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
.magmatic_algebras
import MagmaticAlgebras
10 from sage
.combinat
.free_module
import CombinatorialFreeModule
11 from sage
.matrix
.constructor
import matrix
12 from sage
.matrix
.matrix_space
import MatrixSpace
13 from sage
.misc
.cachefunc
import cached_method
14 from sage
.misc
.prandom
import choice
15 from sage
.misc
.table
import table
16 from sage
.modules
.free_module
import FreeModule
, VectorSpace
17 from sage
.rings
.integer_ring
import ZZ
18 from sage
.rings
.number_field
.number_field
import QuadraticField
19 from sage
.rings
.polynomial
.polynomial_ring_constructor
import PolynomialRing
20 from sage
.rings
.rational_field
import QQ
21 from sage
.structure
.element
import is_Matrix
23 from mjo
.eja
.eja_element
import FiniteDimensionalEuclideanJordanAlgebraElement
24 from mjo
.eja
.eja_utils
import _mat2vec
26 class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule
):
27 # This is an ugly hack needed to prevent the category framework
28 # from implementing a coercion from our base ring (e.g. the
29 # rationals) into the algebra. First of all -- such a coercion is
30 # nonsense to begin with. But more importantly, it tries to do so
31 # in the category of rings, and since our algebras aren't
32 # associative they generally won't be rings.
33 _no_generic_basering_coercion
= True
45 sage: from mjo.eja.eja_algebra import random_eja
49 By definition, Jordan multiplication commutes::
51 sage: set_random_seed()
52 sage: J = random_eja()
53 sage: x = J.random_element()
54 sage: y = J.random_element()
60 self
._natural
_basis
= natural_basis
63 category
= MagmaticAlgebras(field
).FiniteDimensional()
64 category
= category
.WithBasis().Unital()
66 fda
= super(FiniteDimensionalEuclideanJordanAlgebra
, self
)
68 range(len(mult_table
)),
71 self
.print_options(bracket
='')
73 # The multiplication table we're given is necessarily in terms
74 # of vectors, because we don't have an algebra yet for
75 # anything to be an element of. However, it's faster in the
76 # long run to have the multiplication table be in terms of
77 # algebra elements. We do this after calling the superclass
78 # constructor so that from_vector() knows what to do.
79 self
._multiplication
_table
= [ map(lambda x
: self
.from_vector(x
), ls
)
80 for ls
in mult_table
]
83 def _element_constructor_(self
, elt
):
85 Construct an element of this algebra from its natural
88 This gets called only after the parent element _call_ method
89 fails to find a coercion for the argument.
93 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
94 ....: RealCartesianProductEJA,
95 ....: RealSymmetricEJA)
99 The identity in `S^n` is converted to the identity in the EJA::
101 sage: J = RealSymmetricEJA(3)
102 sage: I = matrix.identity(QQ,3)
103 sage: J(I) == J.one()
106 This skew-symmetric matrix can't be represented in the EJA::
108 sage: J = RealSymmetricEJA(3)
109 sage: A = matrix(QQ,3, lambda i,j: i-j)
111 Traceback (most recent call last):
113 ArithmeticError: vector is not in free module
117 Ensure that we can convert any element of the two non-matrix
118 simple algebras (whose natural representations are their usual
119 vector representations) back and forth faithfully::
121 sage: set_random_seed()
122 sage: J = RealCartesianProductEJA(5)
123 sage: x = J.random_element()
124 sage: J(x.to_vector().column()) == x
126 sage: J = JordanSpinEJA(5)
127 sage: x = J.random_element()
128 sage: J(x.to_vector().column()) == x
133 # The superclass implementation of random_element()
134 # needs to be able to coerce "0" into the algebra.
137 natural_basis
= self
.natural_basis()
138 if elt
not in natural_basis
[0].matrix_space():
139 raise ValueError("not a naturally-represented algebra element")
141 # Thanks for nothing! Matrix spaces aren't vector
142 # spaces in Sage, so we have to figure out its
143 # natural-basis coordinates ourselves.
144 V
= VectorSpace(elt
.base_ring(), elt
.nrows()*elt
.ncols())
145 W
= V
.span_of_basis( _mat2vec(s
) for s
in natural_basis
)
146 coords
= W
.coordinate_vector(_mat2vec(elt
))
147 return self
.from_vector(coords
)
152 Return a string representation of ``self``.
156 sage: from mjo.eja.eja_algebra import JordanSpinEJA
160 Ensure that it says what we think it says::
162 sage: JordanSpinEJA(2, field=QQ)
163 Euclidean Jordan algebra of dimension 2 over Rational Field
164 sage: JordanSpinEJA(3, field=RDF)
165 Euclidean Jordan algebra of dimension 3 over Real Double Field
168 fmt
= "Euclidean Jordan algebra of dimension {} over {}"
169 return fmt
.format(self
.dimension(), self
.base_ring())
171 def product_on_basis(self
, i
, j
):
172 return self
._multiplication
_table
[i
][j
]
174 def _a_regular_element(self
):
176 Guess a regular element. Needed to compute the basis for our
177 characteristic polynomial coefficients.
181 sage: from mjo.eja.eja_algebra import random_eja
185 Ensure that this hacky method succeeds for every algebra that we
186 know how to construct::
188 sage: set_random_seed()
189 sage: J = random_eja()
190 sage: J._a_regular_element().is_regular()
195 z
= self
.sum( (i
+1)*gs
[i
] for i
in range(len(gs
)) )
196 if not z
.is_regular():
197 raise ValueError("don't know a regular element")
202 def _charpoly_basis_space(self
):
204 Return the vector space spanned by the basis used in our
205 characteristic polynomial coefficients. This is used not only to
206 compute those coefficients, but also any time we need to
207 evaluate the coefficients (like when we compute the trace or
210 z
= self
._a
_regular
_element
()
211 # Don't use the parent vector space directly here in case this
212 # happens to be a subalgebra. In that case, we would be e.g.
213 # two-dimensional but span_of_basis() would expect three
215 V
= VectorSpace(self
.base_ring(), self
.vector_space().dimension())
216 basis
= [ (z
**k
).to_vector() for k
in range(self
.rank()) ]
217 V1
= V
.span_of_basis( basis
)
218 b
= (V1
.basis() + V1
.complement().basis())
219 return V
.span_of_basis(b
)
223 def _charpoly_coeff(self
, i
):
225 Return the coefficient polynomial "a_{i}" of this algebra's
226 general characteristic polynomial.
228 Having this be a separate cached method lets us compute and
229 store the trace/determinant (a_{r-1} and a_{0} respectively)
230 separate from the entire characteristic polynomial.
232 (A_of_x
, x
, xr
, detA
) = self
._charpoly
_matrix
_system
()
233 R
= A_of_x
.base_ring()
235 # Guaranteed by theory
238 # Danger: the in-place modification is done for performance
239 # reasons (reconstructing a matrix with huge polynomial
240 # entries is slow), but I don't know how cached_method works,
241 # so it's highly possible that we're modifying some global
242 # list variable by reference, here. In other words, you
243 # probably shouldn't call this method twice on the same
244 # algebra, at the same time, in two threads
245 Ai_orig
= A_of_x
.column(i
)
246 A_of_x
.set_column(i
,xr
)
247 numerator
= A_of_x
.det()
248 A_of_x
.set_column(i
,Ai_orig
)
250 # We're relying on the theory here to ensure that each a_i is
251 # indeed back in R, and the added negative signs are to make
252 # the whole charpoly expression sum to zero.
253 return R(-numerator
/detA
)
257 def _charpoly_matrix_system(self
):
259 Compute the matrix whose entries A_ij are polynomials in
260 X1,...,XN, the vector ``x`` of variables X1,...,XN, the vector
261 corresponding to `x^r` and the determinent of the matrix A =
262 [A_ij]. In other words, all of the fixed (cachable) data needed
263 to compute the coefficients of the characteristic polynomial.
268 # Turn my vector space into a module so that "vectors" can
269 # have multivatiate polynomial entries.
270 names
= tuple('X' + str(i
) for i
in range(1,n
+1))
271 R
= PolynomialRing(self
.base_ring(), names
)
273 # Using change_ring() on the parent's vector space doesn't work
274 # here because, in a subalgebra, that vector space has a basis
275 # and change_ring() tries to bring the basis along with it. And
276 # that doesn't work unless the new ring is a PID, which it usually
280 # Now let x = (X1,X2,...,Xn) be the vector whose entries are
284 # And figure out the "left multiplication by x" matrix in
287 monomial_matrices
= [ self
.monomial(i
).operator().matrix()
288 for i
in range(n
) ] # don't recompute these!
290 ek
= self
.monomial(k
).to_vector()
292 sum( x
[i
]*(monomial_matrices
[i
]*ek
)
293 for i
in range(n
) ) )
294 Lx
= matrix
.column(R
, lmbx_cols
)
296 # Now we can compute powers of x "symbolically"
297 x_powers
= [self
.one().to_vector(), x
]
298 for d
in range(2, r
+1):
299 x_powers
.append( Lx
*(x_powers
[-1]) )
301 idmat
= matrix
.identity(R
, n
)
303 W
= self
._charpoly
_basis
_space
()
304 W
= W
.change_ring(R
.fraction_field())
306 # Starting with the standard coordinates x = (X1,X2,...,Xn)
307 # and then converting the entries to W-coordinates allows us
308 # to pass in the standard coordinates to the charpoly and get
309 # back the right answer. Specifically, with x = (X1,X2,...,Xn),
312 # W.coordinates(x^2) eval'd at (standard z-coords)
316 # W-coords of (standard coords of x^2 eval'd at std-coords of z)
318 # We want the middle equivalent thing in our matrix, but use
319 # the first equivalent thing instead so that we can pass in
320 # standard coordinates.
321 x_powers
= [ W
.coordinate_vector(xp
) for xp
in x_powers
]
322 l2
= [idmat
.column(k
-1) for k
in range(r
+1, n
+1)]
323 A_of_x
= matrix
.column(R
, n
, (x_powers
[:r
] + l2
))
324 return (A_of_x
, x
, x_powers
[r
], A_of_x
.det())
328 def characteristic_polynomial(self
):
330 Return a characteristic polynomial that works for all elements
333 The resulting polynomial has `n+1` variables, where `n` is the
334 dimension of this algebra. The first `n` variables correspond to
335 the coordinates of an algebra element: when evaluated at the
336 coordinates of an algebra element with respect to a certain
337 basis, the result is a univariate polynomial (in the one
338 remaining variable ``t``), namely the characteristic polynomial
343 sage: from mjo.eja.eja_algebra import JordanSpinEJA
347 The characteristic polynomial in the spin algebra is given in
348 Alizadeh, Example 11.11::
350 sage: J = JordanSpinEJA(3)
351 sage: p = J.characteristic_polynomial(); p
352 X1^2 - X2^2 - X3^2 + (-2*t)*X1 + t^2
353 sage: xvec = J.one().to_vector()
361 # The list of coefficient polynomials a_1, a_2, ..., a_n.
362 a
= [ self
._charpoly
_coeff
(i
) for i
in range(n
) ]
364 # We go to a bit of trouble here to reorder the
365 # indeterminates, so that it's easier to evaluate the
366 # characteristic polynomial at x's coordinates and get back
367 # something in terms of t, which is what we want.
369 S
= PolynomialRing(self
.base_ring(),'t')
371 S
= PolynomialRing(S
, R
.variable_names())
374 # Note: all entries past the rth should be zero. The
375 # coefficient of the highest power (x^r) is 1, but it doesn't
376 # appear in the solution vector which contains coefficients
377 # for the other powers (to make them sum to x^r).
379 a
[r
] = 1 # corresponds to x^r
381 # When the rank is equal to the dimension, trying to
382 # assign a[r] goes out-of-bounds.
383 a
.append(1) # corresponds to x^r
385 return sum( a
[k
]*(t
**k
) for k
in range(len(a
)) )
388 def inner_product(self
, x
, y
):
390 The inner product associated with this Euclidean Jordan algebra.
392 Defaults to the trace inner product, but can be overridden by
393 subclasses if they are sure that the necessary properties are
398 sage: from mjo.eja.eja_algebra import random_eja
402 The inner product must satisfy its axiom for this algebra to truly
403 be a Euclidean Jordan Algebra::
405 sage: set_random_seed()
406 sage: J = random_eja()
407 sage: x = J.random_element()
408 sage: y = J.random_element()
409 sage: z = J.random_element()
410 sage: (x*y).inner_product(z) == y.inner_product(x*z)
414 if (not x
in self
) or (not y
in self
):
415 raise TypeError("arguments must live in this algebra")
416 return x
.trace_inner_product(y
)
419 def is_trivial(self
):
421 Return whether or not this algebra is trivial.
423 A trivial algebra contains only the zero element.
427 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
431 sage: J = ComplexHermitianEJA(3)
434 sage: A = J.zero().subalgebra_generated_by()
439 return self
.dimension() == 0
442 def multiplication_table(self
):
444 Return a visual representation of this algebra's multiplication
445 table (on basis elements).
449 sage: from mjo.eja.eja_algebra import JordanSpinEJA
453 sage: J = JordanSpinEJA(4)
454 sage: J.multiplication_table()
455 +----++----+----+----+----+
456 | * || e0 | e1 | e2 | e3 |
457 +====++====+====+====+====+
458 | e0 || e0 | e1 | e2 | e3 |
459 +----++----+----+----+----+
460 | e1 || e1 | e0 | 0 | 0 |
461 +----++----+----+----+----+
462 | e2 || e2 | 0 | e0 | 0 |
463 +----++----+----+----+----+
464 | e3 || e3 | 0 | 0 | e0 |
465 +----++----+----+----+----+
468 M
= list(self
._multiplication
_table
) # copy
469 for i
in range(len(M
)):
470 # M had better be "square"
471 M
[i
] = [self
.monomial(i
)] + M
[i
]
472 M
= [["*"] + list(self
.gens())] + M
473 return table(M
, header_row
=True, header_column
=True, frame
=True)
476 def natural_basis(self
):
478 Return a more-natural representation of this algebra's basis.
480 Every finite-dimensional Euclidean Jordan Algebra is a direct
481 sum of five simple algebras, four of which comprise Hermitian
482 matrices. This method returns the original "natural" basis
483 for our underlying vector space. (Typically, the natural basis
484 is used to construct the multiplication table in the first place.)
486 Note that this will always return a matrix. The standard basis
487 in `R^n` will be returned as `n`-by-`1` column matrices.
491 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
492 ....: RealSymmetricEJA)
496 sage: J = RealSymmetricEJA(2)
498 Finite family {0: e0, 1: e1, 2: e2}
499 sage: J.natural_basis()
507 sage: J = JordanSpinEJA(2)
509 Finite family {0: e0, 1: e1}
510 sage: J.natural_basis()
517 if self
._natural
_basis
is None:
518 M
= self
.natural_basis_space()
519 return tuple( M(b
.to_vector()) for b
in self
.basis() )
521 return self
._natural
_basis
524 def natural_basis_space(self
):
526 Return the matrix space in which this algebra's natural basis
529 if self
._natural
_basis
is None or len(self
._natural
_basis
) == 0:
530 return MatrixSpace(self
.base_ring(), self
.dimension(), 1)
532 return self
._natural
_basis
[0].matrix_space()
538 Return the unit element of this algebra.
542 sage: from mjo.eja.eja_algebra import (RealCartesianProductEJA,
547 sage: J = RealCartesianProductEJA(5)
549 e0 + e1 + e2 + e3 + e4
553 The identity element acts like the identity::
555 sage: set_random_seed()
556 sage: J = random_eja()
557 sage: x = J.random_element()
558 sage: J.one()*x == x and x*J.one() == x
561 The matrix of the unit element's operator is the identity::
563 sage: set_random_seed()
564 sage: J = random_eja()
565 sage: actual = J.one().operator().matrix()
566 sage: expected = matrix.identity(J.base_ring(), J.dimension())
567 sage: actual == expected
571 # We can brute-force compute the matrices of the operators
572 # that correspond to the basis elements of this algebra.
573 # If some linear combination of those basis elements is the
574 # algebra identity, then the same linear combination of
575 # their matrices has to be the identity matrix.
577 # Of course, matrices aren't vectors in sage, so we have to
578 # appeal to the "long vectors" isometry.
579 oper_vecs
= [ _mat2vec(g
.operator().matrix()) for g
in self
.gens() ]
581 # Now we use basis linear algebra to find the coefficients,
582 # of the matrices-as-vectors-linear-combination, which should
583 # work for the original algebra basis too.
584 A
= matrix
.column(self
.base_ring(), oper_vecs
)
586 # We used the isometry on the left-hand side already, but we
587 # still need to do it for the right-hand side. Recall that we
588 # wanted something that summed to the identity matrix.
589 b
= _mat2vec( matrix
.identity(self
.base_ring(), self
.dimension()) )
591 # Now if there's an identity element in the algebra, this should work.
592 coeffs
= A
.solve_right(b
)
593 return self
.linear_combination(zip(self
.gens(), coeffs
))
596 def random_element(self
):
597 # Temporary workaround for https://trac.sagemath.org/ticket/28327
598 if self
.is_trivial():
601 s
= super(FiniteDimensionalEuclideanJordanAlgebra
, self
)
602 return s
.random_element()
607 Return the rank of this EJA.
611 The author knows of no algorithm to compute the rank of an EJA
612 where only the multiplication table is known. In lieu of one, we
613 require the rank to be specified when the algebra is created,
614 and simply pass along that number here.
618 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
619 ....: RealSymmetricEJA,
620 ....: ComplexHermitianEJA,
621 ....: QuaternionHermitianEJA,
626 The rank of the Jordan spin algebra is always two::
628 sage: JordanSpinEJA(2).rank()
630 sage: JordanSpinEJA(3).rank()
632 sage: JordanSpinEJA(4).rank()
635 The rank of the `n`-by-`n` Hermitian real, complex, or
636 quaternion matrices is `n`::
638 sage: RealSymmetricEJA(2).rank()
640 sage: ComplexHermitianEJA(2).rank()
642 sage: QuaternionHermitianEJA(2).rank()
644 sage: RealSymmetricEJA(5).rank()
646 sage: ComplexHermitianEJA(5).rank()
648 sage: QuaternionHermitianEJA(5).rank()
653 Ensure that every EJA that we know how to construct has a
654 positive integer rank::
656 sage: set_random_seed()
657 sage: r = random_eja().rank()
658 sage: r in ZZ and r > 0
665 def vector_space(self
):
667 Return the vector space that underlies this algebra.
671 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
675 sage: J = RealSymmetricEJA(2)
676 sage: J.vector_space()
677 Vector space of dimension 3 over Rational Field
680 return self
.zero().to_vector().parent().ambient_vector_space()
683 Element
= FiniteDimensionalEuclideanJordanAlgebraElement
686 class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra
):
688 Return the Euclidean Jordan Algebra corresponding to the set
689 `R^n` under the Hadamard product.
691 Note: this is nothing more than the Cartesian product of ``n``
692 copies of the spin algebra. Once Cartesian product algebras
693 are implemented, this can go.
697 sage: from mjo.eja.eja_algebra import RealCartesianProductEJA
701 This multiplication table can be verified by hand::
703 sage: J = RealCartesianProductEJA(3)
704 sage: e0,e1,e2 = J.gens()
720 We can change the generator prefix::
722 sage: RealCartesianProductEJA(3, prefix='r').gens()
725 Our inner product satisfies the Jordan axiom::
727 sage: set_random_seed()
728 sage: n = ZZ.random_element(1,5)
729 sage: J = RealCartesianProductEJA(n)
730 sage: x = J.random_element()
731 sage: y = J.random_element()
732 sage: z = J.random_element()
733 sage: (x*y).inner_product(z) == y.inner_product(x*z)
737 def __init__(self
, n
, field
=QQ
, **kwargs
):
738 V
= VectorSpace(field
, n
)
739 mult_table
= [ [ V
.gen(i
)*(i
== j
) for j
in range(n
) ]
742 fdeja
= super(RealCartesianProductEJA
, self
)
743 return fdeja
.__init
__(field
, mult_table
, rank
=n
, **kwargs
)
745 def inner_product(self
, x
, y
):
746 return _usual_ip(x
,y
)
751 Return a "random" finite-dimensional Euclidean Jordan Algebra.
755 For now, we choose a random natural number ``n`` (greater than zero)
756 and then give you back one of the following:
758 * The cartesian product of the rational numbers ``n`` times; this is
759 ``QQ^n`` with the Hadamard product.
761 * The Jordan spin algebra on ``QQ^n``.
763 * The ``n``-by-``n`` rational symmetric matrices with the symmetric
766 * The ``n``-by-``n`` complex-rational Hermitian matrices embedded
767 in the space of ``2n``-by-``2n`` real symmetric matrices.
769 * The ``n``-by-``n`` quaternion-rational Hermitian matrices embedded
770 in the space of ``4n``-by-``4n`` real symmetric matrices.
772 Later this might be extended to return Cartesian products of the
777 sage: from mjo.eja.eja_algebra import random_eja
782 Euclidean Jordan algebra of dimension...
786 # The max_n component lets us choose different upper bounds on the
787 # value "n" that gets passed to the constructor. This is needed
788 # because e.g. R^{10} is reasonable to test, while the Hermitian
789 # 10-by-10 quaternion matrices are not.
790 (constructor
, max_n
) = choice([(RealCartesianProductEJA
, 6),
792 (RealSymmetricEJA
, 5),
793 (ComplexHermitianEJA
, 4),
794 (QuaternionHermitianEJA
, 3)])
795 n
= ZZ
.random_element(1, max_n
)
796 return constructor(n
, field
=QQ
)
800 def _real_symmetric_basis(n
, field
=QQ
):
802 Return a basis for the space of real symmetric n-by-n matrices.
804 # The basis of symmetric matrices, as matrices, in their R^(n-by-n)
808 for j
in xrange(i
+1):
809 Eij
= matrix(field
, n
, lambda k
,l
: k
==i
and l
==j
)
813 # Beware, orthogonal but not normalized!
814 Sij
= Eij
+ Eij
.transpose()
819 def _complex_hermitian_basis(n
, field
=QQ
):
821 Returns a basis for the space of complex Hermitian n-by-n matrices.
825 sage: from mjo.eja.eja_algebra import _complex_hermitian_basis
829 sage: set_random_seed()
830 sage: n = ZZ.random_element(1,5)
831 sage: all( M.is_symmetric() for M in _complex_hermitian_basis(n) )
835 F
= QuadraticField(-1, 'I')
838 # This is like the symmetric case, but we need to be careful:
840 # * We want conjugate-symmetry, not just symmetry.
841 # * The diagonal will (as a result) be real.
845 for j
in xrange(i
+1):
846 Eij
= matrix(field
, n
, lambda k
,l
: k
==i
and l
==j
)
848 Sij
= _embed_complex_matrix(Eij
)
851 # Beware, orthogonal but not normalized! The second one
852 # has a minus because it's conjugated.
853 Sij_real
= _embed_complex_matrix(Eij
+ Eij
.transpose())
855 Sij_imag
= _embed_complex_matrix(I
*Eij
- I
*Eij
.transpose())
860 def _quaternion_hermitian_basis(n
, field
=QQ
):
862 Returns a basis for the space of quaternion Hermitian n-by-n matrices.
866 sage: from mjo.eja.eja_algebra import _quaternion_hermitian_basis
870 sage: set_random_seed()
871 sage: n = ZZ.random_element(1,5)
872 sage: all( M.is_symmetric() for M in _quaternion_hermitian_basis(n) )
876 Q
= QuaternionAlgebra(QQ
,-1,-1)
879 # This is like the symmetric case, but we need to be careful:
881 # * We want conjugate-symmetry, not just symmetry.
882 # * The diagonal will (as a result) be real.
886 for j
in xrange(i
+1):
887 Eij
= matrix(Q
, n
, lambda k
,l
: k
==i
and l
==j
)
889 Sij
= _embed_quaternion_matrix(Eij
)
892 # Beware, orthogonal but not normalized! The second,
893 # third, and fourth ones have a minus because they're
895 Sij_real
= _embed_quaternion_matrix(Eij
+ Eij
.transpose())
897 Sij_I
= _embed_quaternion_matrix(I
*Eij
- I
*Eij
.transpose())
899 Sij_J
= _embed_quaternion_matrix(J
*Eij
- J
*Eij
.transpose())
901 Sij_K
= _embed_quaternion_matrix(K
*Eij
- K
*Eij
.transpose())
907 def _multiplication_table_from_matrix_basis(basis
):
909 At least three of the five simple Euclidean Jordan algebras have the
910 symmetric multiplication (A,B) |-> (AB + BA)/2, where the
911 multiplication on the right is matrix multiplication. Given a basis
912 for the underlying matrix space, this function returns a
913 multiplication table (obtained by looping through the basis
914 elements) for an algebra of those matrices.
916 # In S^2, for example, we nominally have four coordinates even
917 # though the space is of dimension three only. The vector space V
918 # is supposed to hold the entire long vector, and the subspace W
919 # of V will be spanned by the vectors that arise from symmetric
920 # matrices. Thus for S^2, dim(V) == 4 and dim(W) == 3.
921 field
= basis
[0].base_ring()
922 dimension
= basis
[0].nrows()
924 V
= VectorSpace(field
, dimension
**2)
925 W
= V
.span_of_basis( _mat2vec(s
) for s
in basis
)
927 mult_table
= [[W
.zero() for j
in range(n
)] for i
in range(n
)]
930 mat_entry
= (basis
[i
]*basis
[j
] + basis
[j
]*basis
[i
])/2
931 mult_table
[i
][j
] = W
.coordinate_vector(_mat2vec(mat_entry
))
936 def _embed_complex_matrix(M
):
938 Embed the n-by-n complex matrix ``M`` into the space of real
939 matrices of size 2n-by-2n via the map the sends each entry `z = a +
940 bi` to the block matrix ``[[a,b],[-b,a]]``.
944 sage: from mjo.eja.eja_algebra import _embed_complex_matrix
948 sage: F = QuadraticField(-1,'i')
949 sage: x1 = F(4 - 2*i)
950 sage: x2 = F(1 + 2*i)
953 sage: M = matrix(F,2,[[x1,x2],[x3,x4]])
954 sage: _embed_complex_matrix(M)
963 Embedding is a homomorphism (isomorphism, in fact)::
965 sage: set_random_seed()
966 sage: n = ZZ.random_element(5)
967 sage: F = QuadraticField(-1, 'i')
968 sage: X = random_matrix(F, n)
969 sage: Y = random_matrix(F, n)
970 sage: actual = _embed_complex_matrix(X) * _embed_complex_matrix(Y)
971 sage: expected = _embed_complex_matrix(X*Y)
972 sage: actual == expected
978 raise ValueError("the matrix 'M' must be square")
979 field
= M
.base_ring()
984 blocks
.append(matrix(field
, 2, [[a
,b
],[-b
,a
]]))
986 # We can drop the imaginaries here.
987 return matrix
.block(field
.base_ring(), n
, blocks
)
990 def _unembed_complex_matrix(M
):
992 The inverse of _embed_complex_matrix().
996 sage: from mjo.eja.eja_algebra import (_embed_complex_matrix,
997 ....: _unembed_complex_matrix)
1001 sage: A = matrix(QQ,[ [ 1, 2, 3, 4],
1002 ....: [-2, 1, -4, 3],
1003 ....: [ 9, 10, 11, 12],
1004 ....: [-10, 9, -12, 11] ])
1005 sage: _unembed_complex_matrix(A)
1007 [ 10*i + 9 12*i + 11]
1011 Unembedding is the inverse of embedding::
1013 sage: set_random_seed()
1014 sage: F = QuadraticField(-1, 'i')
1015 sage: M = random_matrix(F, 3)
1016 sage: _unembed_complex_matrix(_embed_complex_matrix(M)) == M
1022 raise ValueError("the matrix 'M' must be square")
1023 if not n
.mod(2).is_zero():
1024 raise ValueError("the matrix 'M' must be a complex embedding")
1026 F
= QuadraticField(-1, 'i')
1029 # Go top-left to bottom-right (reading order), converting every
1030 # 2-by-2 block we see to a single complex element.
1032 for k
in xrange(n
/2):
1033 for j
in xrange(n
/2):
1034 submat
= M
[2*k
:2*k
+2,2*j
:2*j
+2]
1035 if submat
[0,0] != submat
[1,1]:
1036 raise ValueError('bad on-diagonal submatrix')
1037 if submat
[0,1] != -submat
[1,0]:
1038 raise ValueError('bad off-diagonal submatrix')
1039 z
= submat
[0,0] + submat
[0,1]*i
1042 return matrix(F
, n
/2, elements
)
1045 def _embed_quaternion_matrix(M
):
1047 Embed the n-by-n quaternion matrix ``M`` into the space of real
1048 matrices of size 4n-by-4n by first sending each quaternion entry
1049 `z = a + bi + cj + dk` to the block-complex matrix
1050 ``[[a + bi, c+di],[-c + di, a-bi]]`, and then embedding those into
1055 sage: from mjo.eja.eja_algebra import _embed_quaternion_matrix
1059 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1060 sage: i,j,k = Q.gens()
1061 sage: x = 1 + 2*i + 3*j + 4*k
1062 sage: M = matrix(Q, 1, [[x]])
1063 sage: _embed_quaternion_matrix(M)
1069 Embedding is a homomorphism (isomorphism, in fact)::
1071 sage: set_random_seed()
1072 sage: n = ZZ.random_element(5)
1073 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1074 sage: X = random_matrix(Q, n)
1075 sage: Y = random_matrix(Q, n)
1076 sage: actual = _embed_quaternion_matrix(X)*_embed_quaternion_matrix(Y)
1077 sage: expected = _embed_quaternion_matrix(X*Y)
1078 sage: actual == expected
1082 quaternions
= M
.base_ring()
1085 raise ValueError("the matrix 'M' must be square")
1087 F
= QuadraticField(-1, 'i')
1092 t
= z
.coefficient_tuple()
1097 cplx_matrix
= matrix(F
, 2, [[ a
+ b
*i
, c
+ d
*i
],
1098 [-c
+ d
*i
, a
- b
*i
]])
1099 blocks
.append(_embed_complex_matrix(cplx_matrix
))
1101 # We should have real entries by now, so use the realest field
1102 # we've got for the return value.
1103 return matrix
.block(quaternions
.base_ring(), n
, blocks
)
1106 def _unembed_quaternion_matrix(M
):
1108 The inverse of _embed_quaternion_matrix().
1112 sage: from mjo.eja.eja_algebra import (_embed_quaternion_matrix,
1113 ....: _unembed_quaternion_matrix)
1117 sage: M = matrix(QQ, [[ 1, 2, 3, 4],
1118 ....: [-2, 1, -4, 3],
1119 ....: [-3, 4, 1, -2],
1120 ....: [-4, -3, 2, 1]])
1121 sage: _unembed_quaternion_matrix(M)
1122 [1 + 2*i + 3*j + 4*k]
1126 Unembedding is the inverse of embedding::
1128 sage: set_random_seed()
1129 sage: Q = QuaternionAlgebra(QQ, -1, -1)
1130 sage: M = random_matrix(Q, 3)
1131 sage: _unembed_quaternion_matrix(_embed_quaternion_matrix(M)) == M
1137 raise ValueError("the matrix 'M' must be square")
1138 if not n
.mod(4).is_zero():
1139 raise ValueError("the matrix 'M' must be a complex embedding")
1141 Q
= QuaternionAlgebra(QQ
,-1,-1)
1144 # Go top-left to bottom-right (reading order), converting every
1145 # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1
1148 for l
in xrange(n
/4):
1149 for m
in xrange(n
/4):
1150 submat
= _unembed_complex_matrix(M
[4*l
:4*l
+4,4*m
:4*m
+4])
1151 if submat
[0,0] != submat
[1,1].conjugate():
1152 raise ValueError('bad on-diagonal submatrix')
1153 if submat
[0,1] != -submat
[1,0].conjugate():
1154 raise ValueError('bad off-diagonal submatrix')
1155 z
= submat
[0,0].real() + submat
[0,0].imag()*i
1156 z
+= submat
[0,1].real()*j
+ submat
[0,1].imag()*k
1159 return matrix(Q
, n
/4, elements
)
1162 # The usual inner product on R^n.
1164 return x
.to_vector().inner_product(y
.to_vector())
1166 # The inner product used for the real symmetric simple EJA.
1167 # We keep it as a separate function because e.g. the complex
1168 # algebra uses the same inner product, except divided by 2.
1169 def _matrix_ip(X
,Y
):
1170 X_mat
= X
.natural_representation()
1171 Y_mat
= Y
.natural_representation()
1172 return (X_mat
*Y_mat
).trace()
1175 class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1177 The rank-n simple EJA consisting of real symmetric n-by-n
1178 matrices, the usual symmetric Jordan product, and the trace inner
1179 product. It has dimension `(n^2 + n)/2` over the reals.
1183 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
1187 sage: J = RealSymmetricEJA(2)
1188 sage: e0, e1, e2 = J.gens()
1198 The dimension of this algebra is `(n^2 + n) / 2`::
1200 sage: set_random_seed()
1201 sage: n = ZZ.random_element(1,5)
1202 sage: J = RealSymmetricEJA(n)
1203 sage: J.dimension() == (n^2 + n)/2
1206 The Jordan multiplication is what we think it is::
1208 sage: set_random_seed()
1209 sage: n = ZZ.random_element(1,5)
1210 sage: J = RealSymmetricEJA(n)
1211 sage: x = J.random_element()
1212 sage: y = J.random_element()
1213 sage: actual = (x*y).natural_representation()
1214 sage: X = x.natural_representation()
1215 sage: Y = y.natural_representation()
1216 sage: expected = (X*Y + Y*X)/2
1217 sage: actual == expected
1219 sage: J(expected) == x*y
1222 We can change the generator prefix::
1224 sage: RealSymmetricEJA(3, prefix='q').gens()
1225 (q0, q1, q2, q3, q4, q5)
1227 Our inner product satisfies the Jordan axiom::
1229 sage: set_random_seed()
1230 sage: n = ZZ.random_element(1,5)
1231 sage: J = RealSymmetricEJA(n)
1232 sage: x = J.random_element()
1233 sage: y = J.random_element()
1234 sage: z = J.random_element()
1235 sage: (x*y).inner_product(z) == y.inner_product(x*z)
1239 def __init__(self
, n
, field
=QQ
, **kwargs
):
1240 S
= _real_symmetric_basis(n
, field
=field
)
1241 Qs
= _multiplication_table_from_matrix_basis(S
)
1243 fdeja
= super(RealSymmetricEJA
, self
)
1244 return fdeja
.__init
__(field
,
1250 def inner_product(self
, x
, y
):
1251 return _matrix_ip(x
,y
)
1254 class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1256 The rank-n simple EJA consisting of complex Hermitian n-by-n
1257 matrices over the real numbers, the usual symmetric Jordan product,
1258 and the real-part-of-trace inner product. It has dimension `n^2` over
1263 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
1267 The dimension of this algebra is `n^2`::
1269 sage: set_random_seed()
1270 sage: n = ZZ.random_element(1,5)
1271 sage: J = ComplexHermitianEJA(n)
1272 sage: J.dimension() == n^2
1275 The Jordan multiplication is what we think it is::
1277 sage: set_random_seed()
1278 sage: n = ZZ.random_element(1,5)
1279 sage: J = ComplexHermitianEJA(n)
1280 sage: x = J.random_element()
1281 sage: y = J.random_element()
1282 sage: actual = (x*y).natural_representation()
1283 sage: X = x.natural_representation()
1284 sage: Y = y.natural_representation()
1285 sage: expected = (X*Y + Y*X)/2
1286 sage: actual == expected
1288 sage: J(expected) == x*y
1291 We can change the generator prefix::
1293 sage: ComplexHermitianEJA(2, prefix='z').gens()
1296 Our inner product satisfies the Jordan axiom::
1298 sage: set_random_seed()
1299 sage: n = ZZ.random_element(1,5)
1300 sage: J = ComplexHermitianEJA(n)
1301 sage: x = J.random_element()
1302 sage: y = J.random_element()
1303 sage: z = J.random_element()
1304 sage: (x*y).inner_product(z) == y.inner_product(x*z)
1308 def __init__(self
, n
, field
=QQ
, **kwargs
):
1309 S
= _complex_hermitian_basis(n
)
1310 Qs
= _multiplication_table_from_matrix_basis(S
)
1312 fdeja
= super(ComplexHermitianEJA
, self
)
1313 return fdeja
.__init
__(field
,
1320 def inner_product(self
, x
, y
):
1321 # Since a+bi on the diagonal is represented as
1326 # we'll double-count the "a" entries if we take the trace of
1328 return _matrix_ip(x
,y
)/2
1331 class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1333 The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
1334 matrices, the usual symmetric Jordan product, and the
1335 real-part-of-trace inner product. It has dimension `2n^2 - n` over
1340 sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
1344 The dimension of this algebra is `n^2`::
1346 sage: set_random_seed()
1347 sage: n = ZZ.random_element(1,5)
1348 sage: J = QuaternionHermitianEJA(n)
1349 sage: J.dimension() == 2*(n^2) - n
1352 The Jordan multiplication is what we think it is::
1354 sage: set_random_seed()
1355 sage: n = ZZ.random_element(1,5)
1356 sage: J = QuaternionHermitianEJA(n)
1357 sage: x = J.random_element()
1358 sage: y = J.random_element()
1359 sage: actual = (x*y).natural_representation()
1360 sage: X = x.natural_representation()
1361 sage: Y = y.natural_representation()
1362 sage: expected = (X*Y + Y*X)/2
1363 sage: actual == expected
1365 sage: J(expected) == x*y
1368 We can change the generator prefix::
1370 sage: QuaternionHermitianEJA(2, prefix='a').gens()
1371 (a0, a1, a2, a3, a4, a5)
1373 Our inner product satisfies the Jordan axiom::
1375 sage: set_random_seed()
1376 sage: n = ZZ.random_element(1,5)
1377 sage: J = QuaternionHermitianEJA(n)
1378 sage: x = J.random_element()
1379 sage: y = J.random_element()
1380 sage: z = J.random_element()
1381 sage: (x*y).inner_product(z) == y.inner_product(x*z)
1385 def __init__(self
, n
, field
=QQ
, **kwargs
):
1386 S
= _quaternion_hermitian_basis(n
)
1387 Qs
= _multiplication_table_from_matrix_basis(S
)
1389 fdeja
= super(QuaternionHermitianEJA
, self
)
1390 return fdeja
.__init
__(field
,
1396 def inner_product(self
, x
, y
):
1397 # Since a+bi+cj+dk on the diagonal is represented as
1399 # a + bi +cj + dk = [ a b c d]
1404 # we'll quadruple-count the "a" entries if we take the trace of
1406 return _matrix_ip(x
,y
)/4
1409 class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1411 The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
1412 with the usual inner product and jordan product ``x*y =
1413 (<x_bar,y_bar>, x0*y_bar + y0*x_bar)``. It has dimension `n` over
1418 sage: from mjo.eja.eja_algebra import JordanSpinEJA
1422 This multiplication table can be verified by hand::
1424 sage: J = JordanSpinEJA(4)
1425 sage: e0,e1,e2,e3 = J.gens()
1441 We can change the generator prefix::
1443 sage: JordanSpinEJA(2, prefix='B').gens()
1446 Our inner product satisfies the Jordan axiom::
1448 sage: set_random_seed()
1449 sage: n = ZZ.random_element(1,5)
1450 sage: J = JordanSpinEJA(n)
1451 sage: x = J.random_element()
1452 sage: y = J.random_element()
1453 sage: z = J.random_element()
1454 sage: (x*y).inner_product(z) == y.inner_product(x*z)
1458 def __init__(self
, n
, field
=QQ
, **kwargs
):
1459 V
= VectorSpace(field
, n
)
1460 mult_table
= [[V
.zero() for j
in range(n
)] for i
in range(n
)]
1470 z0
= x
.inner_product(y
)
1471 zbar
= y0
*xbar
+ x0
*ybar
1472 z
= V([z0
] + zbar
.list())
1473 mult_table
[i
][j
] = z
1475 # The rank of the spin algebra is two, unless we're in a
1476 # one-dimensional ambient space (because the rank is bounded by
1477 # the ambient dimension).
1478 fdeja
= super(JordanSpinEJA
, self
)
1479 return fdeja
.__init
__(field
, mult_table
, rank
=min(n
,2), **kwargs
)
1481 def inner_product(self
, x
, y
):
1482 return _usual_ip(x
,y
)