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 sage
.algebras
.quatalg
.quaternion_algebra
import QuaternionAlgebra
9 from sage
.categories
.finite_dimensional_algebras_with_basis
import FiniteDimensionalAlgebrasWithBasis
10 from sage
.combinat
.free_module
import CombinatorialFreeModule
11 from sage
.matrix
.constructor
import matrix
12 from sage
.misc
.cachefunc
import cached_method
13 from sage
.misc
.prandom
import choice
14 from sage
.modules
.free_module
import VectorSpace
15 from sage
.rings
.integer_ring
import ZZ
16 from sage
.rings
.number_field
.number_field
import QuadraticField
17 from sage
.rings
.polynomial
.polynomial_ring_constructor
import PolynomialRing
18 from sage
.rings
.rational_field
import QQ
19 from sage
.structure
.element
import is_Matrix
21 from mjo
.eja
.eja_element
import FiniteDimensionalEuclideanJordanAlgebraElement
22 from mjo
.eja
.eja_utils
import _mat2vec
24 class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule
):
35 sage: from mjo.eja.eja_algebra import random_eja
39 By definition, Jordan multiplication commutes::
41 sage: set_random_seed()
42 sage: J = random_eja()
43 sage: x = J.random_element()
44 sage: y = J.random_element()
50 self
._natural
_basis
= natural_basis
53 category
= FiniteDimensionalAlgebrasWithBasis(field
).Unital()
54 fda
= super(FiniteDimensionalEuclideanJordanAlgebra
, self
)
56 range(len(mult_table
)),
59 self
.print_options(bracket
='')
61 # The multiplication table we're given is necessarily in terms
62 # of vectors, because we don't have an algebra yet for
63 # anything to be an element of. However, it's faster in the
64 # long run to have the multiplication table be in terms of
65 # algebra elements. We do this after calling the superclass
66 # constructor so that from_vector() knows what to do.
67 self
._multiplication
_table
= [ map(lambda x
: self
.from_vector(x
), ls
)
68 for ls
in mult_table
]
71 def _element_constructor_(self
, elt
):
73 Construct an element of this algebra from its natural
76 This gets called only after the parent element _call_ method
77 fails to find a coercion for the argument.
81 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
82 ....: RealCartesianProductEJA,
83 ....: RealSymmetricEJA)
87 The identity in `S^n` is converted to the identity in the EJA::
89 sage: J = RealSymmetricEJA(3)
90 sage: I = matrix.identity(QQ,3)
94 This skew-symmetric matrix can't be represented in the EJA::
96 sage: J = RealSymmetricEJA(3)
97 sage: A = matrix(QQ,3, lambda i,j: i-j)
99 Traceback (most recent call last):
101 ArithmeticError: vector is not in free module
105 Ensure that we can convert any element of the two non-matrix
106 simple algebras (whose natural representations are their usual
107 vector representations) back and forth faithfully::
109 sage: set_random_seed()
110 sage: J = RealCartesianProductEJA(5)
111 sage: x = J.random_element()
112 sage: J(x.to_vector().column()) == x
114 sage: J = JordanSpinEJA(5)
115 sage: x = J.random_element()
116 sage: J(x.to_vector().column()) == x
120 natural_basis
= self
.natural_basis()
121 if elt
not in natural_basis
[0].matrix_space():
122 raise ValueError("not a naturally-represented algebra element")
124 # Thanks for nothing! Matrix spaces aren't vector
125 # spaces in Sage, so we have to figure out its
126 # natural-basis coordinates ourselves.
127 V
= VectorSpace(elt
.base_ring(), elt
.nrows()*elt
.ncols())
128 W
= V
.span_of_basis( _mat2vec(s
) for s
in natural_basis
)
129 coords
= W
.coordinate_vector(_mat2vec(elt
))
130 return self
.from_vector(coords
)
135 Return a string representation of ``self``.
139 sage: from mjo.eja.eja_algebra import JordanSpinEJA
143 Ensure that it says what we think it says::
145 sage: JordanSpinEJA(2, field=QQ)
146 Euclidean Jordan algebra of degree 2 over Rational Field
147 sage: JordanSpinEJA(3, field=RDF)
148 Euclidean Jordan algebra of degree 3 over Real Double Field
151 # TODO: change this to say "dimension" and fix all the tests.
152 fmt
= "Euclidean Jordan algebra of degree {} over {}"
153 return fmt
.format(self
.dimension(), self
.base_ring())
155 def product_on_basis(self
, i
, j
):
156 return self
._multiplication
_table
[i
][j
]
158 def _a_regular_element(self
):
160 Guess a regular element. Needed to compute the basis for our
161 characteristic polynomial coefficients.
165 sage: from mjo.eja.eja_algebra import random_eja
169 Ensure that this hacky method succeeds for every algebra that we
170 know how to construct::
172 sage: set_random_seed()
173 sage: J = random_eja()
174 sage: J._a_regular_element().is_regular()
179 z
= self
.sum( (i
+1)*gs
[i
] for i
in range(len(gs
)) )
180 if not z
.is_regular():
181 raise ValueError("don't know a regular element")
186 def _charpoly_basis_space(self
):
188 Return the vector space spanned by the basis used in our
189 characteristic polynomial coefficients. This is used not only to
190 compute those coefficients, but also any time we need to
191 evaluate the coefficients (like when we compute the trace or
194 z
= self
._a
_regular
_element
()
195 V
= self
.vector_space()
196 V1
= V
.span_of_basis( (z
**k
).to_vector() for k
in range(self
.rank()) )
197 b
= (V1
.basis() + V1
.complement().basis())
198 return V
.span_of_basis(b
)
202 def _charpoly_coeff(self
, i
):
204 Return the coefficient polynomial "a_{i}" of this algebra's
205 general characteristic polynomial.
207 Having this be a separate cached method lets us compute and
208 store the trace/determinant (a_{r-1} and a_{0} respectively)
209 separate from the entire characteristic polynomial.
211 (A_of_x
, x
, xr
, detA
) = self
._charpoly
_matrix
_system
()
212 R
= A_of_x
.base_ring()
214 # Guaranteed by theory
217 # Danger: the in-place modification is done for performance
218 # reasons (reconstructing a matrix with huge polynomial
219 # entries is slow), but I don't know how cached_method works,
220 # so it's highly possible that we're modifying some global
221 # list variable by reference, here. In other words, you
222 # probably shouldn't call this method twice on the same
223 # algebra, at the same time, in two threads
224 Ai_orig
= A_of_x
.column(i
)
225 A_of_x
.set_column(i
,xr
)
226 numerator
= A_of_x
.det()
227 A_of_x
.set_column(i
,Ai_orig
)
229 # We're relying on the theory here to ensure that each a_i is
230 # indeed back in R, and the added negative signs are to make
231 # the whole charpoly expression sum to zero.
232 return R(-numerator
/detA
)
236 def _charpoly_matrix_system(self
):
238 Compute the matrix whose entries A_ij are polynomials in
239 X1,...,XN, the vector ``x`` of variables X1,...,XN, the vector
240 corresponding to `x^r` and the determinent of the matrix A =
241 [A_ij]. In other words, all of the fixed (cachable) data needed
242 to compute the coefficients of the characteristic polynomial.
247 # Construct a new algebra over a multivariate polynomial ring...
248 names
= tuple('X' + str(i
) for i
in range(1,n
+1))
249 R
= PolynomialRing(self
.base_ring(), names
)
250 # Hack around the fact that our multiplication table is in terms of
251 # algebra elements but the constructor wants it in terms of vectors.
252 vmt
= [ tuple(map(lambda x
: x
.to_vector(), ls
))
253 for ls
in self
._multiplication
_table
]
254 J
= FiniteDimensionalEuclideanJordanAlgebra(R
, tuple(vmt
), r
)
256 idmat
= matrix
.identity(J
.base_ring(), n
)
258 W
= self
._charpoly
_basis
_space
()
259 W
= W
.change_ring(R
.fraction_field())
261 # Starting with the standard coordinates x = (X1,X2,...,Xn)
262 # and then converting the entries to W-coordinates allows us
263 # to pass in the standard coordinates to the charpoly and get
264 # back the right answer. Specifically, with x = (X1,X2,...,Xn),
267 # W.coordinates(x^2) eval'd at (standard z-coords)
271 # W-coords of (standard coords of x^2 eval'd at std-coords of z)
273 # We want the middle equivalent thing in our matrix, but use
274 # the first equivalent thing instead so that we can pass in
275 # standard coordinates.
276 x
= J
.from_vector(W(R
.gens()))
278 # Handle the zeroth power separately, because computing
279 # the unit element in J is mathematically suspect.
280 x0
= W
.coordinate_vector(self
.one().to_vector())
282 l1
+= [ W
.coordinate_vector((x
**k
).to_vector()).column()
283 for k
in range(1,r
) ]
284 l2
= [idmat
.column(k
-1).column() for k
in range(r
+1, n
+1)]
285 A_of_x
= matrix
.block(R
, 1, n
, (l1
+ l2
))
286 xr
= W
.coordinate_vector((x
**r
).to_vector())
287 return (A_of_x
, x
, xr
, A_of_x
.det())
291 def characteristic_polynomial(self
):
293 Return a characteristic polynomial that works for all elements
296 The resulting polynomial has `n+1` variables, where `n` is the
297 dimension of this algebra. The first `n` variables correspond to
298 the coordinates of an algebra element: when evaluated at the
299 coordinates of an algebra element with respect to a certain
300 basis, the result is a univariate polynomial (in the one
301 remaining variable ``t``), namely the characteristic polynomial
306 sage: from mjo.eja.eja_algebra import JordanSpinEJA
310 The characteristic polynomial in the spin algebra is given in
311 Alizadeh, Example 11.11::
313 sage: J = JordanSpinEJA(3)
314 sage: p = J.characteristic_polynomial(); p
315 X1^2 - X2^2 - X3^2 + (-2*t)*X1 + t^2
316 sage: xvec = J.one().to_vector()
324 # The list of coefficient polynomials a_1, a_2, ..., a_n.
325 a
= [ self
._charpoly
_coeff
(i
) for i
in range(n
) ]
327 # We go to a bit of trouble here to reorder the
328 # indeterminates, so that it's easier to evaluate the
329 # characteristic polynomial at x's coordinates and get back
330 # something in terms of t, which is what we want.
332 S
= PolynomialRing(self
.base_ring(),'t')
334 S
= PolynomialRing(S
, R
.variable_names())
337 # Note: all entries past the rth should be zero. The
338 # coefficient of the highest power (x^r) is 1, but it doesn't
339 # appear in the solution vector which contains coefficients
340 # for the other powers (to make them sum to x^r).
342 a
[r
] = 1 # corresponds to x^r
344 # When the rank is equal to the dimension, trying to
345 # assign a[r] goes out-of-bounds.
346 a
.append(1) # corresponds to x^r
348 return sum( a
[k
]*(t
**k
) for k
in range(len(a
)) )
351 def inner_product(self
, x
, y
):
353 The inner product associated with this Euclidean Jordan algebra.
355 Defaults to the trace inner product, but can be overridden by
356 subclasses if they are sure that the necessary properties are
361 sage: from mjo.eja.eja_algebra import random_eja
365 The inner product must satisfy its axiom for this algebra to truly
366 be a Euclidean Jordan Algebra::
368 sage: set_random_seed()
369 sage: J = random_eja()
370 sage: x = J.random_element()
371 sage: y = J.random_element()
372 sage: z = J.random_element()
373 sage: (x*y).inner_product(z) == y.inner_product(x*z)
377 if (not x
in self
) or (not y
in self
):
378 raise TypeError("arguments must live in this algebra")
379 return x
.trace_inner_product(y
)
382 def multiplication_table(self
):
384 Return a readable matrix representation of this algebra's
385 multiplication table. The (i,j)th entry in the matrix contains
386 the product of the ith basis element with the jth.
388 This is not extraordinarily useful, but it overrides a superclass
389 method that would otherwise just crash and complain about the
390 algebra being infinite.
394 sage: J = RealCartesianProductEJA(3)
395 sage: J.multiplication_table()
402 sage: J = JordanSpinEJA(3)
403 sage: J.multiplication_table()
409 return matrix(self
._multiplication
_table
)
412 def natural_basis(self
):
414 Return a more-natural representation of this algebra's basis.
416 Every finite-dimensional Euclidean Jordan Algebra is a direct
417 sum of five simple algebras, four of which comprise Hermitian
418 matrices. This method returns the original "natural" basis
419 for our underlying vector space. (Typically, the natural basis
420 is used to construct the multiplication table in the first place.)
422 Note that this will always return a matrix. The standard basis
423 in `R^n` will be returned as `n`-by-`1` column matrices.
427 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
428 ....: RealSymmetricEJA)
432 sage: J = RealSymmetricEJA(2)
434 Finite family {0: e0, 1: e1, 2: e2}
435 sage: J.natural_basis()
443 sage: J = JordanSpinEJA(2)
445 Finite family {0: e0, 1: e1}
446 sage: J.natural_basis()
453 if self
._natural
_basis
is None:
454 return tuple( b
.to_vector().column() for b
in self
.basis() )
456 return self
._natural
_basis
462 Return the unit element of this algebra.
466 sage: from mjo.eja.eja_algebra import (RealCartesianProductEJA,
471 sage: J = RealCartesianProductEJA(5)
473 e0 + e1 + e2 + e3 + e4
477 The identity element acts like the identity::
479 sage: set_random_seed()
480 sage: J = random_eja()
481 sage: x = J.random_element()
482 sage: J.one()*x == x and x*J.one() == x
485 The matrix of the unit element's operator is the identity::
487 sage: set_random_seed()
488 sage: J = random_eja()
489 sage: actual = J.one().operator().matrix()
490 sage: expected = matrix.identity(J.base_ring(), J.dimension())
491 sage: actual == expected
495 # We can brute-force compute the matrices of the operators
496 # that correspond to the basis elements of this algebra.
497 # If some linear combination of those basis elements is the
498 # algebra identity, then the same linear combination of
499 # their matrices has to be the identity matrix.
501 # Of course, matrices aren't vectors in sage, so we have to
502 # appeal to the "long vectors" isometry.
503 oper_vecs
= [ _mat2vec(g
.operator().matrix()) for g
in self
.gens() ]
505 # Now we use basis linear algebra to find the coefficients,
506 # of the matrices-as-vectors-linear-combination, which should
507 # work for the original algebra basis too.
508 A
= matrix
.column(self
.base_ring(), oper_vecs
)
510 # We used the isometry on the left-hand side already, but we
511 # still need to do it for the right-hand side. Recall that we
512 # wanted something that summed to the identity matrix.
513 b
= _mat2vec( matrix
.identity(self
.base_ring(), self
.dimension()) )
515 # Now if there's an identity element in the algebra, this should work.
516 coeffs
= A
.solve_right(b
)
517 return self
.linear_combination(zip(self
.gens(), coeffs
))
522 Return the rank of this EJA.
526 The author knows of no algorithm to compute the rank of an EJA
527 where only the multiplication table is known. In lieu of one, we
528 require the rank to be specified when the algebra is created,
529 and simply pass along that number here.
533 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
534 ....: RealSymmetricEJA,
535 ....: ComplexHermitianEJA,
536 ....: QuaternionHermitianEJA,
541 The rank of the Jordan spin algebra is always two::
543 sage: JordanSpinEJA(2).rank()
545 sage: JordanSpinEJA(3).rank()
547 sage: JordanSpinEJA(4).rank()
550 The rank of the `n`-by-`n` Hermitian real, complex, or
551 quaternion matrices is `n`::
553 sage: RealSymmetricEJA(2).rank()
555 sage: ComplexHermitianEJA(2).rank()
557 sage: QuaternionHermitianEJA(2).rank()
559 sage: RealSymmetricEJA(5).rank()
561 sage: ComplexHermitianEJA(5).rank()
563 sage: QuaternionHermitianEJA(5).rank()
568 Ensure that every EJA that we know how to construct has a
569 positive integer rank::
571 sage: set_random_seed()
572 sage: r = random_eja().rank()
573 sage: r in ZZ and r > 0
580 def vector_space(self
):
582 Return the vector space that underlies this algebra.
586 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
590 sage: J = RealSymmetricEJA(2)
591 sage: J.vector_space()
592 Vector space of dimension 3 over Rational Field
595 return self
.zero().to_vector().parent().ambient_vector_space()
598 Element
= FiniteDimensionalEuclideanJordanAlgebraElement
601 class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra
):
603 Return the Euclidean Jordan Algebra corresponding to the set
604 `R^n` under the Hadamard product.
606 Note: this is nothing more than the Cartesian product of ``n``
607 copies of the spin algebra. Once Cartesian product algebras
608 are implemented, this can go.
612 sage: from mjo.eja.eja_algebra import RealCartesianProductEJA
616 This multiplication table can be verified by hand::
618 sage: J = RealCartesianProductEJA(3)
619 sage: e0,e1,e2 = J.gens()
634 def __init__(self
, n
, field
=QQ
):
635 V
= VectorSpace(field
, n
)
636 mult_table
= [ [ V
.basis()[i
]*(i
== j
) for i
in range(n
) ]
639 fdeja
= super(RealCartesianProductEJA
, self
)
640 return fdeja
.__init
__(field
, mult_table
, rank
=n
)
642 def inner_product(self
, x
, y
):
643 return _usual_ip(x
,y
)
648 Return a "random" finite-dimensional Euclidean Jordan Algebra.
652 For now, we choose a random natural number ``n`` (greater than zero)
653 and then give you back one of the following:
655 * The cartesian product of the rational numbers ``n`` times; this is
656 ``QQ^n`` with the Hadamard product.
658 * The Jordan spin algebra on ``QQ^n``.
660 * The ``n``-by-``n`` rational symmetric matrices with the symmetric
663 * The ``n``-by-``n`` complex-rational Hermitian matrices embedded
664 in the space of ``2n``-by-``2n`` real symmetric matrices.
666 * The ``n``-by-``n`` quaternion-rational Hermitian matrices embedded
667 in the space of ``4n``-by-``4n`` real symmetric matrices.
669 Later this might be extended to return Cartesian products of the
674 sage: from mjo.eja.eja_algebra import random_eja
679 Euclidean Jordan algebra of degree...
683 # The max_n component lets us choose different upper bounds on the
684 # value "n" that gets passed to the constructor. This is needed
685 # because e.g. R^{10} is reasonable to test, while the Hermitian
686 # 10-by-10 quaternion matrices are not.
687 (constructor
, max_n
) = choice([(RealCartesianProductEJA
, 6),
689 (RealSymmetricEJA
, 5),
690 (ComplexHermitianEJA
, 4),
691 (QuaternionHermitianEJA
, 3)])
692 n
= ZZ
.random_element(1, max_n
)
693 return constructor(n
, field
=QQ
)
697 def _real_symmetric_basis(n
, field
=QQ
):
699 Return a basis for the space of real symmetric n-by-n matrices.
701 # The basis of symmetric matrices, as matrices, in their R^(n-by-n)
705 for j
in xrange(i
+1):
706 Eij
= matrix(field
, n
, lambda k
,l
: k
==i
and l
==j
)
710 # Beware, orthogonal but not normalized!
711 Sij
= Eij
+ Eij
.transpose()
716 def _complex_hermitian_basis(n
, field
=QQ
):
718 Returns a basis for the space of complex Hermitian n-by-n matrices.
722 sage: from mjo.eja.eja_algebra import _complex_hermitian_basis
726 sage: set_random_seed()
727 sage: n = ZZ.random_element(1,5)
728 sage: all( M.is_symmetric() for M in _complex_hermitian_basis(n) )
732 F
= QuadraticField(-1, 'I')
735 # This is like the symmetric case, but we need to be careful:
737 # * We want conjugate-symmetry, not just symmetry.
738 # * The diagonal will (as a result) be real.
742 for j
in xrange(i
+1):
743 Eij
= matrix(field
, n
, lambda k
,l
: k
==i
and l
==j
)
745 Sij
= _embed_complex_matrix(Eij
)
748 # Beware, orthogonal but not normalized! The second one
749 # has a minus because it's conjugated.
750 Sij_real
= _embed_complex_matrix(Eij
+ Eij
.transpose())
752 Sij_imag
= _embed_complex_matrix(I
*Eij
- I
*Eij
.transpose())
757 def _quaternion_hermitian_basis(n
, field
=QQ
):
759 Returns a basis for the space of quaternion Hermitian n-by-n matrices.
763 sage: from mjo.eja.eja_algebra import _quaternion_hermitian_basis
767 sage: set_random_seed()
768 sage: n = ZZ.random_element(1,5)
769 sage: all( M.is_symmetric() for M in _quaternion_hermitian_basis(n) )
773 Q
= QuaternionAlgebra(QQ
,-1,-1)
776 # This is like the symmetric case, but we need to be careful:
778 # * We want conjugate-symmetry, not just symmetry.
779 # * The diagonal will (as a result) be real.
783 for j
in xrange(i
+1):
784 Eij
= matrix(Q
, n
, lambda k
,l
: k
==i
and l
==j
)
786 Sij
= _embed_quaternion_matrix(Eij
)
789 # Beware, orthogonal but not normalized! The second,
790 # third, and fourth ones have a minus because they're
792 Sij_real
= _embed_quaternion_matrix(Eij
+ Eij
.transpose())
794 Sij_I
= _embed_quaternion_matrix(I
*Eij
- I
*Eij
.transpose())
796 Sij_J
= _embed_quaternion_matrix(J
*Eij
- J
*Eij
.transpose())
798 Sij_K
= _embed_quaternion_matrix(K
*Eij
- K
*Eij
.transpose())
804 def _multiplication_table_from_matrix_basis(basis
):
806 At least three of the five simple Euclidean Jordan algebras have the
807 symmetric multiplication (A,B) |-> (AB + BA)/2, where the
808 multiplication on the right is matrix multiplication. Given a basis
809 for the underlying matrix space, this function returns a
810 multiplication table (obtained by looping through the basis
811 elements) for an algebra of those matrices.
813 # In S^2, for example, we nominally have four coordinates even
814 # though the space is of dimension three only. The vector space V
815 # is supposed to hold the entire long vector, and the subspace W
816 # of V will be spanned by the vectors that arise from symmetric
817 # matrices. Thus for S^2, dim(V) == 4 and dim(W) == 3.
818 field
= basis
[0].base_ring()
819 dimension
= basis
[0].nrows()
821 V
= VectorSpace(field
, dimension
**2)
822 W
= V
.span_of_basis( _mat2vec(s
) for s
in basis
)
824 mult_table
= [[W
.zero() for i
in range(n
)] for j
in range(n
)]
827 mat_entry
= (basis
[i
]*basis
[j
] + basis
[j
]*basis
[i
])/2
828 mult_table
[i
][j
] = W
.coordinate_vector(_mat2vec(mat_entry
))
833 def _embed_complex_matrix(M
):
835 Embed the n-by-n complex matrix ``M`` into the space of real
836 matrices of size 2n-by-2n via the map the sends each entry `z = a +
837 bi` to the block matrix ``[[a,b],[-b,a]]``.
841 sage: from mjo.eja.eja_algebra import _embed_complex_matrix
845 sage: F = QuadraticField(-1,'i')
846 sage: x1 = F(4 - 2*i)
847 sage: x2 = F(1 + 2*i)
850 sage: M = matrix(F,2,[[x1,x2],[x3,x4]])
851 sage: _embed_complex_matrix(M)
860 Embedding is a homomorphism (isomorphism, in fact)::
862 sage: set_random_seed()
863 sage: n = ZZ.random_element(5)
864 sage: F = QuadraticField(-1, 'i')
865 sage: X = random_matrix(F, n)
866 sage: Y = random_matrix(F, n)
867 sage: actual = _embed_complex_matrix(X) * _embed_complex_matrix(Y)
868 sage: expected = _embed_complex_matrix(X*Y)
869 sage: actual == expected
875 raise ValueError("the matrix 'M' must be square")
876 field
= M
.base_ring()
881 blocks
.append(matrix(field
, 2, [[a
,b
],[-b
,a
]]))
883 # We can drop the imaginaries here.
884 return matrix
.block(field
.base_ring(), n
, blocks
)
887 def _unembed_complex_matrix(M
):
889 The inverse of _embed_complex_matrix().
893 sage: from mjo.eja.eja_algebra import (_embed_complex_matrix,
894 ....: _unembed_complex_matrix)
898 sage: A = matrix(QQ,[ [ 1, 2, 3, 4],
899 ....: [-2, 1, -4, 3],
900 ....: [ 9, 10, 11, 12],
901 ....: [-10, 9, -12, 11] ])
902 sage: _unembed_complex_matrix(A)
904 [ 10*i + 9 12*i + 11]
908 Unembedding is the inverse of embedding::
910 sage: set_random_seed()
911 sage: F = QuadraticField(-1, 'i')
912 sage: M = random_matrix(F, 3)
913 sage: _unembed_complex_matrix(_embed_complex_matrix(M)) == M
919 raise ValueError("the matrix 'M' must be square")
920 if not n
.mod(2).is_zero():
921 raise ValueError("the matrix 'M' must be a complex embedding")
923 F
= QuadraticField(-1, 'i')
926 # Go top-left to bottom-right (reading order), converting every
927 # 2-by-2 block we see to a single complex element.
929 for k
in xrange(n
/2):
930 for j
in xrange(n
/2):
931 submat
= M
[2*k
:2*k
+2,2*j
:2*j
+2]
932 if submat
[0,0] != submat
[1,1]:
933 raise ValueError('bad on-diagonal submatrix')
934 if submat
[0,1] != -submat
[1,0]:
935 raise ValueError('bad off-diagonal submatrix')
936 z
= submat
[0,0] + submat
[0,1]*i
939 return matrix(F
, n
/2, elements
)
942 def _embed_quaternion_matrix(M
):
944 Embed the n-by-n quaternion matrix ``M`` into the space of real
945 matrices of size 4n-by-4n by first sending each quaternion entry
946 `z = a + bi + cj + dk` to the block-complex matrix
947 ``[[a + bi, c+di],[-c + di, a-bi]]`, and then embedding those into
952 sage: from mjo.eja.eja_algebra import _embed_quaternion_matrix
956 sage: Q = QuaternionAlgebra(QQ,-1,-1)
957 sage: i,j,k = Q.gens()
958 sage: x = 1 + 2*i + 3*j + 4*k
959 sage: M = matrix(Q, 1, [[x]])
960 sage: _embed_quaternion_matrix(M)
966 Embedding is a homomorphism (isomorphism, in fact)::
968 sage: set_random_seed()
969 sage: n = ZZ.random_element(5)
970 sage: Q = QuaternionAlgebra(QQ,-1,-1)
971 sage: X = random_matrix(Q, n)
972 sage: Y = random_matrix(Q, n)
973 sage: actual = _embed_quaternion_matrix(X)*_embed_quaternion_matrix(Y)
974 sage: expected = _embed_quaternion_matrix(X*Y)
975 sage: actual == expected
979 quaternions
= M
.base_ring()
982 raise ValueError("the matrix 'M' must be square")
984 F
= QuadraticField(-1, 'i')
989 t
= z
.coefficient_tuple()
994 cplx_matrix
= matrix(F
, 2, [[ a
+ b
*i
, c
+ d
*i
],
995 [-c
+ d
*i
, a
- b
*i
]])
996 blocks
.append(_embed_complex_matrix(cplx_matrix
))
998 # We should have real entries by now, so use the realest field
999 # we've got for the return value.
1000 return matrix
.block(quaternions
.base_ring(), n
, blocks
)
1003 def _unembed_quaternion_matrix(M
):
1005 The inverse of _embed_quaternion_matrix().
1009 sage: from mjo.eja.eja_algebra import (_embed_quaternion_matrix,
1010 ....: _unembed_quaternion_matrix)
1014 sage: M = matrix(QQ, [[ 1, 2, 3, 4],
1015 ....: [-2, 1, -4, 3],
1016 ....: [-3, 4, 1, -2],
1017 ....: [-4, -3, 2, 1]])
1018 sage: _unembed_quaternion_matrix(M)
1019 [1 + 2*i + 3*j + 4*k]
1023 Unembedding is the inverse of embedding::
1025 sage: set_random_seed()
1026 sage: Q = QuaternionAlgebra(QQ, -1, -1)
1027 sage: M = random_matrix(Q, 3)
1028 sage: _unembed_quaternion_matrix(_embed_quaternion_matrix(M)) == M
1034 raise ValueError("the matrix 'M' must be square")
1035 if not n
.mod(4).is_zero():
1036 raise ValueError("the matrix 'M' must be a complex embedding")
1038 Q
= QuaternionAlgebra(QQ
,-1,-1)
1041 # Go top-left to bottom-right (reading order), converting every
1042 # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1
1045 for l
in xrange(n
/4):
1046 for m
in xrange(n
/4):
1047 submat
= _unembed_complex_matrix(M
[4*l
:4*l
+4,4*m
:4*m
+4])
1048 if submat
[0,0] != submat
[1,1].conjugate():
1049 raise ValueError('bad on-diagonal submatrix')
1050 if submat
[0,1] != -submat
[1,0].conjugate():
1051 raise ValueError('bad off-diagonal submatrix')
1052 z
= submat
[0,0].real() + submat
[0,0].imag()*i
1053 z
+= submat
[0,1].real()*j
+ submat
[0,1].imag()*k
1056 return matrix(Q
, n
/4, elements
)
1059 # The usual inner product on R^n.
1061 return x
.to_vector().inner_product(y
.to_vector())
1063 # The inner product used for the real symmetric simple EJA.
1064 # We keep it as a separate function because e.g. the complex
1065 # algebra uses the same inner product, except divided by 2.
1066 def _matrix_ip(X
,Y
):
1067 X_mat
= X
.natural_representation()
1068 Y_mat
= Y
.natural_representation()
1069 return (X_mat
*Y_mat
).trace()
1072 class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1074 The rank-n simple EJA consisting of real symmetric n-by-n
1075 matrices, the usual symmetric Jordan product, and the trace inner
1076 product. It has dimension `(n^2 + n)/2` over the reals.
1080 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
1084 sage: J = RealSymmetricEJA(2)
1085 sage: e0, e1, e2 = J.gens()
1095 The dimension of this algebra is `(n^2 + n) / 2`::
1097 sage: set_random_seed()
1098 sage: n = ZZ.random_element(1,5)
1099 sage: J = RealSymmetricEJA(n)
1100 sage: J.dimension() == (n^2 + n)/2
1103 The Jordan multiplication is what we think it is::
1105 sage: set_random_seed()
1106 sage: n = ZZ.random_element(1,5)
1107 sage: J = RealSymmetricEJA(n)
1108 sage: x = J.random_element()
1109 sage: y = J.random_element()
1110 sage: actual = (x*y).natural_representation()
1111 sage: X = x.natural_representation()
1112 sage: Y = y.natural_representation()
1113 sage: expected = (X*Y + Y*X)/2
1114 sage: actual == expected
1116 sage: J(expected) == x*y
1120 def __init__(self
, n
, field
=QQ
):
1121 S
= _real_symmetric_basis(n
, field
=field
)
1122 Qs
= _multiplication_table_from_matrix_basis(S
)
1124 fdeja
= super(RealSymmetricEJA
, self
)
1125 return fdeja
.__init
__(field
,
1130 def inner_product(self
, x
, y
):
1131 return _matrix_ip(x
,y
)
1134 class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1136 The rank-n simple EJA consisting of complex Hermitian n-by-n
1137 matrices over the real numbers, the usual symmetric Jordan product,
1138 and the real-part-of-trace inner product. It has dimension `n^2` over
1143 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
1147 The dimension of this algebra is `n^2`::
1149 sage: set_random_seed()
1150 sage: n = ZZ.random_element(1,5)
1151 sage: J = ComplexHermitianEJA(n)
1152 sage: J.dimension() == n^2
1155 The Jordan multiplication is what we think it is::
1157 sage: set_random_seed()
1158 sage: n = ZZ.random_element(1,5)
1159 sage: J = ComplexHermitianEJA(n)
1160 sage: x = J.random_element()
1161 sage: y = J.random_element()
1162 sage: actual = (x*y).natural_representation()
1163 sage: X = x.natural_representation()
1164 sage: Y = y.natural_representation()
1165 sage: expected = (X*Y + Y*X)/2
1166 sage: actual == expected
1168 sage: J(expected) == x*y
1172 def __init__(self
, n
, field
=QQ
):
1173 S
= _complex_hermitian_basis(n
)
1174 Qs
= _multiplication_table_from_matrix_basis(S
)
1176 fdeja
= super(ComplexHermitianEJA
, self
)
1177 return fdeja
.__init
__(field
,
1183 def inner_product(self
, x
, y
):
1184 # Since a+bi on the diagonal is represented as
1189 # we'll double-count the "a" entries if we take the trace of
1191 return _matrix_ip(x
,y
)/2
1194 class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1196 The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
1197 matrices, the usual symmetric Jordan product, and the
1198 real-part-of-trace inner product. It has dimension `2n^2 - n` over
1203 sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
1207 The dimension of this algebra is `n^2`::
1209 sage: set_random_seed()
1210 sage: n = ZZ.random_element(1,5)
1211 sage: J = QuaternionHermitianEJA(n)
1212 sage: J.dimension() == 2*(n^2) - n
1215 The Jordan multiplication is what we think it is::
1217 sage: set_random_seed()
1218 sage: n = ZZ.random_element(1,5)
1219 sage: J = QuaternionHermitianEJA(n)
1220 sage: x = J.random_element()
1221 sage: y = J.random_element()
1222 sage: actual = (x*y).natural_representation()
1223 sage: X = x.natural_representation()
1224 sage: Y = y.natural_representation()
1225 sage: expected = (X*Y + Y*X)/2
1226 sage: actual == expected
1228 sage: J(expected) == x*y
1232 def __init__(self
, n
, field
=QQ
):
1233 S
= _quaternion_hermitian_basis(n
)
1234 Qs
= _multiplication_table_from_matrix_basis(S
)
1236 fdeja
= super(QuaternionHermitianEJA
, self
)
1237 return fdeja
.__init
__(field
,
1242 def inner_product(self
, x
, y
):
1243 # Since a+bi+cj+dk on the diagonal is represented as
1245 # a + bi +cj + dk = [ a b c d]
1250 # we'll quadruple-count the "a" entries if we take the trace of
1252 return _matrix_ip(x
,y
)/4
1255 class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1257 The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
1258 with the usual inner product and jordan product ``x*y =
1259 (<x_bar,y_bar>, x0*y_bar + y0*x_bar)``. It has dimension `n` over
1264 sage: from mjo.eja.eja_algebra import JordanSpinEJA
1268 This multiplication table can be verified by hand::
1270 sage: J = JordanSpinEJA(4)
1271 sage: e0,e1,e2,e3 = J.gens()
1288 def __init__(self
, n
, field
=QQ
):
1289 V
= VectorSpace(field
, n
)
1290 mult_table
= [[V
.zero() for i
in range(n
)] for j
in range(n
)]
1300 z0
= x
.inner_product(y
)
1301 zbar
= y0
*xbar
+ x0
*ybar
1302 z
= V([z0
] + zbar
.list())
1303 mult_table
[i
][j
] = z
1305 # The rank of the spin algebra is two, unless we're in a
1306 # one-dimensional ambient space (because the rank is bounded by
1307 # the ambient dimension).
1308 fdeja
= super(JordanSpinEJA
, self
)
1309 return fdeja
.__init
__(field
, mult_table
, rank
=min(n
,2))
1311 def inner_product(self
, x
, y
):
1312 return _usual_ip(x
,y
)