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
, 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=QQ)
211 Euclidean Jordan algebra of dimension 2 over Rational 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 1/2*sqrt2] [0 0]
555 [0 0], [1/2*sqrt2 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().left_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 The author knows of no algorithm to compute the rank of an EJA
804 where only the multiplication table is known. In lieu of one, we
805 require the rank to be specified when the algebra is created,
806 and simply pass along that number here.
810 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
811 ....: RealSymmetricEJA,
812 ....: ComplexHermitianEJA,
813 ....: QuaternionHermitianEJA,
818 The rank of the Jordan spin algebra is always two::
820 sage: JordanSpinEJA(2).rank()
822 sage: JordanSpinEJA(3).rank()
824 sage: JordanSpinEJA(4).rank()
827 The rank of the `n`-by-`n` Hermitian real, complex, or
828 quaternion matrices is `n`::
830 sage: RealSymmetricEJA(4).rank()
832 sage: ComplexHermitianEJA(3).rank()
834 sage: QuaternionHermitianEJA(2).rank()
839 Ensure that every EJA that we know how to construct has a
840 positive integer rank, unless the algebra is trivial in
841 which case its rank will be zero::
843 sage: set_random_seed()
844 sage: J = random_eja()
848 sage: r > 0 or (r == 0 and J.is_trivial())
855 def vector_space(self
):
857 Return the vector space that underlies this algebra.
861 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
865 sage: J = RealSymmetricEJA(2)
866 sage: J.vector_space()
867 Vector space of dimension 3 over...
870 return self
.zero().to_vector().parent().ambient_vector_space()
873 Element
= FiniteDimensionalEuclideanJordanAlgebraElement
876 class KnownRankEJA(object):
878 A class for algebras that we actually know we can construct. The
879 main issue is that, for most of our methods to make sense, we need
880 to know the rank of our algebra. Thus we can't simply generate a
881 "random" algebra, or even check that a given basis and product
882 satisfy the axioms; because even if everything looks OK, we wouldn't
883 know the rank we need to actuallty build the thing.
885 Not really a subclass of FDEJA because doing that causes method
886 resolution errors, e.g.
888 TypeError: Error when calling the metaclass bases
889 Cannot create a consistent method resolution
890 order (MRO) for bases FiniteDimensionalEuclideanJordanAlgebra,
895 def _max_test_case_size():
897 Return an integer "size" that is an upper bound on the size of
898 this algebra when it is used in a random test
899 case. Unfortunately, the term "size" is quite vague -- when
900 dealing with `R^n` under either the Hadamard or Jordan spin
901 product, the "size" refers to the dimension `n`. When dealing
902 with a matrix algebra (real symmetric or complex/quaternion
903 Hermitian), it refers to the size of the matrix, which is
904 far less than the dimension of the underlying vector space.
906 We default to five in this class, which is safe in `R^n`. The
907 matrix algebra subclasses (or any class where the "size" is
908 interpreted to be far less than the dimension) should override
909 with a smaller number.
914 def random_instance(cls
, field
=QQ
, **kwargs
):
916 Return a random instance of this type of algebra.
918 Beware, this will crash for "most instances" because the
919 constructor below looks wrong.
921 if cls
is TrivialEJA
:
922 # The TrivialEJA class doesn't take an "n" argument because
926 n
= ZZ
.random_element(cls
._max
_test
_case
_size
()) + 1
927 return cls(n
, field
, **kwargs
)
930 class HadamardEJA(FiniteDimensionalEuclideanJordanAlgebra
, KnownRankEJA
):
932 Return the Euclidean Jordan Algebra corresponding to the set
933 `R^n` under the Hadamard product.
935 Note: this is nothing more than the Cartesian product of ``n``
936 copies of the spin algebra. Once Cartesian product algebras
937 are implemented, this can go.
941 sage: from mjo.eja.eja_algebra import HadamardEJA
945 This multiplication table can be verified by hand::
947 sage: J = HadamardEJA(3)
948 sage: e0,e1,e2 = J.gens()
964 We can change the generator prefix::
966 sage: HadamardEJA(3, prefix='r').gens()
970 def __init__(self
, n
, field
=QQ
, **kwargs
):
971 V
= VectorSpace(field
, n
)
972 mult_table
= [ [ V
.gen(i
)*(i
== j
) for j
in range(n
) ]
975 fdeja
= super(HadamardEJA
, self
)
976 return fdeja
.__init
__(field
, mult_table
, rank
=n
, **kwargs
)
978 def inner_product(self
, x
, y
):
980 Faster to reimplement than to use natural representations.
984 sage: from mjo.eja.eja_algebra import HadamardEJA
988 Ensure that this is the usual inner product for the algebras
991 sage: set_random_seed()
992 sage: J = HadamardEJA.random_instance()
993 sage: x,y = J.random_elements(2)
994 sage: X = x.natural_representation()
995 sage: Y = y.natural_representation()
996 sage: x.inner_product(y) == J.natural_inner_product(X,Y)
1000 return x
.to_vector().inner_product(y
.to_vector())
1003 def random_eja(field
=QQ
, nontrivial
=False):
1005 Return a "random" finite-dimensional Euclidean Jordan Algebra.
1009 sage: from mjo.eja.eja_algebra import random_eja
1014 Euclidean Jordan algebra of dimension...
1017 eja_classes
= KnownRankEJA
.__subclasses
__()
1019 eja_classes
.remove(TrivialEJA
)
1020 classname
= choice(eja_classes
)
1021 return classname
.random_instance(field
=field
)
1028 class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra
):
1030 def _max_test_case_size():
1031 # Play it safe, since this will be squared and the underlying
1032 # field can have dimension 4 (quaternions) too.
1035 def __init__(self
, field
, basis
, rank
, normalize_basis
=True, **kwargs
):
1037 Compared to the superclass constructor, we take a basis instead of
1038 a multiplication table because the latter can be computed in terms
1039 of the former when the product is known (like it is here).
1041 # Used in this class's fast _charpoly_coeff() override.
1042 self
._basis
_normalizers
= None
1044 # We're going to loop through this a few times, so now's a good
1045 # time to ensure that it isn't a generator expression.
1046 basis
= tuple(basis
)
1048 if rank
> 1 and normalize_basis
:
1049 # We'll need sqrt(2) to normalize the basis, and this
1050 # winds up in the multiplication table, so the whole
1051 # algebra needs to be over the field extension.
1052 R
= PolynomialRing(field
, 'z')
1055 if p
.is_irreducible():
1056 field
= field
.extension(p
, 'sqrt2', embedding
=RLF(2).sqrt())
1057 basis
= tuple( s
.change_ring(field
) for s
in basis
)
1058 self
._basis
_normalizers
= tuple(
1059 ~
(self
.natural_inner_product(s
,s
).sqrt()) for s
in basis
)
1060 basis
= tuple(s
*c
for (s
,c
) in zip(basis
,self
._basis
_normalizers
))
1062 Qs
= self
.multiplication_table_from_matrix_basis(basis
)
1064 fdeja
= super(MatrixEuclideanJordanAlgebra
, self
)
1065 return fdeja
.__init
__(field
,
1068 natural_basis
=basis
,
1073 def _charpoly_coeff(self
, i
):
1075 Override the parent method with something that tries to compute
1076 over a faster (non-extension) field.
1078 if self
._basis
_normalizers
is None:
1079 # We didn't normalize, so assume that the basis we started
1080 # with had entries in a nice field.
1081 return super(MatrixEuclideanJordanAlgebra
, self
)._charpoly
_coeff
(i
)
1083 basis
= ( (b
/n
) for (b
,n
) in zip(self
.natural_basis(),
1084 self
._basis
_normalizers
) )
1086 # Do this over the rationals and convert back at the end.
1087 J
= MatrixEuclideanJordanAlgebra(QQ
,
1090 normalize_basis
=False)
1091 (_
,x
,_
,_
) = J
._charpoly
_matrix
_system
()
1092 p
= J
._charpoly
_coeff
(i
)
1093 # p might be missing some vars, have to substitute "optionally"
1094 pairs
= zip(x
.base_ring().gens(), self
._basis
_normalizers
)
1095 substitutions
= { v: v*c for (v,c) in pairs }
1096 result
= p
.subs(substitutions
)
1098 # The result of "subs" can be either a coefficient-ring
1099 # element or a polynomial. Gotta handle both cases.
1101 return self
.base_ring()(result
)
1103 return result
.change_ring(self
.base_ring())
1107 def multiplication_table_from_matrix_basis(basis
):
1109 At least three of the five simple Euclidean Jordan algebras have the
1110 symmetric multiplication (A,B) |-> (AB + BA)/2, where the
1111 multiplication on the right is matrix multiplication. Given a basis
1112 for the underlying matrix space, this function returns a
1113 multiplication table (obtained by looping through the basis
1114 elements) for an algebra of those matrices.
1116 # In S^2, for example, we nominally have four coordinates even
1117 # though the space is of dimension three only. The vector space V
1118 # is supposed to hold the entire long vector, and the subspace W
1119 # of V will be spanned by the vectors that arise from symmetric
1120 # matrices. Thus for S^2, dim(V) == 4 and dim(W) == 3.
1121 field
= basis
[0].base_ring()
1122 dimension
= basis
[0].nrows()
1124 V
= VectorSpace(field
, dimension
**2)
1125 W
= V
.span_of_basis( _mat2vec(s
) for s
in basis
)
1127 mult_table
= [[W
.zero() for j
in range(n
)] for i
in range(n
)]
1130 mat_entry
= (basis
[i
]*basis
[j
] + basis
[j
]*basis
[i
])/2
1131 mult_table
[i
][j
] = W
.coordinate_vector(_mat2vec(mat_entry
))
1139 Embed the matrix ``M`` into a space of real matrices.
1141 The matrix ``M`` can have entries in any field at the moment:
1142 the real numbers, complex numbers, or quaternions. And although
1143 they are not a field, we can probably support octonions at some
1144 point, too. This function returns a real matrix that "acts like"
1145 the original with respect to matrix multiplication; i.e.
1147 real_embed(M*N) = real_embed(M)*real_embed(N)
1150 raise NotImplementedError
1154 def real_unembed(M
):
1156 The inverse of :meth:`real_embed`.
1158 raise NotImplementedError
1162 def natural_inner_product(cls
,X
,Y
):
1163 Xu
= cls
.real_unembed(X
)
1164 Yu
= cls
.real_unembed(Y
)
1165 tr
= (Xu
*Yu
).trace()
1168 # It's real already.
1171 # Otherwise, try the thing that works for complex numbers; and
1172 # if that doesn't work, the thing that works for quaternions.
1174 return tr
.vector()[0] # real part, imag part is index 1
1175 except AttributeError:
1176 # A quaternions doesn't have a vector() method, but does
1177 # have coefficient_tuple() method that returns the
1178 # coefficients of 1, i, j, and k -- in that order.
1179 return tr
.coefficient_tuple()[0]
1182 class RealMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra
):
1186 The identity function, for embedding real matrices into real
1192 def real_unembed(M
):
1194 The identity function, for unembedding real matrices from real
1200 class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra
, KnownRankEJA
):
1202 The rank-n simple EJA consisting of real symmetric n-by-n
1203 matrices, the usual symmetric Jordan product, and the trace inner
1204 product. It has dimension `(n^2 + n)/2` over the reals.
1208 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
1212 sage: J = RealSymmetricEJA(2)
1213 sage: e0, e1, e2 = J.gens()
1221 In theory, our "field" can be any subfield of the reals::
1223 sage: RealSymmetricEJA(2, AA)
1224 Euclidean Jordan algebra of dimension 3 over Algebraic Real Field
1225 sage: RealSymmetricEJA(2, RR)
1226 Euclidean Jordan algebra of dimension 3 over Real Field with
1227 53 bits of precision
1231 The dimension of this algebra is `(n^2 + n) / 2`::
1233 sage: set_random_seed()
1234 sage: n_max = RealSymmetricEJA._max_test_case_size()
1235 sage: n = ZZ.random_element(1, n_max)
1236 sage: J = RealSymmetricEJA(n)
1237 sage: J.dimension() == (n^2 + n)/2
1240 The Jordan multiplication is what we think it is::
1242 sage: set_random_seed()
1243 sage: J = RealSymmetricEJA.random_instance()
1244 sage: x,y = J.random_elements(2)
1245 sage: actual = (x*y).natural_representation()
1246 sage: X = x.natural_representation()
1247 sage: Y = y.natural_representation()
1248 sage: expected = (X*Y + Y*X)/2
1249 sage: actual == expected
1251 sage: J(expected) == x*y
1254 We can change the generator prefix::
1256 sage: RealSymmetricEJA(3, prefix='q').gens()
1257 (q0, q1, q2, q3, q4, q5)
1259 Our natural basis is normalized with respect to the natural inner
1260 product unless we specify otherwise::
1262 sage: set_random_seed()
1263 sage: J = RealSymmetricEJA.random_instance()
1264 sage: all( b.norm() == 1 for b in J.gens() )
1267 Since our natural basis is normalized with respect to the natural
1268 inner product, and since we know that this algebra is an EJA, any
1269 left-multiplication operator's matrix will be symmetric because
1270 natural->EJA basis representation is an isometry and within the EJA
1271 the operator is self-adjoint by the Jordan axiom::
1273 sage: set_random_seed()
1274 sage: x = RealSymmetricEJA.random_instance().random_element()
1275 sage: x.operator().matrix().is_symmetric()
1280 def _denormalized_basis(cls
, n
, field
):
1282 Return a basis for the space of real symmetric n-by-n matrices.
1286 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
1290 sage: set_random_seed()
1291 sage: n = ZZ.random_element(1,5)
1292 sage: B = RealSymmetricEJA._denormalized_basis(n,QQ)
1293 sage: all( M.is_symmetric() for M in B)
1297 # The basis of symmetric matrices, as matrices, in their R^(n-by-n)
1301 for j
in range(i
+1):
1302 Eij
= matrix(field
, n
, lambda k
,l
: k
==i
and l
==j
)
1306 Sij
= Eij
+ Eij
.transpose()
1312 def _max_test_case_size():
1313 return 4 # Dimension 10
1316 def __init__(self
, n
, field
=QQ
, **kwargs
):
1317 basis
= self
._denormalized
_basis
(n
, field
)
1318 super(RealSymmetricEJA
, self
).__init
__(field
, basis
, n
, **kwargs
)
1321 class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra
):
1325 Embed the n-by-n complex matrix ``M`` into the space of real
1326 matrices of size 2n-by-2n via the map the sends each entry `z = a +
1327 bi` to the block matrix ``[[a,b],[-b,a]]``.
1331 sage: from mjo.eja.eja_algebra import \
1332 ....: ComplexMatrixEuclideanJordanAlgebra
1336 sage: F = QuadraticField(-1, 'i')
1337 sage: x1 = F(4 - 2*i)
1338 sage: x2 = F(1 + 2*i)
1341 sage: M = matrix(F,2,[[x1,x2],[x3,x4]])
1342 sage: ComplexMatrixEuclideanJordanAlgebra.real_embed(M)
1351 Embedding is a homomorphism (isomorphism, in fact)::
1353 sage: set_random_seed()
1354 sage: n_max = ComplexMatrixEuclideanJordanAlgebra._max_test_case_size()
1355 sage: n = ZZ.random_element(n_max)
1356 sage: F = QuadraticField(-1, 'i')
1357 sage: X = random_matrix(F, n)
1358 sage: Y = random_matrix(F, n)
1359 sage: Xe = ComplexMatrixEuclideanJordanAlgebra.real_embed(X)
1360 sage: Ye = ComplexMatrixEuclideanJordanAlgebra.real_embed(Y)
1361 sage: XYe = ComplexMatrixEuclideanJordanAlgebra.real_embed(X*Y)
1368 raise ValueError("the matrix 'M' must be square")
1370 # We don't need any adjoined elements...
1371 field
= M
.base_ring().base_ring()
1375 a
= z
.list()[0] # real part, I guess
1376 b
= z
.list()[1] # imag part, I guess
1377 blocks
.append(matrix(field
, 2, [[a
,b
],[-b
,a
]]))
1379 return matrix
.block(field
, n
, blocks
)
1383 def real_unembed(M
):
1385 The inverse of _embed_complex_matrix().
1389 sage: from mjo.eja.eja_algebra import \
1390 ....: ComplexMatrixEuclideanJordanAlgebra
1394 sage: A = matrix(QQ,[ [ 1, 2, 3, 4],
1395 ....: [-2, 1, -4, 3],
1396 ....: [ 9, 10, 11, 12],
1397 ....: [-10, 9, -12, 11] ])
1398 sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(A)
1400 [ 10*i + 9 12*i + 11]
1404 Unembedding is the inverse of embedding::
1406 sage: set_random_seed()
1407 sage: F = QuadraticField(-1, 'i')
1408 sage: M = random_matrix(F, 3)
1409 sage: Me = ComplexMatrixEuclideanJordanAlgebra.real_embed(M)
1410 sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(Me) == M
1416 raise ValueError("the matrix 'M' must be square")
1417 if not n
.mod(2).is_zero():
1418 raise ValueError("the matrix 'M' must be a complex embedding")
1420 # If "M" was normalized, its base ring might have roots
1421 # adjoined and they can stick around after unembedding.
1422 field
= M
.base_ring()
1423 R
= PolynomialRing(field
, 'z')
1425 F
= field
.extension(z
**2 + 1, 'i', embedding
=CLF(-1).sqrt())
1428 # Go top-left to bottom-right (reading order), converting every
1429 # 2-by-2 block we see to a single complex element.
1431 for k
in range(n
/2):
1432 for j
in range(n
/2):
1433 submat
= M
[2*k
:2*k
+2,2*j
:2*j
+2]
1434 if submat
[0,0] != submat
[1,1]:
1435 raise ValueError('bad on-diagonal submatrix')
1436 if submat
[0,1] != -submat
[1,0]:
1437 raise ValueError('bad off-diagonal submatrix')
1438 z
= submat
[0,0] + submat
[0,1]*i
1441 return matrix(F
, n
/2, elements
)
1445 def natural_inner_product(cls
,X
,Y
):
1447 Compute a natural inner product in this algebra directly from
1452 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
1456 This gives the same answer as the slow, default method implemented
1457 in :class:`MatrixEuclideanJordanAlgebra`::
1459 sage: set_random_seed()
1460 sage: J = ComplexHermitianEJA.random_instance()
1461 sage: x,y = J.random_elements(2)
1462 sage: Xe = x.natural_representation()
1463 sage: Ye = y.natural_representation()
1464 sage: X = ComplexHermitianEJA.real_unembed(Xe)
1465 sage: Y = ComplexHermitianEJA.real_unembed(Ye)
1466 sage: expected = (X*Y).trace().vector()[0]
1467 sage: actual = ComplexHermitianEJA.natural_inner_product(Xe,Ye)
1468 sage: actual == expected
1472 return RealMatrixEuclideanJordanAlgebra
.natural_inner_product(X
,Y
)/2
1475 class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra
, KnownRankEJA
):
1477 The rank-n simple EJA consisting of complex Hermitian n-by-n
1478 matrices over the real numbers, the usual symmetric Jordan product,
1479 and the real-part-of-trace inner product. It has dimension `n^2` over
1484 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
1488 In theory, our "field" can be any subfield of the reals::
1490 sage: ComplexHermitianEJA(2, AA)
1491 Euclidean Jordan algebra of dimension 4 over Algebraic Real Field
1492 sage: ComplexHermitianEJA(2, RR)
1493 Euclidean Jordan algebra of dimension 4 over Real Field with
1494 53 bits of precision
1498 The dimension of this algebra is `n^2`::
1500 sage: set_random_seed()
1501 sage: n_max = ComplexHermitianEJA._max_test_case_size()
1502 sage: n = ZZ.random_element(1, n_max)
1503 sage: J = ComplexHermitianEJA(n)
1504 sage: J.dimension() == n^2
1507 The Jordan multiplication is what we think it is::
1509 sage: set_random_seed()
1510 sage: J = ComplexHermitianEJA.random_instance()
1511 sage: x,y = J.random_elements(2)
1512 sage: actual = (x*y).natural_representation()
1513 sage: X = x.natural_representation()
1514 sage: Y = y.natural_representation()
1515 sage: expected = (X*Y + Y*X)/2
1516 sage: actual == expected
1518 sage: J(expected) == x*y
1521 We can change the generator prefix::
1523 sage: ComplexHermitianEJA(2, prefix='z').gens()
1526 Our natural basis is normalized with respect to the natural inner
1527 product unless we specify otherwise::
1529 sage: set_random_seed()
1530 sage: J = ComplexHermitianEJA.random_instance()
1531 sage: all( b.norm() == 1 for b in J.gens() )
1534 Since our natural basis is normalized with respect to the natural
1535 inner product, and since we know that this algebra is an EJA, any
1536 left-multiplication operator's matrix will be symmetric because
1537 natural->EJA basis representation is an isometry and within the EJA
1538 the operator is self-adjoint by the Jordan axiom::
1540 sage: set_random_seed()
1541 sage: x = ComplexHermitianEJA.random_instance().random_element()
1542 sage: x.operator().matrix().is_symmetric()
1548 def _denormalized_basis(cls
, n
, field
):
1550 Returns a basis for the space of complex Hermitian n-by-n matrices.
1552 Why do we embed these? Basically, because all of numerical linear
1553 algebra assumes that you're working with vectors consisting of `n`
1554 entries from a field and scalars from the same field. There's no way
1555 to tell SageMath that (for example) the vectors contain complex
1556 numbers, while the scalar field is real.
1560 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
1564 sage: set_random_seed()
1565 sage: n = ZZ.random_element(1,5)
1566 sage: field = QuadraticField(2, 'sqrt2')
1567 sage: B = ComplexHermitianEJA._denormalized_basis(n, field)
1568 sage: all( M.is_symmetric() for M in B)
1572 R
= PolynomialRing(field
, 'z')
1574 F
= field
.extension(z
**2 + 1, 'I')
1577 # This is like the symmetric case, but we need to be careful:
1579 # * We want conjugate-symmetry, not just symmetry.
1580 # * The diagonal will (as a result) be real.
1584 for j
in range(i
+1):
1585 Eij
= matrix(F
, n
, lambda k
,l
: k
==i
and l
==j
)
1587 Sij
= cls
.real_embed(Eij
)
1590 # The second one has a minus because it's conjugated.
1591 Sij_real
= cls
.real_embed(Eij
+ Eij
.transpose())
1593 Sij_imag
= cls
.real_embed(I
*Eij
- I
*Eij
.transpose())
1596 # Since we embedded these, we can drop back to the "field" that we
1597 # started with instead of the complex extension "F".
1598 return ( s
.change_ring(field
) for s
in S
)
1601 def __init__(self
, n
, field
=QQ
, **kwargs
):
1602 basis
= self
._denormalized
_basis
(n
,field
)
1603 super(ComplexHermitianEJA
,self
).__init
__(field
, basis
, n
, **kwargs
)
1606 class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra
):
1610 Embed the n-by-n quaternion matrix ``M`` into the space of real
1611 matrices of size 4n-by-4n by first sending each quaternion entry `z
1612 = a + bi + cj + dk` to the block-complex matrix ``[[a + bi,
1613 c+di],[-c + di, a-bi]]`, and then embedding those into a real
1618 sage: from mjo.eja.eja_algebra import \
1619 ....: QuaternionMatrixEuclideanJordanAlgebra
1623 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1624 sage: i,j,k = Q.gens()
1625 sage: x = 1 + 2*i + 3*j + 4*k
1626 sage: M = matrix(Q, 1, [[x]])
1627 sage: QuaternionMatrixEuclideanJordanAlgebra.real_embed(M)
1633 Embedding is a homomorphism (isomorphism, in fact)::
1635 sage: set_random_seed()
1636 sage: n_max = QuaternionMatrixEuclideanJordanAlgebra._max_test_case_size()
1637 sage: n = ZZ.random_element(n_max)
1638 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1639 sage: X = random_matrix(Q, n)
1640 sage: Y = random_matrix(Q, n)
1641 sage: Xe = QuaternionMatrixEuclideanJordanAlgebra.real_embed(X)
1642 sage: Ye = QuaternionMatrixEuclideanJordanAlgebra.real_embed(Y)
1643 sage: XYe = QuaternionMatrixEuclideanJordanAlgebra.real_embed(X*Y)
1648 quaternions
= M
.base_ring()
1651 raise ValueError("the matrix 'M' must be square")
1653 F
= QuadraticField(-1, 'i')
1658 t
= z
.coefficient_tuple()
1663 cplxM
= matrix(F
, 2, [[ a
+ b
*i
, c
+ d
*i
],
1664 [-c
+ d
*i
, a
- b
*i
]])
1665 realM
= ComplexMatrixEuclideanJordanAlgebra
.real_embed(cplxM
)
1666 blocks
.append(realM
)
1668 # We should have real entries by now, so use the realest field
1669 # we've got for the return value.
1670 return matrix
.block(quaternions
.base_ring(), n
, blocks
)
1675 def real_unembed(M
):
1677 The inverse of _embed_quaternion_matrix().
1681 sage: from mjo.eja.eja_algebra import \
1682 ....: QuaternionMatrixEuclideanJordanAlgebra
1686 sage: M = matrix(QQ, [[ 1, 2, 3, 4],
1687 ....: [-2, 1, -4, 3],
1688 ....: [-3, 4, 1, -2],
1689 ....: [-4, -3, 2, 1]])
1690 sage: QuaternionMatrixEuclideanJordanAlgebra.real_unembed(M)
1691 [1 + 2*i + 3*j + 4*k]
1695 Unembedding is the inverse of embedding::
1697 sage: set_random_seed()
1698 sage: Q = QuaternionAlgebra(QQ, -1, -1)
1699 sage: M = random_matrix(Q, 3)
1700 sage: Me = QuaternionMatrixEuclideanJordanAlgebra.real_embed(M)
1701 sage: QuaternionMatrixEuclideanJordanAlgebra.real_unembed(Me) == M
1707 raise ValueError("the matrix 'M' must be square")
1708 if not n
.mod(4).is_zero():
1709 raise ValueError("the matrix 'M' must be a quaternion embedding")
1711 # Use the base ring of the matrix to ensure that its entries can be
1712 # multiplied by elements of the quaternion algebra.
1713 field
= M
.base_ring()
1714 Q
= QuaternionAlgebra(field
,-1,-1)
1717 # Go top-left to bottom-right (reading order), converting every
1718 # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1
1721 for l
in range(n
/4):
1722 for m
in range(n
/4):
1723 submat
= ComplexMatrixEuclideanJordanAlgebra
.real_unembed(
1724 M
[4*l
:4*l
+4,4*m
:4*m
+4] )
1725 if submat
[0,0] != submat
[1,1].conjugate():
1726 raise ValueError('bad on-diagonal submatrix')
1727 if submat
[0,1] != -submat
[1,0].conjugate():
1728 raise ValueError('bad off-diagonal submatrix')
1729 z
= submat
[0,0].vector()[0] # real part
1730 z
+= submat
[0,0].vector()[1]*i
# imag part
1731 z
+= submat
[0,1].vector()[0]*j
# real part
1732 z
+= submat
[0,1].vector()[1]*k
# imag part
1735 return matrix(Q
, n
/4, elements
)
1739 def natural_inner_product(cls
,X
,Y
):
1741 Compute a natural inner product in this algebra directly from
1746 sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
1750 This gives the same answer as the slow, default method implemented
1751 in :class:`MatrixEuclideanJordanAlgebra`::
1753 sage: set_random_seed()
1754 sage: J = QuaternionHermitianEJA.random_instance()
1755 sage: x,y = J.random_elements(2)
1756 sage: Xe = x.natural_representation()
1757 sage: Ye = y.natural_representation()
1758 sage: X = QuaternionHermitianEJA.real_unembed(Xe)
1759 sage: Y = QuaternionHermitianEJA.real_unembed(Ye)
1760 sage: expected = (X*Y).trace().coefficient_tuple()[0]
1761 sage: actual = QuaternionHermitianEJA.natural_inner_product(Xe,Ye)
1762 sage: actual == expected
1766 return RealMatrixEuclideanJordanAlgebra
.natural_inner_product(X
,Y
)/4
1769 class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra
,
1772 The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
1773 matrices, the usual symmetric Jordan product, and the
1774 real-part-of-trace inner product. It has dimension `2n^2 - n` over
1779 sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
1783 In theory, our "field" can be any subfield of the reals::
1785 sage: QuaternionHermitianEJA(2, AA)
1786 Euclidean Jordan algebra of dimension 6 over Algebraic Real Field
1787 sage: QuaternionHermitianEJA(2, RR)
1788 Euclidean Jordan algebra of dimension 6 over Real Field with
1789 53 bits of precision
1793 The dimension of this algebra is `2*n^2 - n`::
1795 sage: set_random_seed()
1796 sage: n_max = QuaternionHermitianEJA._max_test_case_size()
1797 sage: n = ZZ.random_element(1, n_max)
1798 sage: J = QuaternionHermitianEJA(n)
1799 sage: J.dimension() == 2*(n^2) - n
1802 The Jordan multiplication is what we think it is::
1804 sage: set_random_seed()
1805 sage: J = QuaternionHermitianEJA.random_instance()
1806 sage: x,y = J.random_elements(2)
1807 sage: actual = (x*y).natural_representation()
1808 sage: X = x.natural_representation()
1809 sage: Y = y.natural_representation()
1810 sage: expected = (X*Y + Y*X)/2
1811 sage: actual == expected
1813 sage: J(expected) == x*y
1816 We can change the generator prefix::
1818 sage: QuaternionHermitianEJA(2, prefix='a').gens()
1819 (a0, a1, a2, a3, a4, a5)
1821 Our natural basis is normalized with respect to the natural inner
1822 product unless we specify otherwise::
1824 sage: set_random_seed()
1825 sage: J = QuaternionHermitianEJA.random_instance()
1826 sage: all( b.norm() == 1 for b in J.gens() )
1829 Since our natural basis is normalized with respect to the natural
1830 inner product, and since we know that this algebra is an EJA, any
1831 left-multiplication operator's matrix will be symmetric because
1832 natural->EJA basis representation is an isometry and within the EJA
1833 the operator is self-adjoint by the Jordan axiom::
1835 sage: set_random_seed()
1836 sage: x = QuaternionHermitianEJA.random_instance().random_element()
1837 sage: x.operator().matrix().is_symmetric()
1842 def _denormalized_basis(cls
, n
, field
):
1844 Returns a basis for the space of quaternion Hermitian n-by-n matrices.
1846 Why do we embed these? Basically, because all of numerical
1847 linear algebra assumes that you're working with vectors consisting
1848 of `n` entries from a field and scalars from the same field. There's
1849 no way to tell SageMath that (for example) the vectors contain
1850 complex numbers, while the scalar field is real.
1854 sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
1858 sage: set_random_seed()
1859 sage: n = ZZ.random_element(1,5)
1860 sage: B = QuaternionHermitianEJA._denormalized_basis(n,QQ)
1861 sage: all( M.is_symmetric() for M in B )
1865 Q
= QuaternionAlgebra(QQ
,-1,-1)
1868 # This is like the symmetric case, but we need to be careful:
1870 # * We want conjugate-symmetry, not just symmetry.
1871 # * The diagonal will (as a result) be real.
1875 for j
in range(i
+1):
1876 Eij
= matrix(Q
, n
, lambda k
,l
: k
==i
and l
==j
)
1878 Sij
= cls
.real_embed(Eij
)
1881 # The second, third, and fourth ones have a minus
1882 # because they're conjugated.
1883 Sij_real
= cls
.real_embed(Eij
+ Eij
.transpose())
1885 Sij_I
= cls
.real_embed(I
*Eij
- I
*Eij
.transpose())
1887 Sij_J
= cls
.real_embed(J
*Eij
- J
*Eij
.transpose())
1889 Sij_K
= cls
.real_embed(K
*Eij
- K
*Eij
.transpose())
1892 # Since we embedded these, we can drop back to the "field" that we
1893 # started with instead of the quaternion algebra "Q".
1894 return ( s
.change_ring(field
) for s
in S
)
1897 def __init__(self
, n
, field
=QQ
, **kwargs
):
1898 basis
= self
._denormalized
_basis
(n
,field
)
1899 super(QuaternionHermitianEJA
,self
).__init
__(field
, basis
, n
, **kwargs
)
1902 class BilinearFormEJA(FiniteDimensionalEuclideanJordanAlgebra
, KnownRankEJA
):
1904 The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
1905 with the half-trace inner product and jordan product ``x*y =
1906 (x0*y0 + <B*x_bar,y_bar>, x0*y_bar + y0*x_bar)`` where ``B`` is a
1907 symmetric positive-definite "bilinear form" matrix. It has
1908 dimension `n` over the reals, and reduces to the ``JordanSpinEJA``
1909 when ``B`` is the identity matrix of order ``n-1``.
1913 sage: from mjo.eja.eja_algebra import (BilinearFormEJA,
1914 ....: JordanSpinEJA)
1918 When no bilinear form is specified, the identity matrix is used,
1919 and the resulting algebra is the Jordan spin algebra::
1921 sage: J0 = BilinearFormEJA(3)
1922 sage: J1 = JordanSpinEJA(3)
1923 sage: J0.multiplication_table() == J0.multiplication_table()
1928 We can create a zero-dimensional algebra::
1930 sage: J = BilinearFormEJA(0)
1934 We can check the multiplication condition given in the Jordan, von
1935 Neumann, and Wigner paper (and also discussed on my "On the
1936 symmetry..." paper). Note that this relies heavily on the standard
1937 choice of basis, as does anything utilizing the bilinear form matrix::
1939 sage: set_random_seed()
1940 sage: n = ZZ.random_element(5)
1941 sage: M = matrix.random(QQ, max(0,n-1), algorithm='unimodular')
1942 sage: B = M.transpose()*M
1943 sage: J = BilinearFormEJA(n, B=B)
1944 sage: eis = VectorSpace(M.base_ring(), M.ncols()).basis()
1945 sage: V = J.vector_space()
1946 sage: sis = [ J.from_vector(V([0] + (M.inverse()*ei).list()))
1947 ....: for ei in eis ]
1948 sage: actual = [ sis[i]*sis[j]
1949 ....: for i in range(n-1)
1950 ....: for j in range(n-1) ]
1951 sage: expected = [ J.one() if i == j else J.zero()
1952 ....: for i in range(n-1)
1953 ....: for j in range(n-1) ]
1954 sage: actual == expected
1957 def __init__(self
, n
, field
=QQ
, B
=None, **kwargs
):
1959 self
._B
= matrix
.identity(field
, max(0,n
-1))
1963 V
= VectorSpace(field
, n
)
1964 mult_table
= [[V
.zero() for j
in range(n
)] for i
in range(n
)]
1973 z0
= x0
*y0
+ (self
._B
*xbar
).inner_product(ybar
)
1974 zbar
= y0
*xbar
+ x0
*ybar
1975 z
= V([z0
] + zbar
.list())
1976 mult_table
[i
][j
] = z
1978 # The rank of this algebra is two, unless we're in a
1979 # one-dimensional ambient space (because the rank is bounded
1980 # by the ambient dimension).
1981 fdeja
= super(BilinearFormEJA
, self
)
1982 return fdeja
.__init
__(field
, mult_table
, rank
=min(n
,2), **kwargs
)
1984 def inner_product(self
, x
, y
):
1986 Half of the trace inner product.
1988 This is defined so that the special case of the Jordan spin
1989 algebra gets the usual inner product.
1993 sage: from mjo.eja.eja_algebra import BilinearFormEJA
1997 Ensure that this is one-half of the trace inner-product::
1999 sage: set_random_seed()
2000 sage: n = ZZ.random_element(5)
2001 sage: M = matrix.random(QQ, n-1, algorithm='unimodular')
2002 sage: B = M.transpose()*M
2003 sage: J = BilinearFormEJA(n, B=B)
2004 sage: eis = VectorSpace(M.base_ring(), M.ncols()).basis()
2005 sage: V = J.vector_space()
2006 sage: sis = [ J.from_vector(V([0] + (M.inverse()*ei).list()))
2007 ....: for ei in eis ]
2008 sage: actual = [ sis[i]*sis[j]
2009 ....: for i in range(n-1)
2010 ....: for j in range(n-1) ]
2011 sage: expected = [ J.one() if i == j else J.zero()
2012 ....: for i in range(n-1)
2013 ....: for j in range(n-1) ]
2016 xvec
= x
.to_vector()
2018 yvec
= y
.to_vector()
2020 return x
[0]*y
[0] + (self
._B
*xbar
).inner_product(ybar
)
2022 class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra
, KnownRankEJA
):
2024 The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
2025 with the usual inner product and jordan product ``x*y =
2026 (<x,y>, x0*y_bar + y0*x_bar)``. It has dimension `n` over
2031 sage: from mjo.eja.eja_algebra import JordanSpinEJA
2035 This multiplication table can be verified by hand::
2037 sage: J = JordanSpinEJA(4)
2038 sage: e0,e1,e2,e3 = J.gens()
2054 We can change the generator prefix::
2056 sage: JordanSpinEJA(2, prefix='B').gens()
2060 def __init__(self
, n
, field
=QQ
, **kwargs
):
2061 V
= VectorSpace(field
, n
)
2062 mult_table
= [[V
.zero() for j
in range(n
)] for i
in range(n
)]
2072 z0
= x
.inner_product(y
)
2073 zbar
= y0
*xbar
+ x0
*ybar
2074 z
= V([z0
] + zbar
.list())
2075 mult_table
[i
][j
] = z
2077 # The rank of the spin algebra is two, unless we're in a
2078 # one-dimensional ambient space (because the rank is bounded by
2079 # the ambient dimension).
2080 fdeja
= super(JordanSpinEJA
, self
)
2081 return fdeja
.__init
__(field
, mult_table
, rank
=min(n
,2), **kwargs
)
2083 def inner_product(self
, x
, y
):
2085 Faster to reimplement than to use natural representations.
2089 sage: from mjo.eja.eja_algebra import JordanSpinEJA
2093 Ensure that this is the usual inner product for the algebras
2096 sage: set_random_seed()
2097 sage: J = JordanSpinEJA.random_instance()
2098 sage: x,y = J.random_elements(2)
2099 sage: X = x.natural_representation()
2100 sage: Y = y.natural_representation()
2101 sage: x.inner_product(y) == J.natural_inner_product(X,Y)
2105 return x
.to_vector().inner_product(y
.to_vector())
2108 class TrivialEJA(FiniteDimensionalEuclideanJordanAlgebra
, KnownRankEJA
):
2110 The trivial Euclidean Jordan algebra consisting of only a zero element.
2114 sage: from mjo.eja.eja_algebra import TrivialEJA
2118 sage: J = TrivialEJA()
2125 sage: 7*J.one()*12*J.one()
2127 sage: J.one().inner_product(J.one())
2129 sage: J.one().norm()
2131 sage: J.one().subalgebra_generated_by()
2132 Euclidean Jordan algebra of dimension 0 over Rational Field
2137 def __init__(self
, field
=QQ
, **kwargs
):
2139 fdeja
= super(TrivialEJA
, self
)
2140 # The rank is zero using my definition, namely the dimension of the
2141 # largest subalgebra generated by any element.
2142 return fdeja
.__init
__(field
, mult_table
, rank
=0, **kwargs
)