(QQ, [ [1,0,0],
708 sage: J0,J5,J1 = J.peirce_decomposition(c)
710 Euclidean Jordan algebra of dimension 1...
712 Vector space of degree 6 and dimension 2...
714 Euclidean Jordan algebra of dimension 3...
718 Every algebra decomposes trivially with respect to its identity
721 sage: set_random_seed()
722 sage: J = random_eja()
723 sage: J0,J5,J1 = J.peirce_decomposition(J.one())
724 sage: J0.dimension() == 0 and J5.dimension() == 0
726 sage: J1.superalgebra() == J and J1.dimension() == J.dimension()
729 The identity elements in the two subalgebras are the
730 projections onto their respective subspaces of the
731 superalgebra's identity element::
733 sage: set_random_seed()
734 sage: J = random_eja()
735 sage: x = J.random_element()
736 sage: if not J.is_trivial():
737 ....: while x.is_nilpotent():
738 ....: x = J.random_element()
739 sage: c = x.subalgebra_idempotent()
740 sage: J0,J5,J1 = J.peirce_decomposition(c)
741 sage: J1(c) == J1.one()
743 sage: J0(J.one() - c) == J0.one()
747 if not c
.is_idempotent():
748 raise ValueError("element is not idempotent: %s" % c
)
750 # Default these to what they should be if they turn out to be
751 # trivial, because eigenspaces_left() won't return eigenvalues
752 # corresponding to trivial spaces (e.g. it returns only the
753 # eigenspace corresponding to lambda=1 if you take the
754 # decomposition relative to the identity element).
755 trivial
= FiniteDimensionalEuclideanJordanSubalgebra(self
, ())
756 J0
= trivial
# eigenvalue zero
757 J5
= VectorSpace(self
.base_ring(), 0) # eigenvalue one-half
758 J1
= trivial
# eigenvalue one
760 for (eigval
, eigspace
) in c
.operator().matrix().right_eigenspaces():
761 if eigval
== ~
(self
.base_ring()(2)):
764 gens
= tuple( self
.from_vector(b
) for b
in eigspace
.basis() )
765 subalg
= FiniteDimensionalEuclideanJordanSubalgebra(self
, gens
)
771 raise ValueError("unexpected eigenvalue: %s" % eigval
)
776 def random_elements(self
, count
):
778 Return ``count`` random elements as a tuple.
782 sage: from mjo.eja.eja_algebra import JordanSpinEJA
786 sage: J = JordanSpinEJA(3)
787 sage: x,y,z = J.random_elements(3)
788 sage: all( [ x in J, y in J, z in J ])
790 sage: len( J.random_elements(10) ) == 10
794 return tuple( self
.random_element() for idx
in range(count
) )
799 Return the rank of this EJA.
803 We first compute the polynomial "column matrices" `p_{k}` that
804 evaluate to `x^k` on the coordinates of `x`. Then, we begin
805 adding them to a matrix one at a time, and trying to solve the
806 system that makes `p_{0}`,`p_{1}`,..., `p_{s-1}` add up to
807 `p_{s}`. This will succeed only when `s` is the rank of the
808 algebra, as proven in a recent draft paper of mine.
812 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
813 ....: RealSymmetricEJA,
814 ....: ComplexHermitianEJA,
815 ....: QuaternionHermitianEJA,
820 The rank of the Jordan spin algebra is always two::
822 sage: JordanSpinEJA(2).rank()
824 sage: JordanSpinEJA(3).rank()
826 sage: JordanSpinEJA(4).rank()
829 The rank of the `n`-by-`n` Hermitian real, complex, or
830 quaternion matrices is `n`::
832 sage: RealSymmetricEJA(4).rank()
834 sage: ComplexHermitianEJA(3).rank()
836 sage: QuaternionHermitianEJA(2).rank()
841 Ensure that every EJA that we know how to construct has a
842 positive integer rank, unless the algebra is trivial in
843 which case its rank will be zero::
845 sage: set_random_seed()
846 sage: J = random_eja()
850 sage: r > 0 or (r == 0 and J.is_trivial())
853 Ensure that computing the rank actually works, since the ranks
854 of all simple algebras are known and will be cached by default::
856 sage: J = HadamardEJA(4)
857 sage: J.rank.clear_cache()
863 sage: J = JordanSpinEJA(4)
864 sage: J.rank.clear_cache()
870 sage: J = RealSymmetricEJA(3)
871 sage: J.rank.clear_cache()
877 sage: J = ComplexHermitianEJA(2)
878 sage: J.rank.clear_cache()
884 sage: J = QuaternionHermitianEJA(2)
885 sage: J.rank.clear_cache()
896 var_names
= [ "X" + str(z
) for z
in range(1,n
+1) ]
897 R
= PolynomialRing(self
.base_ring(), var_names
)
901 # From a result in my book, these are the entries of the
902 # basis representation of L_x.
903 return sum( vars[k
]*self
.monomial(k
).operator().matrix()[i
,j
]
906 L_x
= matrix(R
, n
, n
, L_x_i_j
)
907 x_powers
= [ vars[k
]*(L_x
**k
)*self
.one().to_vector()
911 M
= matrix([x_powers
[0]])
915 M
= matrix(M
.rows() + [x_powers
[d
]])
917 # TODO: we've basically solved the system here.
918 # We should save the echelonized matrix somehow
919 # so that it can be reused in the charpoly method.
921 if new_rank
== old_rank
:
929 def vector_space(self
):
931 Return the vector space that underlies this algebra.
935 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
939 sage: J = RealSymmetricEJA(2)
940 sage: J.vector_space()
941 Vector space of dimension 3 over...
944 return self
.zero().to_vector().parent().ambient_vector_space()
947 Element
= FiniteDimensionalEuclideanJordanAlgebraElement
950 class KnownRankEJA(object):
952 A class for algebras that we actually know we can construct. The
953 main issue is that, for most of our methods to make sense, we need
954 to know the rank of our algebra. Thus we can't simply generate a
955 "random" algebra, or even check that a given basis and product
956 satisfy the axioms; because even if everything looks OK, we wouldn't
957 know the rank we need to actuallty build the thing.
959 Not really a subclass of FDEJA because doing that causes method
960 resolution errors, e.g.
962 TypeError: Error when calling the metaclass bases
963 Cannot create a consistent method resolution
964 order (MRO) for bases FiniteDimensionalEuclideanJordanAlgebra,
969 def _max_test_case_size():
971 Return an integer "size" that is an upper bound on the size of
972 this algebra when it is used in a random test
973 case. Unfortunately, the term "size" is quite vague -- when
974 dealing with `R^n` under either the Hadamard or Jordan spin
975 product, the "size" refers to the dimension `n`. When dealing
976 with a matrix algebra (real symmetric or complex/quaternion
977 Hermitian), it refers to the size of the matrix, which is
978 far less than the dimension of the underlying vector space.
980 We default to five in this class, which is safe in `R^n`. The
981 matrix algebra subclasses (or any class where the "size" is
982 interpreted to be far less than the dimension) should override
983 with a smaller number.
988 def random_instance(cls
, field
=AA
, **kwargs
):
990 Return a random instance of this type of algebra.
992 Beware, this will crash for "most instances" because the
993 constructor below looks wrong.
995 if cls
is TrivialEJA
:
996 # The TrivialEJA class doesn't take an "n" argument because
1000 n
= ZZ
.random_element(cls
._max
_test
_case
_size
()) + 1
1001 return cls(n
, field
, **kwargs
)
1004 class HadamardEJA(FiniteDimensionalEuclideanJordanAlgebra
, KnownRankEJA
):
1006 Return the Euclidean Jordan Algebra corresponding to the set
1007 `R^n` under the Hadamard product.
1009 Note: this is nothing more than the Cartesian product of ``n``
1010 copies of the spin algebra. Once Cartesian product algebras
1011 are implemented, this can go.
1015 sage: from mjo.eja.eja_algebra import HadamardEJA
1019 This multiplication table can be verified by hand::
1021 sage: J = HadamardEJA(3)
1022 sage: e0,e1,e2 = J.gens()
1038 We can change the generator prefix::
1040 sage: HadamardEJA(3, prefix='r').gens()
1044 def __init__(self
, n
, field
=AA
, **kwargs
):
1045 V
= VectorSpace(field
, n
)
1046 mult_table
= [ [ V
.gen(i
)*(i
== j
) for j
in range(n
) ]
1049 fdeja
= super(HadamardEJA
, self
)
1050 return fdeja
.__init
__(field
, mult_table
, rank
=n
, **kwargs
)
1052 def inner_product(self
, x
, y
):
1054 Faster to reimplement than to use natural representations.
1058 sage: from mjo.eja.eja_algebra import HadamardEJA
1062 Ensure that this is the usual inner product for the algebras
1065 sage: set_random_seed()
1066 sage: J = HadamardEJA.random_instance()
1067 sage: x,y = J.random_elements(2)
1068 sage: X = x.natural_representation()
1069 sage: Y = y.natural_representation()
1070 sage: x.inner_product(y) == J.natural_inner_product(X,Y)
1074 return x
.to_vector().inner_product(y
.to_vector())
1077 def random_eja(field
=AA
, nontrivial
=False):
1079 Return a "random" finite-dimensional Euclidean Jordan Algebra.
1083 sage: from mjo.eja.eja_algebra import random_eja
1088 Euclidean Jordan algebra of dimension...
1091 eja_classes
= KnownRankEJA
.__subclasses
__()
1093 eja_classes
.remove(TrivialEJA
)
1094 classname
= choice(eja_classes
)
1095 return classname
.random_instance(field
=field
)
1102 class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra
):
1104 def _max_test_case_size():
1105 # Play it safe, since this will be squared and the underlying
1106 # field can have dimension 4 (quaternions) too.
1109 def __init__(self
, field
, basis
, rank
, normalize_basis
=True, **kwargs
):
1111 Compared to the superclass constructor, we take a basis instead of
1112 a multiplication table because the latter can be computed in terms
1113 of the former when the product is known (like it is here).
1115 # Used in this class's fast _charpoly_coeff() override.
1116 self
._basis
_normalizers
= None
1118 # We're going to loop through this a few times, so now's a good
1119 # time to ensure that it isn't a generator expression.
1120 basis
= tuple(basis
)
1122 if rank
> 1 and normalize_basis
:
1123 # We'll need sqrt(2) to normalize the basis, and this
1124 # winds up in the multiplication table, so the whole
1125 # algebra needs to be over the field extension.
1126 R
= PolynomialRing(field
, 'z')
1129 if p
.is_irreducible():
1130 field
= field
.extension(p
, 'sqrt2', embedding
=RLF(2).sqrt())
1131 basis
= tuple( s
.change_ring(field
) for s
in basis
)
1132 self
._basis
_normalizers
= tuple(
1133 ~
(self
.natural_inner_product(s
,s
).sqrt()) for s
in basis
)
1134 basis
= tuple(s
*c
for (s
,c
) in zip(basis
,self
._basis
_normalizers
))
1136 Qs
= self
.multiplication_table_from_matrix_basis(basis
)
1138 fdeja
= super(MatrixEuclideanJordanAlgebra
, self
)
1139 return fdeja
.__init
__(field
,
1142 natural_basis
=basis
,
1146 def _rank_computation(self
):
1148 Override the parent method with something that tries to compute
1149 over a faster (non-extension) field.
1151 if self
._basis
_normalizers
is None:
1152 # We didn't normalize, so assume that the basis we started
1153 # with had entries in a nice field.
1154 return super(MatrixEuclideanJordanAlgebra
, self
)._rank
_computation
()
1156 basis
= ( (b
/n
) for (b
,n
) in zip(self
.natural_basis(),
1157 self
._basis
_normalizers
) )
1159 # Do this over the rationals and convert back at the end.
1160 # Only works because we know the entries of the basis are
1162 J
= MatrixEuclideanJordanAlgebra(QQ
,
1165 normalize_basis
=False)
1166 return J
._rank
_computation
()
1169 def _charpoly_coeff(self
, i
):
1171 Override the parent method with something that tries to compute
1172 over a faster (non-extension) field.
1174 if self
._basis
_normalizers
is None:
1175 # We didn't normalize, so assume that the basis we started
1176 # with had entries in a nice field.
1177 return super(MatrixEuclideanJordanAlgebra
, self
)._charpoly
_coeff
(i
)
1179 basis
= ( (b
/n
) for (b
,n
) in zip(self
.natural_basis(),
1180 self
._basis
_normalizers
) )
1182 # Do this over the rationals and convert back at the end.
1183 J
= MatrixEuclideanJordanAlgebra(QQ
,
1186 normalize_basis
=False)
1187 (_
,x
,_
,_
) = J
._charpoly
_matrix
_system
()
1188 p
= J
._charpoly
_coeff
(i
)
1189 # p might be missing some vars, have to substitute "optionally"
1190 pairs
= zip(x
.base_ring().gens(), self
._basis
_normalizers
)
1191 substitutions
= { v: v*c for (v,c) in pairs }
1192 result
= p
.subs(substitutions
)
1194 # The result of "subs" can be either a coefficient-ring
1195 # element or a polynomial. Gotta handle both cases.
1197 return self
.base_ring()(result
)
1199 return result
.change_ring(self
.base_ring())
1203 def multiplication_table_from_matrix_basis(basis
):
1205 At least three of the five simple Euclidean Jordan algebras have the
1206 symmetric multiplication (A,B) |-> (AB + BA)/2, where the
1207 multiplication on the right is matrix multiplication. Given a basis
1208 for the underlying matrix space, this function returns a
1209 multiplication table (obtained by looping through the basis
1210 elements) for an algebra of those matrices.
1212 # In S^2, for example, we nominally have four coordinates even
1213 # though the space is of dimension three only. The vector space V
1214 # is supposed to hold the entire long vector, and the subspace W
1215 # of V will be spanned by the vectors that arise from symmetric
1216 # matrices. Thus for S^2, dim(V) == 4 and dim(W) == 3.
1217 field
= basis
[0].base_ring()
1218 dimension
= basis
[0].nrows()
1220 V
= VectorSpace(field
, dimension
**2)
1221 W
= V
.span_of_basis( _mat2vec(s
) for s
in basis
)
1223 mult_table
= [[W
.zero() for j
in range(n
)] for i
in range(n
)]
1226 mat_entry
= (basis
[i
]*basis
[j
] + basis
[j
]*basis
[i
])/2
1227 mult_table
[i
][j
] = W
.coordinate_vector(_mat2vec(mat_entry
))
1235 Embed the matrix ``M`` into a space of real matrices.
1237 The matrix ``M`` can have entries in any field at the moment:
1238 the real numbers, complex numbers, or quaternions. And although
1239 they are not a field, we can probably support octonions at some
1240 point, too. This function returns a real matrix that "acts like"
1241 the original with respect to matrix multiplication; i.e.
1243 real_embed(M*N) = real_embed(M)*real_embed(N)
1246 raise NotImplementedError
1250 def real_unembed(M
):
1252 The inverse of :meth:`real_embed`.
1254 raise NotImplementedError
1258 def natural_inner_product(cls
,X
,Y
):
1259 Xu
= cls
.real_unembed(X
)
1260 Yu
= cls
.real_unembed(Y
)
1261 tr
= (Xu
*Yu
).trace()
1264 # It's real already.
1267 # Otherwise, try the thing that works for complex numbers; and
1268 # if that doesn't work, the thing that works for quaternions.
1270 return tr
.vector()[0] # real part, imag part is index 1
1271 except AttributeError:
1272 # A quaternions doesn't have a vector() method, but does
1273 # have coefficient_tuple() method that returns the
1274 # coefficients of 1, i, j, and k -- in that order.
1275 return tr
.coefficient_tuple()[0]
1278 class RealMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra
):
1282 The identity function, for embedding real matrices into real
1288 def real_unembed(M
):
1290 The identity function, for unembedding real matrices from real
1296 class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra
, KnownRankEJA
):
1298 The rank-n simple EJA consisting of real symmetric n-by-n
1299 matrices, the usual symmetric Jordan product, and the trace inner
1300 product. It has dimension `(n^2 + n)/2` over the reals.
1304 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
1308 sage: J = RealSymmetricEJA(2)
1309 sage: e0, e1, e2 = J.gens()
1317 In theory, our "field" can be any subfield of the reals::
1319 sage: RealSymmetricEJA(2, RDF)
1320 Euclidean Jordan algebra of dimension 3 over Real Double Field
1321 sage: RealSymmetricEJA(2, RR)
1322 Euclidean Jordan algebra of dimension 3 over Real Field with
1323 53 bits of precision
1327 The dimension of this algebra is `(n^2 + n) / 2`::
1329 sage: set_random_seed()
1330 sage: n_max = RealSymmetricEJA._max_test_case_size()
1331 sage: n = ZZ.random_element(1, n_max)
1332 sage: J = RealSymmetricEJA(n)
1333 sage: J.dimension() == (n^2 + n)/2
1336 The Jordan multiplication is what we think it is::
1338 sage: set_random_seed()
1339 sage: J = RealSymmetricEJA.random_instance()
1340 sage: x,y = J.random_elements(2)
1341 sage: actual = (x*y).natural_representation()
1342 sage: X = x.natural_representation()
1343 sage: Y = y.natural_representation()
1344 sage: expected = (X*Y + Y*X)/2
1345 sage: actual == expected
1347 sage: J(expected) == x*y
1350 We can change the generator prefix::
1352 sage: RealSymmetricEJA(3, prefix='q').gens()
1353 (q0, q1, q2, q3, q4, q5)
1355 Our natural basis is normalized with respect to the natural inner
1356 product unless we specify otherwise::
1358 sage: set_random_seed()
1359 sage: J = RealSymmetricEJA.random_instance()
1360 sage: all( b.norm() == 1 for b in J.gens() )
1363 Since our natural basis is normalized with respect to the natural
1364 inner product, and since we know that this algebra is an EJA, any
1365 left-multiplication operator's matrix will be symmetric because
1366 natural->EJA basis representation is an isometry and within the EJA
1367 the operator is self-adjoint by the Jordan axiom::
1369 sage: set_random_seed()
1370 sage: x = RealSymmetricEJA.random_instance().random_element()
1371 sage: x.operator().matrix().is_symmetric()
1376 def _denormalized_basis(cls
, n
, field
):
1378 Return a basis for the space of real symmetric n-by-n matrices.
1382 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
1386 sage: set_random_seed()
1387 sage: n = ZZ.random_element(1,5)
1388 sage: B = RealSymmetricEJA._denormalized_basis(n,QQ)
1389 sage: all( M.is_symmetric() for M in B)
1393 # The basis of symmetric matrices, as matrices, in their R^(n-by-n)
1397 for j
in range(i
+1):
1398 Eij
= matrix(field
, n
, lambda k
,l
: k
==i
and l
==j
)
1402 Sij
= Eij
+ Eij
.transpose()
1408 def _max_test_case_size():
1409 return 4 # Dimension 10
1412 def __init__(self
, n
, field
=AA
, **kwargs
):
1413 basis
= self
._denormalized
_basis
(n
, field
)
1414 super(RealSymmetricEJA
, self
).__init
__(field
, basis
, n
, **kwargs
)
1417 class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra
):
1421 Embed the n-by-n complex matrix ``M`` into the space of real
1422 matrices of size 2n-by-2n via the map the sends each entry `z = a +
1423 bi` to the block matrix ``[[a,b],[-b,a]]``.
1427 sage: from mjo.eja.eja_algebra import \
1428 ....: ComplexMatrixEuclideanJordanAlgebra
1432 sage: F = QuadraticField(-1, 'I')
1433 sage: x1 = F(4 - 2*i)
1434 sage: x2 = F(1 + 2*i)
1437 sage: M = matrix(F,2,[[x1,x2],[x3,x4]])
1438 sage: ComplexMatrixEuclideanJordanAlgebra.real_embed(M)
1447 Embedding is a homomorphism (isomorphism, in fact)::
1449 sage: set_random_seed()
1450 sage: n_max = ComplexMatrixEuclideanJordanAlgebra._max_test_case_size()
1451 sage: n = ZZ.random_element(n_max)
1452 sage: F = QuadraticField(-1, 'I')
1453 sage: X = random_matrix(F, n)
1454 sage: Y = random_matrix(F, n)
1455 sage: Xe = ComplexMatrixEuclideanJordanAlgebra.real_embed(X)
1456 sage: Ye = ComplexMatrixEuclideanJordanAlgebra.real_embed(Y)
1457 sage: XYe = ComplexMatrixEuclideanJordanAlgebra.real_embed(X*Y)
1464 raise ValueError("the matrix 'M' must be square")
1466 # We don't need any adjoined elements...
1467 field
= M
.base_ring().base_ring()
1471 a
= z
.list()[0] # real part, I guess
1472 b
= z
.list()[1] # imag part, I guess
1473 blocks
.append(matrix(field
, 2, [[a
,b
],[-b
,a
]]))
1475 return matrix
.block(field
, n
, blocks
)
1479 def real_unembed(M
):
1481 The inverse of _embed_complex_matrix().
1485 sage: from mjo.eja.eja_algebra import \
1486 ....: ComplexMatrixEuclideanJordanAlgebra
1490 sage: A = matrix(QQ,[ [ 1, 2, 3, 4],
1491 ....: [-2, 1, -4, 3],
1492 ....: [ 9, 10, 11, 12],
1493 ....: [-10, 9, -12, 11] ])
1494 sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(A)
1496 [ 10*I + 9 12*I + 11]
1500 Unembedding is the inverse of embedding::
1502 sage: set_random_seed()
1503 sage: F = QuadraticField(-1, 'I')
1504 sage: M = random_matrix(F, 3)
1505 sage: Me = ComplexMatrixEuclideanJordanAlgebra.real_embed(M)
1506 sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(Me) == M
1512 raise ValueError("the matrix 'M' must be square")
1513 if not n
.mod(2).is_zero():
1514 raise ValueError("the matrix 'M' must be a complex embedding")
1516 # If "M" was normalized, its base ring might have roots
1517 # adjoined and they can stick around after unembedding.
1518 field
= M
.base_ring()
1519 R
= PolynomialRing(field
, 'z')
1522 # Sage doesn't know how to embed AA into QQbar, i.e. how
1523 # to adjoin sqrt(-1) to AA.
1526 F
= field
.extension(z
**2 + 1, 'I', embedding
=CLF(-1).sqrt())
1529 # Go top-left to bottom-right (reading order), converting every
1530 # 2-by-2 block we see to a single complex element.
1532 for k
in range(n
/2):
1533 for j
in range(n
/2):
1534 submat
= M
[2*k
:2*k
+2,2*j
:2*j
+2]
1535 if submat
[0,0] != submat
[1,1]:
1536 raise ValueError('bad on-diagonal submatrix')
1537 if submat
[0,1] != -submat
[1,0]:
1538 raise ValueError('bad off-diagonal submatrix')
1539 z
= submat
[0,0] + submat
[0,1]*i
1542 return matrix(F
, n
/2, elements
)
1546 def natural_inner_product(cls
,X
,Y
):
1548 Compute a natural inner product in this algebra directly from
1553 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
1557 This gives the same answer as the slow, default method implemented
1558 in :class:`MatrixEuclideanJordanAlgebra`::
1560 sage: set_random_seed()
1561 sage: J = ComplexHermitianEJA.random_instance()
1562 sage: x,y = J.random_elements(2)
1563 sage: Xe = x.natural_representation()
1564 sage: Ye = y.natural_representation()
1565 sage: X = ComplexHermitianEJA.real_unembed(Xe)
1566 sage: Y = ComplexHermitianEJA.real_unembed(Ye)
1567 sage: expected = (X*Y).trace().real()
1568 sage: actual = ComplexHermitianEJA.natural_inner_product(Xe,Ye)
1569 sage: actual == expected
1573 return RealMatrixEuclideanJordanAlgebra
.natural_inner_product(X
,Y
)/2
1576 class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra
, KnownRankEJA
):
1578 The rank-n simple EJA consisting of complex Hermitian n-by-n
1579 matrices over the real numbers, the usual symmetric Jordan product,
1580 and the real-part-of-trace inner product. It has dimension `n^2` over
1585 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
1589 In theory, our "field" can be any subfield of the reals::
1591 sage: ComplexHermitianEJA(2, RDF)
1592 Euclidean Jordan algebra of dimension 4 over Real Double Field
1593 sage: ComplexHermitianEJA(2, RR)
1594 Euclidean Jordan algebra of dimension 4 over Real Field with
1595 53 bits of precision
1599 The dimension of this algebra is `n^2`::
1601 sage: set_random_seed()
1602 sage: n_max = ComplexHermitianEJA._max_test_case_size()
1603 sage: n = ZZ.random_element(1, n_max)
1604 sage: J = ComplexHermitianEJA(n)
1605 sage: J.dimension() == n^2
1608 The Jordan multiplication is what we think it is::
1610 sage: set_random_seed()
1611 sage: J = ComplexHermitianEJA.random_instance()
1612 sage: x,y = J.random_elements(2)
1613 sage: actual = (x*y).natural_representation()
1614 sage: X = x.natural_representation()
1615 sage: Y = y.natural_representation()
1616 sage: expected = (X*Y + Y*X)/2
1617 sage: actual == expected
1619 sage: J(expected) == x*y
1622 We can change the generator prefix::
1624 sage: ComplexHermitianEJA(2, prefix='z').gens()
1627 Our natural basis is normalized with respect to the natural inner
1628 product unless we specify otherwise::
1630 sage: set_random_seed()
1631 sage: J = ComplexHermitianEJA.random_instance()
1632 sage: all( b.norm() == 1 for b in J.gens() )
1635 Since our natural basis is normalized with respect to the natural
1636 inner product, and since we know that this algebra is an EJA, any
1637 left-multiplication operator's matrix will be symmetric because
1638 natural->EJA basis representation is an isometry and within the EJA
1639 the operator is self-adjoint by the Jordan axiom::
1641 sage: set_random_seed()
1642 sage: x = ComplexHermitianEJA.random_instance().random_element()
1643 sage: x.operator().matrix().is_symmetric()
1649 def _denormalized_basis(cls
, n
, field
):
1651 Returns a basis for the space of complex Hermitian n-by-n matrices.
1653 Why do we embed these? Basically, because all of numerical linear
1654 algebra assumes that you're working with vectors consisting of `n`
1655 entries from a field and scalars from the same field. There's no way
1656 to tell SageMath that (for example) the vectors contain complex
1657 numbers, while the scalar field is real.
1661 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
1665 sage: set_random_seed()
1666 sage: n = ZZ.random_element(1,5)
1667 sage: field = QuadraticField(2, 'sqrt2')
1668 sage: B = ComplexHermitianEJA._denormalized_basis(n, field)
1669 sage: all( M.is_symmetric() for M in B)
1673 R
= PolynomialRing(field
, 'z')
1675 F
= field
.extension(z
**2 + 1, 'I')
1678 # This is like the symmetric case, but we need to be careful:
1680 # * We want conjugate-symmetry, not just symmetry.
1681 # * The diagonal will (as a result) be real.
1685 for j
in range(i
+1):
1686 Eij
= matrix(F
, n
, lambda k
,l
: k
==i
and l
==j
)
1688 Sij
= cls
.real_embed(Eij
)
1691 # The second one has a minus because it's conjugated.
1692 Sij_real
= cls
.real_embed(Eij
+ Eij
.transpose())
1694 Sij_imag
= cls
.real_embed(I
*Eij
- I
*Eij
.transpose())
1697 # Since we embedded these, we can drop back to the "field" that we
1698 # started with instead of the complex extension "F".
1699 return ( s
.change_ring(field
) for s
in S
)
1702 def __init__(self
, n
, field
=AA
, **kwargs
):
1703 basis
= self
._denormalized
_basis
(n
,field
)
1704 super(ComplexHermitianEJA
,self
).__init
__(field
, basis
, n
, **kwargs
)
1707 class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra
):
1711 Embed the n-by-n quaternion matrix ``M`` into the space of real
1712 matrices of size 4n-by-4n by first sending each quaternion entry `z
1713 = a + bi + cj + dk` to the block-complex matrix ``[[a + bi,
1714 c+di],[-c + di, a-bi]]`, and then embedding those into a real
1719 sage: from mjo.eja.eja_algebra import \
1720 ....: QuaternionMatrixEuclideanJordanAlgebra
1724 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1725 sage: i,j,k = Q.gens()
1726 sage: x = 1 + 2*i + 3*j + 4*k
1727 sage: M = matrix(Q, 1, [[x]])
1728 sage: QuaternionMatrixEuclideanJordanAlgebra.real_embed(M)
1734 Embedding is a homomorphism (isomorphism, in fact)::
1736 sage: set_random_seed()
1737 sage: n_max = QuaternionMatrixEuclideanJordanAlgebra._max_test_case_size()
1738 sage: n = ZZ.random_element(n_max)
1739 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1740 sage: X = random_matrix(Q, n)
1741 sage: Y = random_matrix(Q, n)
1742 sage: Xe = QuaternionMatrixEuclideanJordanAlgebra.real_embed(X)
1743 sage: Ye = QuaternionMatrixEuclideanJordanAlgebra.real_embed(Y)
1744 sage: XYe = QuaternionMatrixEuclideanJordanAlgebra.real_embed(X*Y)
1749 quaternions
= M
.base_ring()
1752 raise ValueError("the matrix 'M' must be square")
1754 F
= QuadraticField(-1, 'I')
1759 t
= z
.coefficient_tuple()
1764 cplxM
= matrix(F
, 2, [[ a
+ b
*i
, c
+ d
*i
],
1765 [-c
+ d
*i
, a
- b
*i
]])
1766 realM
= ComplexMatrixEuclideanJordanAlgebra
.real_embed(cplxM
)
1767 blocks
.append(realM
)
1769 # We should have real entries by now, so use the realest field
1770 # we've got for the return value.
1771 return matrix
.block(quaternions
.base_ring(), n
, blocks
)
1776 def real_unembed(M
):
1778 The inverse of _embed_quaternion_matrix().
1782 sage: from mjo.eja.eja_algebra import \
1783 ....: QuaternionMatrixEuclideanJordanAlgebra
1787 sage: M = matrix(QQ, [[ 1, 2, 3, 4],
1788 ....: [-2, 1, -4, 3],
1789 ....: [-3, 4, 1, -2],
1790 ....: [-4, -3, 2, 1]])
1791 sage: QuaternionMatrixEuclideanJordanAlgebra.real_unembed(M)
1792 [1 + 2*i + 3*j + 4*k]
1796 Unembedding is the inverse of embedding::
1798 sage: set_random_seed()
1799 sage: Q = QuaternionAlgebra(QQ, -1, -1)
1800 sage: M = random_matrix(Q, 3)
1801 sage: Me = QuaternionMatrixEuclideanJordanAlgebra.real_embed(M)
1802 sage: QuaternionMatrixEuclideanJordanAlgebra.real_unembed(Me) == M
1808 raise ValueError("the matrix 'M' must be square")
1809 if not n
.mod(4).is_zero():
1810 raise ValueError("the matrix 'M' must be a quaternion embedding")
1812 # Use the base ring of the matrix to ensure that its entries can be
1813 # multiplied by elements of the quaternion algebra.
1814 field
= M
.base_ring()
1815 Q
= QuaternionAlgebra(field
,-1,-1)
1818 # Go top-left to bottom-right (reading order), converting every
1819 # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1
1822 for l
in range(n
/4):
1823 for m
in range(n
/4):
1824 submat
= ComplexMatrixEuclideanJordanAlgebra
.real_unembed(
1825 M
[4*l
:4*l
+4,4*m
:4*m
+4] )
1826 if submat
[0,0] != submat
[1,1].conjugate():
1827 raise ValueError('bad on-diagonal submatrix')
1828 if submat
[0,1] != -submat
[1,0].conjugate():
1829 raise ValueError('bad off-diagonal submatrix')
1830 z
= submat
[0,0].real()
1831 z
+= submat
[0,0].imag()*i
1832 z
+= submat
[0,1].real()*j
1833 z
+= submat
[0,1].imag()*k
1836 return matrix(Q
, n
/4, elements
)
1840 def natural_inner_product(cls
,X
,Y
):
1842 Compute a natural inner product in this algebra directly from
1847 sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
1851 This gives the same answer as the slow, default method implemented
1852 in :class:`MatrixEuclideanJordanAlgebra`::
1854 sage: set_random_seed()
1855 sage: J = QuaternionHermitianEJA.random_instance()
1856 sage: x,y = J.random_elements(2)
1857 sage: Xe = x.natural_representation()
1858 sage: Ye = y.natural_representation()
1859 sage: X = QuaternionHermitianEJA.real_unembed(Xe)
1860 sage: Y = QuaternionHermitianEJA.real_unembed(Ye)
1861 sage: expected = (X*Y).trace().coefficient_tuple()[0]
1862 sage: actual = QuaternionHermitianEJA.natural_inner_product(Xe,Ye)
1863 sage: actual == expected
1867 return RealMatrixEuclideanJordanAlgebra
.natural_inner_product(X
,Y
)/4
1870 class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra
,
1873 The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
1874 matrices, the usual symmetric Jordan product, and the
1875 real-part-of-trace inner product. It has dimension `2n^2 - n` over
1880 sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
1884 In theory, our "field" can be any subfield of the reals::
1886 sage: QuaternionHermitianEJA(2, RDF)
1887 Euclidean Jordan algebra of dimension 6 over Real Double Field
1888 sage: QuaternionHermitianEJA(2, RR)
1889 Euclidean Jordan algebra of dimension 6 over Real Field with
1890 53 bits of precision
1894 The dimension of this algebra is `2*n^2 - n`::
1896 sage: set_random_seed()
1897 sage: n_max = QuaternionHermitianEJA._max_test_case_size()
1898 sage: n = ZZ.random_element(1, n_max)
1899 sage: J = QuaternionHermitianEJA(n)
1900 sage: J.dimension() == 2*(n^2) - n
1903 The Jordan multiplication is what we think it is::
1905 sage: set_random_seed()
1906 sage: J = QuaternionHermitianEJA.random_instance()
1907 sage: x,y = J.random_elements(2)
1908 sage: actual = (x*y).natural_representation()
1909 sage: X = x.natural_representation()
1910 sage: Y = y.natural_representation()
1911 sage: expected = (X*Y + Y*X)/2
1912 sage: actual == expected
1914 sage: J(expected) == x*y
1917 We can change the generator prefix::
1919 sage: QuaternionHermitianEJA(2, prefix='a').gens()
1920 (a0, a1, a2, a3, a4, a5)
1922 Our natural basis is normalized with respect to the natural inner
1923 product unless we specify otherwise::
1925 sage: set_random_seed()
1926 sage: J = QuaternionHermitianEJA.random_instance()
1927 sage: all( b.norm() == 1 for b in J.gens() )
1930 Since our natural basis is normalized with respect to the natural
1931 inner product, and since we know that this algebra is an EJA, any
1932 left-multiplication operator's matrix will be symmetric because
1933 natural->EJA basis representation is an isometry and within the EJA
1934 the operator is self-adjoint by the Jordan axiom::
1936 sage: set_random_seed()
1937 sage: x = QuaternionHermitianEJA.random_instance().random_element()
1938 sage: x.operator().matrix().is_symmetric()
1943 def _denormalized_basis(cls
, n
, field
):
1945 Returns a basis for the space of quaternion Hermitian n-by-n matrices.
1947 Why do we embed these? Basically, because all of numerical
1948 linear algebra assumes that you're working with vectors consisting
1949 of `n` entries from a field and scalars from the same field. There's
1950 no way to tell SageMath that (for example) the vectors contain
1951 complex numbers, while the scalar field is real.
1955 sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
1959 sage: set_random_seed()
1960 sage: n = ZZ.random_element(1,5)
1961 sage: B = QuaternionHermitianEJA._denormalized_basis(n,QQ)
1962 sage: all( M.is_symmetric() for M in B )
1966 Q
= QuaternionAlgebra(QQ
,-1,-1)
1969 # This is like the symmetric case, but we need to be careful:
1971 # * We want conjugate-symmetry, not just symmetry.
1972 # * The diagonal will (as a result) be real.
1976 for j
in range(i
+1):
1977 Eij
= matrix(Q
, n
, lambda k
,l
: k
==i
and l
==j
)
1979 Sij
= cls
.real_embed(Eij
)
1982 # The second, third, and fourth ones have a minus
1983 # because they're conjugated.
1984 Sij_real
= cls
.real_embed(Eij
+ Eij
.transpose())
1986 Sij_I
= cls
.real_embed(I
*Eij
- I
*Eij
.transpose())
1988 Sij_J
= cls
.real_embed(J
*Eij
- J
*Eij
.transpose())
1990 Sij_K
= cls
.real_embed(K
*Eij
- K
*Eij
.transpose())
1993 # Since we embedded these, we can drop back to the "field" that we
1994 # started with instead of the quaternion algebra "Q".
1995 return ( s
.change_ring(field
) for s
in S
)
1998 def __init__(self
, n
, field
=AA
, **kwargs
):
1999 basis
= self
._denormalized
_basis
(n
,field
)
2000 super(QuaternionHermitianEJA
,self
).__init
__(field
, basis
, n
, **kwargs
)
2003 class BilinearFormEJA(FiniteDimensionalEuclideanJordanAlgebra
, KnownRankEJA
):
2005 The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
2006 with the half-trace inner product and jordan product ``x*y =
2007 (x0*y0 + <B*x_bar,y_bar>, x0*y_bar + y0*x_bar)`` where ``B`` is a
2008 symmetric positive-definite "bilinear form" matrix. It has
2009 dimension `n` over the reals, and reduces to the ``JordanSpinEJA``
2010 when ``B`` is the identity matrix of order ``n-1``.
2014 sage: from mjo.eja.eja_algebra import (BilinearFormEJA,
2015 ....: JordanSpinEJA)
2019 When no bilinear form is specified, the identity matrix is used,
2020 and the resulting algebra is the Jordan spin algebra::
2022 sage: J0 = BilinearFormEJA(3)
2023 sage: J1 = JordanSpinEJA(3)
2024 sage: J0.multiplication_table() == J0.multiplication_table()
2029 We can create a zero-dimensional algebra::
2031 sage: J = BilinearFormEJA(0)
2035 We can check the multiplication condition given in the Jordan, von
2036 Neumann, and Wigner paper (and also discussed on my "On the
2037 symmetry..." paper). Note that this relies heavily on the standard
2038 choice of basis, as does anything utilizing the bilinear form matrix::
2040 sage: set_random_seed()
2041 sage: n = ZZ.random_element(5)
2042 sage: M = matrix.random(QQ, max(0,n-1), algorithm='unimodular')
2043 sage: B = M.transpose()*M
2044 sage: J = BilinearFormEJA(n, B=B)
2045 sage: eis = VectorSpace(M.base_ring(), M.ncols()).basis()
2046 sage: V = J.vector_space()
2047 sage: sis = [ J.from_vector(V([0] + (M.inverse()*ei).list()))
2048 ....: for ei in eis ]
2049 sage: actual = [ sis[i]*sis[j]
2050 ....: for i in range(n-1)
2051 ....: for j in range(n-1) ]
2052 sage: expected = [ J.one() if i == j else J.zero()
2053 ....: for i in range(n-1)
2054 ....: for j in range(n-1) ]
2055 sage: actual == expected
2058 def __init__(self
, n
, field
=AA
, B
=None, **kwargs
):
2060 self
._B
= matrix
.identity(field
, max(0,n
-1))
2064 V
= VectorSpace(field
, n
)
2065 mult_table
= [[V
.zero() for j
in range(n
)] for i
in range(n
)]
2074 z0
= x0
*y0
+ (self
._B
*xbar
).inner_product(ybar
)
2075 zbar
= y0
*xbar
+ x0
*ybar
2076 z
= V([z0
] + zbar
.list())
2077 mult_table
[i
][j
] = z
2079 # The rank of this algebra is two, unless we're in a
2080 # one-dimensional ambient space (because the rank is bounded
2081 # by the ambient dimension).
2082 fdeja
= super(BilinearFormEJA
, self
)
2083 return fdeja
.__init
__(field
, mult_table
, rank
=min(n
,2), **kwargs
)
2085 def inner_product(self
, x
, y
):
2087 Half of the trace inner product.
2089 This is defined so that the special case of the Jordan spin
2090 algebra gets the usual inner product.
2094 sage: from mjo.eja.eja_algebra import BilinearFormEJA
2098 Ensure that this is one-half of the trace inner-product when
2099 the algebra isn't just the reals (when ``n`` isn't one). This
2100 is in Faraut and Koranyi, and also my "On the symmetry..."
2103 sage: set_random_seed()
2104 sage: n = ZZ.random_element(2,5)
2105 sage: M = matrix.random(QQ, max(0,n-1), algorithm='unimodular')
2106 sage: B = M.transpose()*M
2107 sage: J = BilinearFormEJA(n, B=B)
2108 sage: x = J.random_element()
2109 sage: y = J.random_element()
2110 sage: x.inner_product(y) == (x*y).trace()/2
2114 xvec
= x
.to_vector()
2116 yvec
= y
.to_vector()
2118 return x
[0]*y
[0] + (self
._B
*xbar
).inner_product(ybar
)
2121 class JordanSpinEJA(BilinearFormEJA
):
2123 The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
2124 with the usual inner product and jordan product ``x*y =
2125 (<x,y>, x0*y_bar + y0*x_bar)``. It has dimension `n` over
2130 sage: from mjo.eja.eja_algebra import JordanSpinEJA
2134 This multiplication table can be verified by hand::
2136 sage: J = JordanSpinEJA(4)
2137 sage: e0,e1,e2,e3 = J.gens()
2153 We can change the generator prefix::
2155 sage: JordanSpinEJA(2, prefix='B').gens()
2160 Ensure that we have the usual inner product on `R^n`::
2162 sage: set_random_seed()
2163 sage: J = JordanSpinEJA.random_instance()
2164 sage: x,y = J.random_elements(2)
2165 sage: X = x.natural_representation()
2166 sage: Y = y.natural_representation()
2167 sage: x.inner_product(y) == J.natural_inner_product(X,Y)
2171 def __init__(self
, n
, field
=AA
, **kwargs
):
2172 # This is a special case of the BilinearFormEJA with the identity
2173 # matrix as its bilinear form.
2174 return super(JordanSpinEJA
, self
).__init
__(n
, field
, **kwargs
)
2177 class TrivialEJA(FiniteDimensionalEuclideanJordanAlgebra
, KnownRankEJA
):
2179 The trivial Euclidean Jordan algebra consisting of only a zero element.
2183 sage: from mjo.eja.eja_algebra import TrivialEJA
2187 sage: J = TrivialEJA()
2194 sage: 7*J.one()*12*J.one()
2196 sage: J.one().inner_product(J.one())
2198 sage: J.one().norm()
2200 sage: J.one().subalgebra_generated_by()
2201 Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
2206 def __init__(self
, field
=AA
, **kwargs
):
2208 fdeja
= super(TrivialEJA
, self
)
2209 # The rank is zero using my definition, namely the dimension of the
2210 # largest subalgebra generated by any element.
2211 return fdeja
.__init
__(field
, mult_table
, rank
=0, **kwargs
)