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
) )
797 def _rank_computation(self
):
799 Compute the rank of this algebra.
803 sage: from mjo.eja.eja_algebra import (HadamardEJA,
805 ....: RealSymmetricEJA,
806 ....: ComplexHermitianEJA,
807 ....: QuaternionHermitianEJA)
811 sage: J = HadamardEJA(4)
812 sage: J._rank_computation() == J.rank()
814 sage: J = JordanSpinEJA(4)
815 sage: J._rank_computation() == J.rank()
817 sage: J = RealSymmetricEJA(3)
818 sage: J._rank_computation() == J.rank()
820 sage: J = ComplexHermitianEJA(2)
821 sage: J._rank_computation() == J.rank()
823 sage: J = QuaternionHermitianEJA(2)
824 sage: J._rank_computation() == J.rank()
834 var_names
= [ "X" + str(z
) for z
in range(1,n
+1) ]
835 R
= PolynomialRing(self
.base_ring(), var_names
)
839 # From a result in my book, these are the entries of the
840 # basis representation of L_x.
841 return sum( vars[k
]*self
.monomial(k
).operator().matrix()[i
,j
]
844 L_x
= matrix(R
, n
, n
, L_x_i_j
)
845 x_powers
= [ vars[k
]*(L_x
**k
)*self
.one().to_vector()
849 M
= matrix([x_powers
[0]])
853 M
= matrix(M
.rows() + [x_powers
[d
]])
856 if new_rank
== old_rank
:
865 Return the rank of this EJA.
869 The author knows of no algorithm to compute the rank of an EJA
870 where only the multiplication table is known. In lieu of one, we
871 require the rank to be specified when the algebra is created,
872 and simply pass along that number here.
876 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
877 ....: RealSymmetricEJA,
878 ....: ComplexHermitianEJA,
879 ....: QuaternionHermitianEJA,
884 The rank of the Jordan spin algebra is always two::
886 sage: JordanSpinEJA(2).rank()
888 sage: JordanSpinEJA(3).rank()
890 sage: JordanSpinEJA(4).rank()
893 The rank of the `n`-by-`n` Hermitian real, complex, or
894 quaternion matrices is `n`::
896 sage: RealSymmetricEJA(4).rank()
898 sage: ComplexHermitianEJA(3).rank()
900 sage: QuaternionHermitianEJA(2).rank()
905 Ensure that every EJA that we know how to construct has a
906 positive integer rank, unless the algebra is trivial in
907 which case its rank will be zero::
909 sage: set_random_seed()
910 sage: J = random_eja()
914 sage: r > 0 or (r == 0 and J.is_trivial())
921 def vector_space(self
):
923 Return the vector space that underlies this algebra.
927 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
931 sage: J = RealSymmetricEJA(2)
932 sage: J.vector_space()
933 Vector space of dimension 3 over...
936 return self
.zero().to_vector().parent().ambient_vector_space()
939 Element
= FiniteDimensionalEuclideanJordanAlgebraElement
942 class KnownRankEJA(object):
944 A class for algebras that we actually know we can construct. The
945 main issue is that, for most of our methods to make sense, we need
946 to know the rank of our algebra. Thus we can't simply generate a
947 "random" algebra, or even check that a given basis and product
948 satisfy the axioms; because even if everything looks OK, we wouldn't
949 know the rank we need to actuallty build the thing.
951 Not really a subclass of FDEJA because doing that causes method
952 resolution errors, e.g.
954 TypeError: Error when calling the metaclass bases
955 Cannot create a consistent method resolution
956 order (MRO) for bases FiniteDimensionalEuclideanJordanAlgebra,
961 def _max_test_case_size():
963 Return an integer "size" that is an upper bound on the size of
964 this algebra when it is used in a random test
965 case. Unfortunately, the term "size" is quite vague -- when
966 dealing with `R^n` under either the Hadamard or Jordan spin
967 product, the "size" refers to the dimension `n`. When dealing
968 with a matrix algebra (real symmetric or complex/quaternion
969 Hermitian), it refers to the size of the matrix, which is
970 far less than the dimension of the underlying vector space.
972 We default to five in this class, which is safe in `R^n`. The
973 matrix algebra subclasses (or any class where the "size" is
974 interpreted to be far less than the dimension) should override
975 with a smaller number.
980 def random_instance(cls
, field
=AA
, **kwargs
):
982 Return a random instance of this type of algebra.
984 Beware, this will crash for "most instances" because the
985 constructor below looks wrong.
987 if cls
is TrivialEJA
:
988 # The TrivialEJA class doesn't take an "n" argument because
992 n
= ZZ
.random_element(cls
._max
_test
_case
_size
()) + 1
993 return cls(n
, field
, **kwargs
)
996 class HadamardEJA(FiniteDimensionalEuclideanJordanAlgebra
, KnownRankEJA
):
998 Return the Euclidean Jordan Algebra corresponding to the set
999 `R^n` under the Hadamard product.
1001 Note: this is nothing more than the Cartesian product of ``n``
1002 copies of the spin algebra. Once Cartesian product algebras
1003 are implemented, this can go.
1007 sage: from mjo.eja.eja_algebra import HadamardEJA
1011 This multiplication table can be verified by hand::
1013 sage: J = HadamardEJA(3)
1014 sage: e0,e1,e2 = J.gens()
1030 We can change the generator prefix::
1032 sage: HadamardEJA(3, prefix='r').gens()
1036 def __init__(self
, n
, field
=AA
, **kwargs
):
1037 V
= VectorSpace(field
, n
)
1038 mult_table
= [ [ V
.gen(i
)*(i
== j
) for j
in range(n
) ]
1041 fdeja
= super(HadamardEJA
, self
)
1042 return fdeja
.__init
__(field
, mult_table
, rank
=n
, **kwargs
)
1044 def inner_product(self
, x
, y
):
1046 Faster to reimplement than to use natural representations.
1050 sage: from mjo.eja.eja_algebra import HadamardEJA
1054 Ensure that this is the usual inner product for the algebras
1057 sage: set_random_seed()
1058 sage: J = HadamardEJA.random_instance()
1059 sage: x,y = J.random_elements(2)
1060 sage: X = x.natural_representation()
1061 sage: Y = y.natural_representation()
1062 sage: x.inner_product(y) == J.natural_inner_product(X,Y)
1066 return x
.to_vector().inner_product(y
.to_vector())
1069 def random_eja(field
=AA
, nontrivial
=False):
1071 Return a "random" finite-dimensional Euclidean Jordan Algebra.
1075 sage: from mjo.eja.eja_algebra import random_eja
1080 Euclidean Jordan algebra of dimension...
1083 eja_classes
= KnownRankEJA
.__subclasses
__()
1085 eja_classes
.remove(TrivialEJA
)
1086 classname
= choice(eja_classes
)
1087 return classname
.random_instance(field
=field
)
1094 class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra
):
1096 def _max_test_case_size():
1097 # Play it safe, since this will be squared and the underlying
1098 # field can have dimension 4 (quaternions) too.
1101 def __init__(self
, field
, basis
, rank
, normalize_basis
=True, **kwargs
):
1103 Compared to the superclass constructor, we take a basis instead of
1104 a multiplication table because the latter can be computed in terms
1105 of the former when the product is known (like it is here).
1107 # Used in this class's fast _charpoly_coeff() override.
1108 self
._basis
_normalizers
= None
1110 # We're going to loop through this a few times, so now's a good
1111 # time to ensure that it isn't a generator expression.
1112 basis
= tuple(basis
)
1114 if rank
> 1 and normalize_basis
:
1115 # We'll need sqrt(2) to normalize the basis, and this
1116 # winds up in the multiplication table, so the whole
1117 # algebra needs to be over the field extension.
1118 R
= PolynomialRing(field
, 'z')
1121 if p
.is_irreducible():
1122 field
= field
.extension(p
, 'sqrt2', embedding
=RLF(2).sqrt())
1123 basis
= tuple( s
.change_ring(field
) for s
in basis
)
1124 self
._basis
_normalizers
= tuple(
1125 ~
(self
.natural_inner_product(s
,s
).sqrt()) for s
in basis
)
1126 basis
= tuple(s
*c
for (s
,c
) in zip(basis
,self
._basis
_normalizers
))
1128 Qs
= self
.multiplication_table_from_matrix_basis(basis
)
1130 fdeja
= super(MatrixEuclideanJordanAlgebra
, self
)
1131 return fdeja
.__init
__(field
,
1134 natural_basis
=basis
,
1138 def _rank_computation(self
):
1140 Override the parent method with something that tries to compute
1141 over a faster (non-extension) field.
1143 if self
._basis
_normalizers
is None:
1144 # We didn't normalize, so assume that the basis we started
1145 # with had entries in a nice field.
1146 return super(MatrixEuclideanJordanAlgebra
, self
)._rank
_computation
()
1148 basis
= ( (b
/n
) for (b
,n
) in zip(self
.natural_basis(),
1149 self
._basis
_normalizers
) )
1151 # Do this over the rationals and convert back at the end.
1152 # Only works because we know the entries of the basis are
1154 J
= MatrixEuclideanJordanAlgebra(QQ
,
1157 normalize_basis
=False)
1158 return J
._rank
_computation
()
1161 def _charpoly_coeff(self
, i
):
1163 Override the parent method with something that tries to compute
1164 over a faster (non-extension) field.
1166 if self
._basis
_normalizers
is None:
1167 # We didn't normalize, so assume that the basis we started
1168 # with had entries in a nice field.
1169 return super(MatrixEuclideanJordanAlgebra
, self
)._charpoly
_coeff
(i
)
1171 basis
= ( (b
/n
) for (b
,n
) in zip(self
.natural_basis(),
1172 self
._basis
_normalizers
) )
1174 # Do this over the rationals and convert back at the end.
1175 J
= MatrixEuclideanJordanAlgebra(QQ
,
1178 normalize_basis
=False)
1179 (_
,x
,_
,_
) = J
._charpoly
_matrix
_system
()
1180 p
= J
._charpoly
_coeff
(i
)
1181 # p might be missing some vars, have to substitute "optionally"
1182 pairs
= zip(x
.base_ring().gens(), self
._basis
_normalizers
)
1183 substitutions
= { v: v*c for (v,c) in pairs }
1184 result
= p
.subs(substitutions
)
1186 # The result of "subs" can be either a coefficient-ring
1187 # element or a polynomial. Gotta handle both cases.
1189 return self
.base_ring()(result
)
1191 return result
.change_ring(self
.base_ring())
1195 def multiplication_table_from_matrix_basis(basis
):
1197 At least three of the five simple Euclidean Jordan algebras have the
1198 symmetric multiplication (A,B) |-> (AB + BA)/2, where the
1199 multiplication on the right is matrix multiplication. Given a basis
1200 for the underlying matrix space, this function returns a
1201 multiplication table (obtained by looping through the basis
1202 elements) for an algebra of those matrices.
1204 # In S^2, for example, we nominally have four coordinates even
1205 # though the space is of dimension three only. The vector space V
1206 # is supposed to hold the entire long vector, and the subspace W
1207 # of V will be spanned by the vectors that arise from symmetric
1208 # matrices. Thus for S^2, dim(V) == 4 and dim(W) == 3.
1209 field
= basis
[0].base_ring()
1210 dimension
= basis
[0].nrows()
1212 V
= VectorSpace(field
, dimension
**2)
1213 W
= V
.span_of_basis( _mat2vec(s
) for s
in basis
)
1215 mult_table
= [[W
.zero() for j
in range(n
)] for i
in range(n
)]
1218 mat_entry
= (basis
[i
]*basis
[j
] + basis
[j
]*basis
[i
])/2
1219 mult_table
[i
][j
] = W
.coordinate_vector(_mat2vec(mat_entry
))
1227 Embed the matrix ``M`` into a space of real matrices.
1229 The matrix ``M`` can have entries in any field at the moment:
1230 the real numbers, complex numbers, or quaternions. And although
1231 they are not a field, we can probably support octonions at some
1232 point, too. This function returns a real matrix that "acts like"
1233 the original with respect to matrix multiplication; i.e.
1235 real_embed(M*N) = real_embed(M)*real_embed(N)
1238 raise NotImplementedError
1242 def real_unembed(M
):
1244 The inverse of :meth:`real_embed`.
1246 raise NotImplementedError
1250 def natural_inner_product(cls
,X
,Y
):
1251 Xu
= cls
.real_unembed(X
)
1252 Yu
= cls
.real_unembed(Y
)
1253 tr
= (Xu
*Yu
).trace()
1256 # It's real already.
1259 # Otherwise, try the thing that works for complex numbers; and
1260 # if that doesn't work, the thing that works for quaternions.
1262 return tr
.vector()[0] # real part, imag part is index 1
1263 except AttributeError:
1264 # A quaternions doesn't have a vector() method, but does
1265 # have coefficient_tuple() method that returns the
1266 # coefficients of 1, i, j, and k -- in that order.
1267 return tr
.coefficient_tuple()[0]
1270 class RealMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra
):
1274 The identity function, for embedding real matrices into real
1280 def real_unembed(M
):
1282 The identity function, for unembedding real matrices from real
1288 class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra
, KnownRankEJA
):
1290 The rank-n simple EJA consisting of real symmetric n-by-n
1291 matrices, the usual symmetric Jordan product, and the trace inner
1292 product. It has dimension `(n^2 + n)/2` over the reals.
1296 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
1300 sage: J = RealSymmetricEJA(2)
1301 sage: e0, e1, e2 = J.gens()
1309 In theory, our "field" can be any subfield of the reals::
1311 sage: RealSymmetricEJA(2, RDF)
1312 Euclidean Jordan algebra of dimension 3 over Real Double Field
1313 sage: RealSymmetricEJA(2, RR)
1314 Euclidean Jordan algebra of dimension 3 over Real Field with
1315 53 bits of precision
1319 The dimension of this algebra is `(n^2 + n) / 2`::
1321 sage: set_random_seed()
1322 sage: n_max = RealSymmetricEJA._max_test_case_size()
1323 sage: n = ZZ.random_element(1, n_max)
1324 sage: J = RealSymmetricEJA(n)
1325 sage: J.dimension() == (n^2 + n)/2
1328 The Jordan multiplication is what we think it is::
1330 sage: set_random_seed()
1331 sage: J = RealSymmetricEJA.random_instance()
1332 sage: x,y = J.random_elements(2)
1333 sage: actual = (x*y).natural_representation()
1334 sage: X = x.natural_representation()
1335 sage: Y = y.natural_representation()
1336 sage: expected = (X*Y + Y*X)/2
1337 sage: actual == expected
1339 sage: J(expected) == x*y
1342 We can change the generator prefix::
1344 sage: RealSymmetricEJA(3, prefix='q').gens()
1345 (q0, q1, q2, q3, q4, q5)
1347 Our natural basis is normalized with respect to the natural inner
1348 product unless we specify otherwise::
1350 sage: set_random_seed()
1351 sage: J = RealSymmetricEJA.random_instance()
1352 sage: all( b.norm() == 1 for b in J.gens() )
1355 Since our natural basis is normalized with respect to the natural
1356 inner product, and since we know that this algebra is an EJA, any
1357 left-multiplication operator's matrix will be symmetric because
1358 natural->EJA basis representation is an isometry and within the EJA
1359 the operator is self-adjoint by the Jordan axiom::
1361 sage: set_random_seed()
1362 sage: x = RealSymmetricEJA.random_instance().random_element()
1363 sage: x.operator().matrix().is_symmetric()
1368 def _denormalized_basis(cls
, n
, field
):
1370 Return a basis for the space of real symmetric n-by-n matrices.
1374 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
1378 sage: set_random_seed()
1379 sage: n = ZZ.random_element(1,5)
1380 sage: B = RealSymmetricEJA._denormalized_basis(n,QQ)
1381 sage: all( M.is_symmetric() for M in B)
1385 # The basis of symmetric matrices, as matrices, in their R^(n-by-n)
1389 for j
in range(i
+1):
1390 Eij
= matrix(field
, n
, lambda k
,l
: k
==i
and l
==j
)
1394 Sij
= Eij
+ Eij
.transpose()
1400 def _max_test_case_size():
1401 return 4 # Dimension 10
1404 def __init__(self
, n
, field
=AA
, **kwargs
):
1405 basis
= self
._denormalized
_basis
(n
, field
)
1406 super(RealSymmetricEJA
, self
).__init
__(field
, basis
, n
, **kwargs
)
1409 class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra
):
1413 Embed the n-by-n complex matrix ``M`` into the space of real
1414 matrices of size 2n-by-2n via the map the sends each entry `z = a +
1415 bi` to the block matrix ``[[a,b],[-b,a]]``.
1419 sage: from mjo.eja.eja_algebra import \
1420 ....: ComplexMatrixEuclideanJordanAlgebra
1424 sage: F = QuadraticField(-1, 'I')
1425 sage: x1 = F(4 - 2*i)
1426 sage: x2 = F(1 + 2*i)
1429 sage: M = matrix(F,2,[[x1,x2],[x3,x4]])
1430 sage: ComplexMatrixEuclideanJordanAlgebra.real_embed(M)
1439 Embedding is a homomorphism (isomorphism, in fact)::
1441 sage: set_random_seed()
1442 sage: n_max = ComplexMatrixEuclideanJordanAlgebra._max_test_case_size()
1443 sage: n = ZZ.random_element(n_max)
1444 sage: F = QuadraticField(-1, 'I')
1445 sage: X = random_matrix(F, n)
1446 sage: Y = random_matrix(F, n)
1447 sage: Xe = ComplexMatrixEuclideanJordanAlgebra.real_embed(X)
1448 sage: Ye = ComplexMatrixEuclideanJordanAlgebra.real_embed(Y)
1449 sage: XYe = ComplexMatrixEuclideanJordanAlgebra.real_embed(X*Y)
1456 raise ValueError("the matrix 'M' must be square")
1458 # We don't need any adjoined elements...
1459 field
= M
.base_ring().base_ring()
1463 a
= z
.list()[0] # real part, I guess
1464 b
= z
.list()[1] # imag part, I guess
1465 blocks
.append(matrix(field
, 2, [[a
,b
],[-b
,a
]]))
1467 return matrix
.block(field
, n
, blocks
)
1471 def real_unembed(M
):
1473 The inverse of _embed_complex_matrix().
1477 sage: from mjo.eja.eja_algebra import \
1478 ....: ComplexMatrixEuclideanJordanAlgebra
1482 sage: A = matrix(QQ,[ [ 1, 2, 3, 4],
1483 ....: [-2, 1, -4, 3],
1484 ....: [ 9, 10, 11, 12],
1485 ....: [-10, 9, -12, 11] ])
1486 sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(A)
1488 [ 10*I + 9 12*I + 11]
1492 Unembedding is the inverse of embedding::
1494 sage: set_random_seed()
1495 sage: F = QuadraticField(-1, 'I')
1496 sage: M = random_matrix(F, 3)
1497 sage: Me = ComplexMatrixEuclideanJordanAlgebra.real_embed(M)
1498 sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(Me) == M
1504 raise ValueError("the matrix 'M' must be square")
1505 if not n
.mod(2).is_zero():
1506 raise ValueError("the matrix 'M' must be a complex embedding")
1508 # If "M" was normalized, its base ring might have roots
1509 # adjoined and they can stick around after unembedding.
1510 field
= M
.base_ring()
1511 R
= PolynomialRing(field
, 'z')
1514 # Sage doesn't know how to embed AA into QQbar, i.e. how
1515 # to adjoin sqrt(-1) to AA.
1518 F
= field
.extension(z
**2 + 1, 'I', embedding
=CLF(-1).sqrt())
1521 # Go top-left to bottom-right (reading order), converting every
1522 # 2-by-2 block we see to a single complex element.
1524 for k
in range(n
/2):
1525 for j
in range(n
/2):
1526 submat
= M
[2*k
:2*k
+2,2*j
:2*j
+2]
1527 if submat
[0,0] != submat
[1,1]:
1528 raise ValueError('bad on-diagonal submatrix')
1529 if submat
[0,1] != -submat
[1,0]:
1530 raise ValueError('bad off-diagonal submatrix')
1531 z
= submat
[0,0] + submat
[0,1]*i
1534 return matrix(F
, n
/2, elements
)
1538 def natural_inner_product(cls
,X
,Y
):
1540 Compute a natural inner product in this algebra directly from
1545 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
1549 This gives the same answer as the slow, default method implemented
1550 in :class:`MatrixEuclideanJordanAlgebra`::
1552 sage: set_random_seed()
1553 sage: J = ComplexHermitianEJA.random_instance()
1554 sage: x,y = J.random_elements(2)
1555 sage: Xe = x.natural_representation()
1556 sage: Ye = y.natural_representation()
1557 sage: X = ComplexHermitianEJA.real_unembed(Xe)
1558 sage: Y = ComplexHermitianEJA.real_unembed(Ye)
1559 sage: expected = (X*Y).trace().real()
1560 sage: actual = ComplexHermitianEJA.natural_inner_product(Xe,Ye)
1561 sage: actual == expected
1565 return RealMatrixEuclideanJordanAlgebra
.natural_inner_product(X
,Y
)/2
1568 class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra
, KnownRankEJA
):
1570 The rank-n simple EJA consisting of complex Hermitian n-by-n
1571 matrices over the real numbers, the usual symmetric Jordan product,
1572 and the real-part-of-trace inner product. It has dimension `n^2` over
1577 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
1581 In theory, our "field" can be any subfield of the reals::
1583 sage: ComplexHermitianEJA(2, RDF)
1584 Euclidean Jordan algebra of dimension 4 over Real Double Field
1585 sage: ComplexHermitianEJA(2, RR)
1586 Euclidean Jordan algebra of dimension 4 over Real Field with
1587 53 bits of precision
1591 The dimension of this algebra is `n^2`::
1593 sage: set_random_seed()
1594 sage: n_max = ComplexHermitianEJA._max_test_case_size()
1595 sage: n = ZZ.random_element(1, n_max)
1596 sage: J = ComplexHermitianEJA(n)
1597 sage: J.dimension() == n^2
1600 The Jordan multiplication is what we think it is::
1602 sage: set_random_seed()
1603 sage: J = ComplexHermitianEJA.random_instance()
1604 sage: x,y = J.random_elements(2)
1605 sage: actual = (x*y).natural_representation()
1606 sage: X = x.natural_representation()
1607 sage: Y = y.natural_representation()
1608 sage: expected = (X*Y + Y*X)/2
1609 sage: actual == expected
1611 sage: J(expected) == x*y
1614 We can change the generator prefix::
1616 sage: ComplexHermitianEJA(2, prefix='z').gens()
1619 Our natural basis is normalized with respect to the natural inner
1620 product unless we specify otherwise::
1622 sage: set_random_seed()
1623 sage: J = ComplexHermitianEJA.random_instance()
1624 sage: all( b.norm() == 1 for b in J.gens() )
1627 Since our natural basis is normalized with respect to the natural
1628 inner product, and since we know that this algebra is an EJA, any
1629 left-multiplication operator's matrix will be symmetric because
1630 natural->EJA basis representation is an isometry and within the EJA
1631 the operator is self-adjoint by the Jordan axiom::
1633 sage: set_random_seed()
1634 sage: x = ComplexHermitianEJA.random_instance().random_element()
1635 sage: x.operator().matrix().is_symmetric()
1641 def _denormalized_basis(cls
, n
, field
):
1643 Returns a basis for the space of complex Hermitian n-by-n matrices.
1645 Why do we embed these? Basically, because all of numerical linear
1646 algebra assumes that you're working with vectors consisting of `n`
1647 entries from a field and scalars from the same field. There's no way
1648 to tell SageMath that (for example) the vectors contain complex
1649 numbers, while the scalar field is real.
1653 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
1657 sage: set_random_seed()
1658 sage: n = ZZ.random_element(1,5)
1659 sage: field = QuadraticField(2, 'sqrt2')
1660 sage: B = ComplexHermitianEJA._denormalized_basis(n, field)
1661 sage: all( M.is_symmetric() for M in B)
1665 R
= PolynomialRing(field
, 'z')
1667 F
= field
.extension(z
**2 + 1, 'I')
1670 # This is like the symmetric case, but we need to be careful:
1672 # * We want conjugate-symmetry, not just symmetry.
1673 # * The diagonal will (as a result) be real.
1677 for j
in range(i
+1):
1678 Eij
= matrix(F
, n
, lambda k
,l
: k
==i
and l
==j
)
1680 Sij
= cls
.real_embed(Eij
)
1683 # The second one has a minus because it's conjugated.
1684 Sij_real
= cls
.real_embed(Eij
+ Eij
.transpose())
1686 Sij_imag
= cls
.real_embed(I
*Eij
- I
*Eij
.transpose())
1689 # Since we embedded these, we can drop back to the "field" that we
1690 # started with instead of the complex extension "F".
1691 return ( s
.change_ring(field
) for s
in S
)
1694 def __init__(self
, n
, field
=AA
, **kwargs
):
1695 basis
= self
._denormalized
_basis
(n
,field
)
1696 super(ComplexHermitianEJA
,self
).__init
__(field
, basis
, n
, **kwargs
)
1699 class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra
):
1703 Embed the n-by-n quaternion matrix ``M`` into the space of real
1704 matrices of size 4n-by-4n by first sending each quaternion entry `z
1705 = a + bi + cj + dk` to the block-complex matrix ``[[a + bi,
1706 c+di],[-c + di, a-bi]]`, and then embedding those into a real
1711 sage: from mjo.eja.eja_algebra import \
1712 ....: QuaternionMatrixEuclideanJordanAlgebra
1716 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1717 sage: i,j,k = Q.gens()
1718 sage: x = 1 + 2*i + 3*j + 4*k
1719 sage: M = matrix(Q, 1, [[x]])
1720 sage: QuaternionMatrixEuclideanJordanAlgebra.real_embed(M)
1726 Embedding is a homomorphism (isomorphism, in fact)::
1728 sage: set_random_seed()
1729 sage: n_max = QuaternionMatrixEuclideanJordanAlgebra._max_test_case_size()
1730 sage: n = ZZ.random_element(n_max)
1731 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1732 sage: X = random_matrix(Q, n)
1733 sage: Y = random_matrix(Q, n)
1734 sage: Xe = QuaternionMatrixEuclideanJordanAlgebra.real_embed(X)
1735 sage: Ye = QuaternionMatrixEuclideanJordanAlgebra.real_embed(Y)
1736 sage: XYe = QuaternionMatrixEuclideanJordanAlgebra.real_embed(X*Y)
1741 quaternions
= M
.base_ring()
1744 raise ValueError("the matrix 'M' must be square")
1746 F
= QuadraticField(-1, 'I')
1751 t
= z
.coefficient_tuple()
1756 cplxM
= matrix(F
, 2, [[ a
+ b
*i
, c
+ d
*i
],
1757 [-c
+ d
*i
, a
- b
*i
]])
1758 realM
= ComplexMatrixEuclideanJordanAlgebra
.real_embed(cplxM
)
1759 blocks
.append(realM
)
1761 # We should have real entries by now, so use the realest field
1762 # we've got for the return value.
1763 return matrix
.block(quaternions
.base_ring(), n
, blocks
)
1768 def real_unembed(M
):
1770 The inverse of _embed_quaternion_matrix().
1774 sage: from mjo.eja.eja_algebra import \
1775 ....: QuaternionMatrixEuclideanJordanAlgebra
1779 sage: M = matrix(QQ, [[ 1, 2, 3, 4],
1780 ....: [-2, 1, -4, 3],
1781 ....: [-3, 4, 1, -2],
1782 ....: [-4, -3, 2, 1]])
1783 sage: QuaternionMatrixEuclideanJordanAlgebra.real_unembed(M)
1784 [1 + 2*i + 3*j + 4*k]
1788 Unembedding is the inverse of embedding::
1790 sage: set_random_seed()
1791 sage: Q = QuaternionAlgebra(QQ, -1, -1)
1792 sage: M = random_matrix(Q, 3)
1793 sage: Me = QuaternionMatrixEuclideanJordanAlgebra.real_embed(M)
1794 sage: QuaternionMatrixEuclideanJordanAlgebra.real_unembed(Me) == M
1800 raise ValueError("the matrix 'M' must be square")
1801 if not n
.mod(4).is_zero():
1802 raise ValueError("the matrix 'M' must be a quaternion embedding")
1804 # Use the base ring of the matrix to ensure that its entries can be
1805 # multiplied by elements of the quaternion algebra.
1806 field
= M
.base_ring()
1807 Q
= QuaternionAlgebra(field
,-1,-1)
1810 # Go top-left to bottom-right (reading order), converting every
1811 # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1
1814 for l
in range(n
/4):
1815 for m
in range(n
/4):
1816 submat
= ComplexMatrixEuclideanJordanAlgebra
.real_unembed(
1817 M
[4*l
:4*l
+4,4*m
:4*m
+4] )
1818 if submat
[0,0] != submat
[1,1].conjugate():
1819 raise ValueError('bad on-diagonal submatrix')
1820 if submat
[0,1] != -submat
[1,0].conjugate():
1821 raise ValueError('bad off-diagonal submatrix')
1822 z
= submat
[0,0].real()
1823 z
+= submat
[0,0].imag()*i
1824 z
+= submat
[0,1].real()*j
1825 z
+= submat
[0,1].imag()*k
1828 return matrix(Q
, n
/4, elements
)
1832 def natural_inner_product(cls
,X
,Y
):
1834 Compute a natural inner product in this algebra directly from
1839 sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
1843 This gives the same answer as the slow, default method implemented
1844 in :class:`MatrixEuclideanJordanAlgebra`::
1846 sage: set_random_seed()
1847 sage: J = QuaternionHermitianEJA.random_instance()
1848 sage: x,y = J.random_elements(2)
1849 sage: Xe = x.natural_representation()
1850 sage: Ye = y.natural_representation()
1851 sage: X = QuaternionHermitianEJA.real_unembed(Xe)
1852 sage: Y = QuaternionHermitianEJA.real_unembed(Ye)
1853 sage: expected = (X*Y).trace().coefficient_tuple()[0]
1854 sage: actual = QuaternionHermitianEJA.natural_inner_product(Xe,Ye)
1855 sage: actual == expected
1859 return RealMatrixEuclideanJordanAlgebra
.natural_inner_product(X
,Y
)/4
1862 class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra
,
1865 The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
1866 matrices, the usual symmetric Jordan product, and the
1867 real-part-of-trace inner product. It has dimension `2n^2 - n` over
1872 sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
1876 In theory, our "field" can be any subfield of the reals::
1878 sage: QuaternionHermitianEJA(2, RDF)
1879 Euclidean Jordan algebra of dimension 6 over Real Double Field
1880 sage: QuaternionHermitianEJA(2, RR)
1881 Euclidean Jordan algebra of dimension 6 over Real Field with
1882 53 bits of precision
1886 The dimension of this algebra is `2*n^2 - n`::
1888 sage: set_random_seed()
1889 sage: n_max = QuaternionHermitianEJA._max_test_case_size()
1890 sage: n = ZZ.random_element(1, n_max)
1891 sage: J = QuaternionHermitianEJA(n)
1892 sage: J.dimension() == 2*(n^2) - n
1895 The Jordan multiplication is what we think it is::
1897 sage: set_random_seed()
1898 sage: J = QuaternionHermitianEJA.random_instance()
1899 sage: x,y = J.random_elements(2)
1900 sage: actual = (x*y).natural_representation()
1901 sage: X = x.natural_representation()
1902 sage: Y = y.natural_representation()
1903 sage: expected = (X*Y + Y*X)/2
1904 sage: actual == expected
1906 sage: J(expected) == x*y
1909 We can change the generator prefix::
1911 sage: QuaternionHermitianEJA(2, prefix='a').gens()
1912 (a0, a1, a2, a3, a4, a5)
1914 Our natural basis is normalized with respect to the natural inner
1915 product unless we specify otherwise::
1917 sage: set_random_seed()
1918 sage: J = QuaternionHermitianEJA.random_instance()
1919 sage: all( b.norm() == 1 for b in J.gens() )
1922 Since our natural basis is normalized with respect to the natural
1923 inner product, and since we know that this algebra is an EJA, any
1924 left-multiplication operator's matrix will be symmetric because
1925 natural->EJA basis representation is an isometry and within the EJA
1926 the operator is self-adjoint by the Jordan axiom::
1928 sage: set_random_seed()
1929 sage: x = QuaternionHermitianEJA.random_instance().random_element()
1930 sage: x.operator().matrix().is_symmetric()
1935 def _denormalized_basis(cls
, n
, field
):
1937 Returns a basis for the space of quaternion Hermitian n-by-n matrices.
1939 Why do we embed these? Basically, because all of numerical
1940 linear algebra assumes that you're working with vectors consisting
1941 of `n` entries from a field and scalars from the same field. There's
1942 no way to tell SageMath that (for example) the vectors contain
1943 complex numbers, while the scalar field is real.
1947 sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
1951 sage: set_random_seed()
1952 sage: n = ZZ.random_element(1,5)
1953 sage: B = QuaternionHermitianEJA._denormalized_basis(n,QQ)
1954 sage: all( M.is_symmetric() for M in B )
1958 Q
= QuaternionAlgebra(QQ
,-1,-1)
1961 # This is like the symmetric case, but we need to be careful:
1963 # * We want conjugate-symmetry, not just symmetry.
1964 # * The diagonal will (as a result) be real.
1968 for j
in range(i
+1):
1969 Eij
= matrix(Q
, n
, lambda k
,l
: k
==i
and l
==j
)
1971 Sij
= cls
.real_embed(Eij
)
1974 # The second, third, and fourth ones have a minus
1975 # because they're conjugated.
1976 Sij_real
= cls
.real_embed(Eij
+ Eij
.transpose())
1978 Sij_I
= cls
.real_embed(I
*Eij
- I
*Eij
.transpose())
1980 Sij_J
= cls
.real_embed(J
*Eij
- J
*Eij
.transpose())
1982 Sij_K
= cls
.real_embed(K
*Eij
- K
*Eij
.transpose())
1985 # Since we embedded these, we can drop back to the "field" that we
1986 # started with instead of the quaternion algebra "Q".
1987 return ( s
.change_ring(field
) for s
in S
)
1990 def __init__(self
, n
, field
=AA
, **kwargs
):
1991 basis
= self
._denormalized
_basis
(n
,field
)
1992 super(QuaternionHermitianEJA
,self
).__init
__(field
, basis
, n
, **kwargs
)
1995 class BilinearFormEJA(FiniteDimensionalEuclideanJordanAlgebra
, KnownRankEJA
):
1997 The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
1998 with the half-trace inner product and jordan product ``x*y =
1999 (x0*y0 + <B*x_bar,y_bar>, x0*y_bar + y0*x_bar)`` where ``B`` is a
2000 symmetric positive-definite "bilinear form" matrix. It has
2001 dimension `n` over the reals, and reduces to the ``JordanSpinEJA``
2002 when ``B`` is the identity matrix of order ``n-1``.
2006 sage: from mjo.eja.eja_algebra import (BilinearFormEJA,
2007 ....: JordanSpinEJA)
2011 When no bilinear form is specified, the identity matrix is used,
2012 and the resulting algebra is the Jordan spin algebra::
2014 sage: J0 = BilinearFormEJA(3)
2015 sage: J1 = JordanSpinEJA(3)
2016 sage: J0.multiplication_table() == J0.multiplication_table()
2021 We can create a zero-dimensional algebra::
2023 sage: J = BilinearFormEJA(0)
2027 We can check the multiplication condition given in the Jordan, von
2028 Neumann, and Wigner paper (and also discussed on my "On the
2029 symmetry..." paper). Note that this relies heavily on the standard
2030 choice of basis, as does anything utilizing the bilinear form matrix::
2032 sage: set_random_seed()
2033 sage: n = ZZ.random_element(5)
2034 sage: M = matrix.random(QQ, max(0,n-1), algorithm='unimodular')
2035 sage: B = M.transpose()*M
2036 sage: J = BilinearFormEJA(n, B=B)
2037 sage: eis = VectorSpace(M.base_ring(), M.ncols()).basis()
2038 sage: V = J.vector_space()
2039 sage: sis = [ J.from_vector(V([0] + (M.inverse()*ei).list()))
2040 ....: for ei in eis ]
2041 sage: actual = [ sis[i]*sis[j]
2042 ....: for i in range(n-1)
2043 ....: for j in range(n-1) ]
2044 sage: expected = [ J.one() if i == j else J.zero()
2045 ....: for i in range(n-1)
2046 ....: for j in range(n-1) ]
2047 sage: actual == expected
2050 def __init__(self
, n
, field
=AA
, B
=None, **kwargs
):
2052 self
._B
= matrix
.identity(field
, max(0,n
-1))
2056 V
= VectorSpace(field
, n
)
2057 mult_table
= [[V
.zero() for j
in range(n
)] for i
in range(n
)]
2066 z0
= x0
*y0
+ (self
._B
*xbar
).inner_product(ybar
)
2067 zbar
= y0
*xbar
+ x0
*ybar
2068 z
= V([z0
] + zbar
.list())
2069 mult_table
[i
][j
] = z
2071 # The rank of this algebra is two, unless we're in a
2072 # one-dimensional ambient space (because the rank is bounded
2073 # by the ambient dimension).
2074 fdeja
= super(BilinearFormEJA
, self
)
2075 return fdeja
.__init
__(field
, mult_table
, rank
=min(n
,2), **kwargs
)
2077 def inner_product(self
, x
, y
):
2079 Half of the trace inner product.
2081 This is defined so that the special case of the Jordan spin
2082 algebra gets the usual inner product.
2086 sage: from mjo.eja.eja_algebra import BilinearFormEJA
2090 Ensure that this is one-half of the trace inner-product when
2091 the algebra isn't just the reals (when ``n`` isn't one). This
2092 is in Faraut and Koranyi, and also my "On the symmetry..."
2095 sage: set_random_seed()
2096 sage: n = ZZ.random_element(2,5)
2097 sage: M = matrix.random(QQ, max(0,n-1), algorithm='unimodular')
2098 sage: B = M.transpose()*M
2099 sage: J = BilinearFormEJA(n, B=B)
2100 sage: x = J.random_element()
2101 sage: y = J.random_element()
2102 sage: x.inner_product(y) == (x*y).trace()/2
2106 xvec
= x
.to_vector()
2108 yvec
= y
.to_vector()
2110 return x
[0]*y
[0] + (self
._B
*xbar
).inner_product(ybar
)
2113 class JordanSpinEJA(BilinearFormEJA
):
2115 The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
2116 with the usual inner product and jordan product ``x*y =
2117 (<x,y>, x0*y_bar + y0*x_bar)``. It has dimension `n` over
2122 sage: from mjo.eja.eja_algebra import JordanSpinEJA
2126 This multiplication table can be verified by hand::
2128 sage: J = JordanSpinEJA(4)
2129 sage: e0,e1,e2,e3 = J.gens()
2145 We can change the generator prefix::
2147 sage: JordanSpinEJA(2, prefix='B').gens()
2152 Ensure that we have the usual inner product on `R^n`::
2154 sage: set_random_seed()
2155 sage: J = JordanSpinEJA.random_instance()
2156 sage: x,y = J.random_elements(2)
2157 sage: X = x.natural_representation()
2158 sage: Y = y.natural_representation()
2159 sage: x.inner_product(y) == J.natural_inner_product(X,Y)
2163 def __init__(self
, n
, field
=AA
, **kwargs
):
2164 # This is a special case of the BilinearFormEJA with the identity
2165 # matrix as its bilinear form.
2166 return super(JordanSpinEJA
, self
).__init
__(n
, field
, **kwargs
)
2169 class TrivialEJA(FiniteDimensionalEuclideanJordanAlgebra
, KnownRankEJA
):
2171 The trivial Euclidean Jordan algebra consisting of only a zero element.
2175 sage: from mjo.eja.eja_algebra import TrivialEJA
2179 sage: J = TrivialEJA()
2186 sage: 7*J.one()*12*J.one()
2188 sage: J.one().inner_product(J.one())
2190 sage: J.one().norm()
2192 sage: J.one().subalgebra_generated_by()
2193 Euclidean Jordan algebra of dimension 0 over Algebraic Real Field
2198 def __init__(self
, field
=AA
, **kwargs
):
2200 fdeja
= super(TrivialEJA
, self
)
2201 # The rank is zero using my definition, namely the dimension of the
2202 # largest subalgebra generated by any element.
2203 return fdeja
.__init
__(field
, mult_table
, rank
=0, **kwargs
)