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
.misc
.cachefunc
import cached_method
13 from sage
.misc
.prandom
import choice
14 from sage
.misc
.table
import table
15 from sage
.modules
.free_module
import VectorSpace
16 from sage
.rings
.integer_ring
import ZZ
17 from sage
.rings
.number_field
.number_field
import QuadraticField
18 from sage
.rings
.polynomial
.polynomial_ring_constructor
import PolynomialRing
19 from sage
.rings
.rational_field
import QQ
20 from sage
.structure
.element
import is_Matrix
22 from mjo
.eja
.eja_element
import FiniteDimensionalEuclideanJordanAlgebraElement
23 from mjo
.eja
.eja_utils
import _mat2vec
25 class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule
):
26 # This is an ugly hack needed to prevent the category framework
27 # from implementing a coercion from our base ring (e.g. the
28 # rationals) into the algebra. First of all -- such a coercion is
29 # nonsense to begin with. But more importantly, it tries to do so
30 # in the category of rings, and since our algebras aren't
31 # associative they generally won't be rings.
32 _no_generic_basering_coercion
= True
44 sage: from mjo.eja.eja_algebra import random_eja
48 By definition, Jordan multiplication commutes::
50 sage: set_random_seed()
51 sage: J = random_eja()
52 sage: x = J.random_element()
53 sage: y = J.random_element()
59 self
._natural
_basis
= natural_basis
62 category
= MagmaticAlgebras(field
).FiniteDimensional()
63 category
= category
.WithBasis().Unital()
65 fda
= super(FiniteDimensionalEuclideanJordanAlgebra
, self
)
67 range(len(mult_table
)),
70 self
.print_options(bracket
='')
72 # The multiplication table we're given is necessarily in terms
73 # of vectors, because we don't have an algebra yet for
74 # anything to be an element of. However, it's faster in the
75 # long run to have the multiplication table be in terms of
76 # algebra elements. We do this after calling the superclass
77 # constructor so that from_vector() knows what to do.
78 self
._multiplication
_table
= [ map(lambda x
: self
.from_vector(x
), ls
)
79 for ls
in mult_table
]
82 def _element_constructor_(self
, elt
):
84 Construct an element of this algebra from its natural
87 This gets called only after the parent element _call_ method
88 fails to find a coercion for the argument.
92 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
93 ....: RealCartesianProductEJA,
94 ....: RealSymmetricEJA)
98 The identity in `S^n` is converted to the identity in the EJA::
100 sage: J = RealSymmetricEJA(3)
101 sage: I = matrix.identity(QQ,3)
102 sage: J(I) == J.one()
105 This skew-symmetric matrix can't be represented in the EJA::
107 sage: J = RealSymmetricEJA(3)
108 sage: A = matrix(QQ,3, lambda i,j: i-j)
110 Traceback (most recent call last):
112 ArithmeticError: vector is not in free module
116 Ensure that we can convert any element of the two non-matrix
117 simple algebras (whose natural representations are their usual
118 vector representations) back and forth faithfully::
120 sage: set_random_seed()
121 sage: J = RealCartesianProductEJA(5)
122 sage: x = J.random_element()
123 sage: J(x.to_vector().column()) == x
125 sage: J = JordanSpinEJA(5)
126 sage: x = J.random_element()
127 sage: J(x.to_vector().column()) == x
132 # The superclass implementation of random_element()
133 # needs to be able to coerce "0" into the algebra.
136 natural_basis
= self
.natural_basis()
137 if elt
not in natural_basis
[0].matrix_space():
138 raise ValueError("not a naturally-represented algebra element")
140 # Thanks for nothing! Matrix spaces aren't vector
141 # spaces in Sage, so we have to figure out its
142 # natural-basis coordinates ourselves.
143 V
= VectorSpace(elt
.base_ring(), elt
.nrows()*elt
.ncols())
144 W
= V
.span_of_basis( _mat2vec(s
) for s
in natural_basis
)
145 coords
= W
.coordinate_vector(_mat2vec(elt
))
146 return self
.from_vector(coords
)
151 Return a string representation of ``self``.
155 sage: from mjo.eja.eja_algebra import JordanSpinEJA
159 Ensure that it says what we think it says::
161 sage: JordanSpinEJA(2, field=QQ)
162 Euclidean Jordan algebra of dimension 2 over Rational Field
163 sage: JordanSpinEJA(3, field=RDF)
164 Euclidean Jordan algebra of dimension 3 over Real Double Field
167 fmt
= "Euclidean Jordan algebra of dimension {} over {}"
168 return fmt
.format(self
.dimension(), self
.base_ring())
170 def product_on_basis(self
, i
, j
):
171 return self
._multiplication
_table
[i
][j
]
173 def _a_regular_element(self
):
175 Guess a regular element. Needed to compute the basis for our
176 characteristic polynomial coefficients.
180 sage: from mjo.eja.eja_algebra import random_eja
184 Ensure that this hacky method succeeds for every algebra that we
185 know how to construct::
187 sage: set_random_seed()
188 sage: J = random_eja()
189 sage: J._a_regular_element().is_regular()
194 z
= self
.sum( (i
+1)*gs
[i
] for i
in range(len(gs
)) )
195 if not z
.is_regular():
196 raise ValueError("don't know a regular element")
201 def _charpoly_basis_space(self
):
203 Return the vector space spanned by the basis used in our
204 characteristic polynomial coefficients. This is used not only to
205 compute those coefficients, but also any time we need to
206 evaluate the coefficients (like when we compute the trace or
209 z
= self
._a
_regular
_element
()
210 V
= self
.vector_space()
211 V1
= V
.span_of_basis( (z
**k
).to_vector() for k
in range(self
.rank()) )
212 b
= (V1
.basis() + V1
.complement().basis())
213 return V
.span_of_basis(b
)
217 def _charpoly_coeff(self
, i
):
219 Return the coefficient polynomial "a_{i}" of this algebra's
220 general characteristic polynomial.
222 Having this be a separate cached method lets us compute and
223 store the trace/determinant (a_{r-1} and a_{0} respectively)
224 separate from the entire characteristic polynomial.
226 (A_of_x
, x
, xr
, detA
) = self
._charpoly
_matrix
_system
()
227 R
= A_of_x
.base_ring()
229 # Guaranteed by theory
232 # Danger: the in-place modification is done for performance
233 # reasons (reconstructing a matrix with huge polynomial
234 # entries is slow), but I don't know how cached_method works,
235 # so it's highly possible that we're modifying some global
236 # list variable by reference, here. In other words, you
237 # probably shouldn't call this method twice on the same
238 # algebra, at the same time, in two threads
239 Ai_orig
= A_of_x
.column(i
)
240 A_of_x
.set_column(i
,xr
)
241 numerator
= A_of_x
.det()
242 A_of_x
.set_column(i
,Ai_orig
)
244 # We're relying on the theory here to ensure that each a_i is
245 # indeed back in R, and the added negative signs are to make
246 # the whole charpoly expression sum to zero.
247 return R(-numerator
/detA
)
251 def _charpoly_matrix_system(self
):
253 Compute the matrix whose entries A_ij are polynomials in
254 X1,...,XN, the vector ``x`` of variables X1,...,XN, the vector
255 corresponding to `x^r` and the determinent of the matrix A =
256 [A_ij]. In other words, all of the fixed (cachable) data needed
257 to compute the coefficients of the characteristic polynomial.
262 # Turn my vector space into a module so that "vectors" can
263 # have multivatiate polynomial entries.
264 names
= tuple('X' + str(i
) for i
in range(1,n
+1))
265 R
= PolynomialRing(self
.base_ring(), names
)
266 V
= self
.vector_space().change_ring(R
)
268 # Now let x = (X1,X2,...,Xn) be the vector whose entries are
272 # And figure out the "left multiplication by x" matrix in
275 monomial_matrices
= [ self
.monomial(i
).operator().matrix()
276 for i
in range(n
) ] # don't recompute these!
278 ek
= self
.monomial(k
).to_vector()
280 sum( x
[i
]*(monomial_matrices
[i
]*ek
)
281 for i
in range(n
) ) )
282 Lx
= matrix
.column(R
, lmbx_cols
)
284 # Now we can compute powers of x "symbolically"
285 x_powers
= [self
.one().to_vector(), x
]
286 for d
in range(2, r
+1):
287 x_powers
.append( Lx
*(x_powers
[-1]) )
289 idmat
= matrix
.identity(R
, n
)
291 W
= self
._charpoly
_basis
_space
()
292 W
= W
.change_ring(R
.fraction_field())
294 # Starting with the standard coordinates x = (X1,X2,...,Xn)
295 # and then converting the entries to W-coordinates allows us
296 # to pass in the standard coordinates to the charpoly and get
297 # back the right answer. Specifically, with x = (X1,X2,...,Xn),
300 # W.coordinates(x^2) eval'd at (standard z-coords)
304 # W-coords of (standard coords of x^2 eval'd at std-coords of z)
306 # We want the middle equivalent thing in our matrix, but use
307 # the first equivalent thing instead so that we can pass in
308 # standard coordinates.
309 x_powers
= [ W
.coordinate_vector(xp
) for xp
in x_powers
]
310 l2
= [idmat
.column(k
-1) for k
in range(r
+1, n
+1)]
311 A_of_x
= matrix
.column(R
, n
, (x_powers
[:r
] + l2
))
312 return (A_of_x
, x
, x_powers
[r
], A_of_x
.det())
316 def characteristic_polynomial(self
):
318 Return a characteristic polynomial that works for all elements
321 The resulting polynomial has `n+1` variables, where `n` is the
322 dimension of this algebra. The first `n` variables correspond to
323 the coordinates of an algebra element: when evaluated at the
324 coordinates of an algebra element with respect to a certain
325 basis, the result is a univariate polynomial (in the one
326 remaining variable ``t``), namely the characteristic polynomial
331 sage: from mjo.eja.eja_algebra import JordanSpinEJA
335 The characteristic polynomial in the spin algebra is given in
336 Alizadeh, Example 11.11::
338 sage: J = JordanSpinEJA(3)
339 sage: p = J.characteristic_polynomial(); p
340 X1^2 - X2^2 - X3^2 + (-2*t)*X1 + t^2
341 sage: xvec = J.one().to_vector()
349 # The list of coefficient polynomials a_1, a_2, ..., a_n.
350 a
= [ self
._charpoly
_coeff
(i
) for i
in range(n
) ]
352 # We go to a bit of trouble here to reorder the
353 # indeterminates, so that it's easier to evaluate the
354 # characteristic polynomial at x's coordinates and get back
355 # something in terms of t, which is what we want.
357 S
= PolynomialRing(self
.base_ring(),'t')
359 S
= PolynomialRing(S
, R
.variable_names())
362 # Note: all entries past the rth should be zero. The
363 # coefficient of the highest power (x^r) is 1, but it doesn't
364 # appear in the solution vector which contains coefficients
365 # for the other powers (to make them sum to x^r).
367 a
[r
] = 1 # corresponds to x^r
369 # When the rank is equal to the dimension, trying to
370 # assign a[r] goes out-of-bounds.
371 a
.append(1) # corresponds to x^r
373 return sum( a
[k
]*(t
**k
) for k
in range(len(a
)) )
376 def inner_product(self
, x
, y
):
378 The inner product associated with this Euclidean Jordan algebra.
380 Defaults to the trace inner product, but can be overridden by
381 subclasses if they are sure that the necessary properties are
386 sage: from mjo.eja.eja_algebra import random_eja
390 The inner product must satisfy its axiom for this algebra to truly
391 be a Euclidean Jordan Algebra::
393 sage: set_random_seed()
394 sage: J = random_eja()
395 sage: x = J.random_element()
396 sage: y = J.random_element()
397 sage: z = J.random_element()
398 sage: (x*y).inner_product(z) == y.inner_product(x*z)
402 if (not x
in self
) or (not y
in self
):
403 raise TypeError("arguments must live in this algebra")
404 return x
.trace_inner_product(y
)
407 def multiplication_table(self
):
409 Return a visual representation of this algebra's multiplication
410 table (on basis elements).
414 sage: from mjo.eja.eja_algebra import JordanSpinEJA
418 sage: J = JordanSpinEJA(4)
419 sage: J.multiplication_table()
420 +----++----+----+----+----+
421 | * || e0 | e1 | e2 | e3 |
422 +====++====+====+====+====+
423 | e0 || e0 | e1 | e2 | e3 |
424 +----++----+----+----+----+
425 | e1 || e1 | e0 | 0 | 0 |
426 +----++----+----+----+----+
427 | e2 || e2 | 0 | e0 | 0 |
428 +----++----+----+----+----+
429 | e3 || e3 | 0 | 0 | e0 |
430 +----++----+----+----+----+
433 M
= list(self
._multiplication
_table
) # copy
434 for i
in range(len(M
)):
435 # M had better be "square"
436 M
[i
] = [self
.monomial(i
)] + M
[i
]
437 M
= [["*"] + list(self
.gens())] + M
438 return table(M
, header_row
=True, header_column
=True, frame
=True)
441 def natural_basis(self
):
443 Return a more-natural representation of this algebra's basis.
445 Every finite-dimensional Euclidean Jordan Algebra is a direct
446 sum of five simple algebras, four of which comprise Hermitian
447 matrices. This method returns the original "natural" basis
448 for our underlying vector space. (Typically, the natural basis
449 is used to construct the multiplication table in the first place.)
451 Note that this will always return a matrix. The standard basis
452 in `R^n` will be returned as `n`-by-`1` column matrices.
456 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
457 ....: RealSymmetricEJA)
461 sage: J = RealSymmetricEJA(2)
463 Finite family {0: e0, 1: e1, 2: e2}
464 sage: J.natural_basis()
472 sage: J = JordanSpinEJA(2)
474 Finite family {0: e0, 1: e1}
475 sage: J.natural_basis()
482 if self
._natural
_basis
is None:
483 return tuple( b
.to_vector().column() for b
in self
.basis() )
485 return self
._natural
_basis
491 Return the unit element of this algebra.
495 sage: from mjo.eja.eja_algebra import (RealCartesianProductEJA,
500 sage: J = RealCartesianProductEJA(5)
502 e0 + e1 + e2 + e3 + e4
506 The identity element acts like the identity::
508 sage: set_random_seed()
509 sage: J = random_eja()
510 sage: x = J.random_element()
511 sage: J.one()*x == x and x*J.one() == x
514 The matrix of the unit element's operator is the identity::
516 sage: set_random_seed()
517 sage: J = random_eja()
518 sage: actual = J.one().operator().matrix()
519 sage: expected = matrix.identity(J.base_ring(), J.dimension())
520 sage: actual == expected
524 # We can brute-force compute the matrices of the operators
525 # that correspond to the basis elements of this algebra.
526 # If some linear combination of those basis elements is the
527 # algebra identity, then the same linear combination of
528 # their matrices has to be the identity matrix.
530 # Of course, matrices aren't vectors in sage, so we have to
531 # appeal to the "long vectors" isometry.
532 oper_vecs
= [ _mat2vec(g
.operator().matrix()) for g
in self
.gens() ]
534 # Now we use basis linear algebra to find the coefficients,
535 # of the matrices-as-vectors-linear-combination, which should
536 # work for the original algebra basis too.
537 A
= matrix
.column(self
.base_ring(), oper_vecs
)
539 # We used the isometry on the left-hand side already, but we
540 # still need to do it for the right-hand side. Recall that we
541 # wanted something that summed to the identity matrix.
542 b
= _mat2vec( matrix
.identity(self
.base_ring(), self
.dimension()) )
544 # Now if there's an identity element in the algebra, this should work.
545 coeffs
= A
.solve_right(b
)
546 return self
.linear_combination(zip(self
.gens(), coeffs
))
551 Return the rank of this EJA.
555 The author knows of no algorithm to compute the rank of an EJA
556 where only the multiplication table is known. In lieu of one, we
557 require the rank to be specified when the algebra is created,
558 and simply pass along that number here.
562 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
563 ....: RealSymmetricEJA,
564 ....: ComplexHermitianEJA,
565 ....: QuaternionHermitianEJA,
570 The rank of the Jordan spin algebra is always two::
572 sage: JordanSpinEJA(2).rank()
574 sage: JordanSpinEJA(3).rank()
576 sage: JordanSpinEJA(4).rank()
579 The rank of the `n`-by-`n` Hermitian real, complex, or
580 quaternion matrices is `n`::
582 sage: RealSymmetricEJA(2).rank()
584 sage: ComplexHermitianEJA(2).rank()
586 sage: QuaternionHermitianEJA(2).rank()
588 sage: RealSymmetricEJA(5).rank()
590 sage: ComplexHermitianEJA(5).rank()
592 sage: QuaternionHermitianEJA(5).rank()
597 Ensure that every EJA that we know how to construct has a
598 positive integer rank::
600 sage: set_random_seed()
601 sage: r = random_eja().rank()
602 sage: r in ZZ and r > 0
609 def vector_space(self
):
611 Return the vector space that underlies this algebra.
615 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
619 sage: J = RealSymmetricEJA(2)
620 sage: J.vector_space()
621 Vector space of dimension 3 over Rational Field
624 return self
.zero().to_vector().parent().ambient_vector_space()
627 Element
= FiniteDimensionalEuclideanJordanAlgebraElement
630 class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra
):
632 Return the Euclidean Jordan Algebra corresponding to the set
633 `R^n` under the Hadamard product.
635 Note: this is nothing more than the Cartesian product of ``n``
636 copies of the spin algebra. Once Cartesian product algebras
637 are implemented, this can go.
641 sage: from mjo.eja.eja_algebra import RealCartesianProductEJA
645 This multiplication table can be verified by hand::
647 sage: J = RealCartesianProductEJA(3)
648 sage: e0,e1,e2 = J.gens()
663 def __init__(self
, n
, field
=QQ
):
664 V
= VectorSpace(field
, n
)
665 mult_table
= [ [ V
.gen(i
)*(i
== j
) for j
in range(n
) ]
668 fdeja
= super(RealCartesianProductEJA
, self
)
669 return fdeja
.__init
__(field
, mult_table
, rank
=n
)
671 def inner_product(self
, x
, y
):
672 return _usual_ip(x
,y
)
677 Return a "random" finite-dimensional Euclidean Jordan Algebra.
681 For now, we choose a random natural number ``n`` (greater than zero)
682 and then give you back one of the following:
684 * The cartesian product of the rational numbers ``n`` times; this is
685 ``QQ^n`` with the Hadamard product.
687 * The Jordan spin algebra on ``QQ^n``.
689 * The ``n``-by-``n`` rational symmetric matrices with the symmetric
692 * The ``n``-by-``n`` complex-rational Hermitian matrices embedded
693 in the space of ``2n``-by-``2n`` real symmetric matrices.
695 * The ``n``-by-``n`` quaternion-rational Hermitian matrices embedded
696 in the space of ``4n``-by-``4n`` real symmetric matrices.
698 Later this might be extended to return Cartesian products of the
703 sage: from mjo.eja.eja_algebra import random_eja
708 Euclidean Jordan algebra of dimension...
712 # The max_n component lets us choose different upper bounds on the
713 # value "n" that gets passed to the constructor. This is needed
714 # because e.g. R^{10} is reasonable to test, while the Hermitian
715 # 10-by-10 quaternion matrices are not.
716 (constructor
, max_n
) = choice([(RealCartesianProductEJA
, 6),
718 (RealSymmetricEJA
, 5),
719 (ComplexHermitianEJA
, 4),
720 (QuaternionHermitianEJA
, 3)])
721 n
= ZZ
.random_element(1, max_n
)
722 return constructor(n
, field
=QQ
)
726 def _real_symmetric_basis(n
, field
=QQ
):
728 Return a basis for the space of real symmetric n-by-n matrices.
730 # The basis of symmetric matrices, as matrices, in their R^(n-by-n)
734 for j
in xrange(i
+1):
735 Eij
= matrix(field
, n
, lambda k
,l
: k
==i
and l
==j
)
739 # Beware, orthogonal but not normalized!
740 Sij
= Eij
+ Eij
.transpose()
745 def _complex_hermitian_basis(n
, field
=QQ
):
747 Returns a basis for the space of complex Hermitian n-by-n matrices.
751 sage: from mjo.eja.eja_algebra import _complex_hermitian_basis
755 sage: set_random_seed()
756 sage: n = ZZ.random_element(1,5)
757 sage: all( M.is_symmetric() for M in _complex_hermitian_basis(n) )
761 F
= QuadraticField(-1, 'I')
764 # This is like the symmetric case, but we need to be careful:
766 # * We want conjugate-symmetry, not just symmetry.
767 # * The diagonal will (as a result) be real.
771 for j
in xrange(i
+1):
772 Eij
= matrix(field
, n
, lambda k
,l
: k
==i
and l
==j
)
774 Sij
= _embed_complex_matrix(Eij
)
777 # Beware, orthogonal but not normalized! The second one
778 # has a minus because it's conjugated.
779 Sij_real
= _embed_complex_matrix(Eij
+ Eij
.transpose())
781 Sij_imag
= _embed_complex_matrix(I
*Eij
- I
*Eij
.transpose())
786 def _quaternion_hermitian_basis(n
, field
=QQ
):
788 Returns a basis for the space of quaternion Hermitian n-by-n matrices.
792 sage: from mjo.eja.eja_algebra import _quaternion_hermitian_basis
796 sage: set_random_seed()
797 sage: n = ZZ.random_element(1,5)
798 sage: all( M.is_symmetric() for M in _quaternion_hermitian_basis(n) )
802 Q
= QuaternionAlgebra(QQ
,-1,-1)
805 # This is like the symmetric case, but we need to be careful:
807 # * We want conjugate-symmetry, not just symmetry.
808 # * The diagonal will (as a result) be real.
812 for j
in xrange(i
+1):
813 Eij
= matrix(Q
, n
, lambda k
,l
: k
==i
and l
==j
)
815 Sij
= _embed_quaternion_matrix(Eij
)
818 # Beware, orthogonal but not normalized! The second,
819 # third, and fourth ones have a minus because they're
821 Sij_real
= _embed_quaternion_matrix(Eij
+ Eij
.transpose())
823 Sij_I
= _embed_quaternion_matrix(I
*Eij
- I
*Eij
.transpose())
825 Sij_J
= _embed_quaternion_matrix(J
*Eij
- J
*Eij
.transpose())
827 Sij_K
= _embed_quaternion_matrix(K
*Eij
- K
*Eij
.transpose())
833 def _multiplication_table_from_matrix_basis(basis
):
835 At least three of the five simple Euclidean Jordan algebras have the
836 symmetric multiplication (A,B) |-> (AB + BA)/2, where the
837 multiplication on the right is matrix multiplication. Given a basis
838 for the underlying matrix space, this function returns a
839 multiplication table (obtained by looping through the basis
840 elements) for an algebra of those matrices.
842 # In S^2, for example, we nominally have four coordinates even
843 # though the space is of dimension three only. The vector space V
844 # is supposed to hold the entire long vector, and the subspace W
845 # of V will be spanned by the vectors that arise from symmetric
846 # matrices. Thus for S^2, dim(V) == 4 and dim(W) == 3.
847 field
= basis
[0].base_ring()
848 dimension
= basis
[0].nrows()
850 V
= VectorSpace(field
, dimension
**2)
851 W
= V
.span_of_basis( _mat2vec(s
) for s
in basis
)
853 mult_table
= [[W
.zero() for j
in range(n
)] for i
in range(n
)]
856 mat_entry
= (basis
[i
]*basis
[j
] + basis
[j
]*basis
[i
])/2
857 mult_table
[i
][j
] = W
.coordinate_vector(_mat2vec(mat_entry
))
862 def _embed_complex_matrix(M
):
864 Embed the n-by-n complex matrix ``M`` into the space of real
865 matrices of size 2n-by-2n via the map the sends each entry `z = a +
866 bi` to the block matrix ``[[a,b],[-b,a]]``.
870 sage: from mjo.eja.eja_algebra import _embed_complex_matrix
874 sage: F = QuadraticField(-1,'i')
875 sage: x1 = F(4 - 2*i)
876 sage: x2 = F(1 + 2*i)
879 sage: M = matrix(F,2,[[x1,x2],[x3,x4]])
880 sage: _embed_complex_matrix(M)
889 Embedding is a homomorphism (isomorphism, in fact)::
891 sage: set_random_seed()
892 sage: n = ZZ.random_element(5)
893 sage: F = QuadraticField(-1, 'i')
894 sage: X = random_matrix(F, n)
895 sage: Y = random_matrix(F, n)
896 sage: actual = _embed_complex_matrix(X) * _embed_complex_matrix(Y)
897 sage: expected = _embed_complex_matrix(X*Y)
898 sage: actual == expected
904 raise ValueError("the matrix 'M' must be square")
905 field
= M
.base_ring()
910 blocks
.append(matrix(field
, 2, [[a
,b
],[-b
,a
]]))
912 # We can drop the imaginaries here.
913 return matrix
.block(field
.base_ring(), n
, blocks
)
916 def _unembed_complex_matrix(M
):
918 The inverse of _embed_complex_matrix().
922 sage: from mjo.eja.eja_algebra import (_embed_complex_matrix,
923 ....: _unembed_complex_matrix)
927 sage: A = matrix(QQ,[ [ 1, 2, 3, 4],
928 ....: [-2, 1, -4, 3],
929 ....: [ 9, 10, 11, 12],
930 ....: [-10, 9, -12, 11] ])
931 sage: _unembed_complex_matrix(A)
933 [ 10*i + 9 12*i + 11]
937 Unembedding is the inverse of embedding::
939 sage: set_random_seed()
940 sage: F = QuadraticField(-1, 'i')
941 sage: M = random_matrix(F, 3)
942 sage: _unembed_complex_matrix(_embed_complex_matrix(M)) == M
948 raise ValueError("the matrix 'M' must be square")
949 if not n
.mod(2).is_zero():
950 raise ValueError("the matrix 'M' must be a complex embedding")
952 F
= QuadraticField(-1, 'i')
955 # Go top-left to bottom-right (reading order), converting every
956 # 2-by-2 block we see to a single complex element.
958 for k
in xrange(n
/2):
959 for j
in xrange(n
/2):
960 submat
= M
[2*k
:2*k
+2,2*j
:2*j
+2]
961 if submat
[0,0] != submat
[1,1]:
962 raise ValueError('bad on-diagonal submatrix')
963 if submat
[0,1] != -submat
[1,0]:
964 raise ValueError('bad off-diagonal submatrix')
965 z
= submat
[0,0] + submat
[0,1]*i
968 return matrix(F
, n
/2, elements
)
971 def _embed_quaternion_matrix(M
):
973 Embed the n-by-n quaternion matrix ``M`` into the space of real
974 matrices of size 4n-by-4n by first sending each quaternion entry
975 `z = a + bi + cj + dk` to the block-complex matrix
976 ``[[a + bi, c+di],[-c + di, a-bi]]`, and then embedding those into
981 sage: from mjo.eja.eja_algebra import _embed_quaternion_matrix
985 sage: Q = QuaternionAlgebra(QQ,-1,-1)
986 sage: i,j,k = Q.gens()
987 sage: x = 1 + 2*i + 3*j + 4*k
988 sage: M = matrix(Q, 1, [[x]])
989 sage: _embed_quaternion_matrix(M)
995 Embedding is a homomorphism (isomorphism, in fact)::
997 sage: set_random_seed()
998 sage: n = ZZ.random_element(5)
999 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1000 sage: X = random_matrix(Q, n)
1001 sage: Y = random_matrix(Q, n)
1002 sage: actual = _embed_quaternion_matrix(X)*_embed_quaternion_matrix(Y)
1003 sage: expected = _embed_quaternion_matrix(X*Y)
1004 sage: actual == expected
1008 quaternions
= M
.base_ring()
1011 raise ValueError("the matrix 'M' must be square")
1013 F
= QuadraticField(-1, 'i')
1018 t
= z
.coefficient_tuple()
1023 cplx_matrix
= matrix(F
, 2, [[ a
+ b
*i
, c
+ d
*i
],
1024 [-c
+ d
*i
, a
- b
*i
]])
1025 blocks
.append(_embed_complex_matrix(cplx_matrix
))
1027 # We should have real entries by now, so use the realest field
1028 # we've got for the return value.
1029 return matrix
.block(quaternions
.base_ring(), n
, blocks
)
1032 def _unembed_quaternion_matrix(M
):
1034 The inverse of _embed_quaternion_matrix().
1038 sage: from mjo.eja.eja_algebra import (_embed_quaternion_matrix,
1039 ....: _unembed_quaternion_matrix)
1043 sage: M = matrix(QQ, [[ 1, 2, 3, 4],
1044 ....: [-2, 1, -4, 3],
1045 ....: [-3, 4, 1, -2],
1046 ....: [-4, -3, 2, 1]])
1047 sage: _unembed_quaternion_matrix(M)
1048 [1 + 2*i + 3*j + 4*k]
1052 Unembedding is the inverse of embedding::
1054 sage: set_random_seed()
1055 sage: Q = QuaternionAlgebra(QQ, -1, -1)
1056 sage: M = random_matrix(Q, 3)
1057 sage: _unembed_quaternion_matrix(_embed_quaternion_matrix(M)) == M
1063 raise ValueError("the matrix 'M' must be square")
1064 if not n
.mod(4).is_zero():
1065 raise ValueError("the matrix 'M' must be a complex embedding")
1067 Q
= QuaternionAlgebra(QQ
,-1,-1)
1070 # Go top-left to bottom-right (reading order), converting every
1071 # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1
1074 for l
in xrange(n
/4):
1075 for m
in xrange(n
/4):
1076 submat
= _unembed_complex_matrix(M
[4*l
:4*l
+4,4*m
:4*m
+4])
1077 if submat
[0,0] != submat
[1,1].conjugate():
1078 raise ValueError('bad on-diagonal submatrix')
1079 if submat
[0,1] != -submat
[1,0].conjugate():
1080 raise ValueError('bad off-diagonal submatrix')
1081 z
= submat
[0,0].real() + submat
[0,0].imag()*i
1082 z
+= submat
[0,1].real()*j
+ submat
[0,1].imag()*k
1085 return matrix(Q
, n
/4, elements
)
1088 # The usual inner product on R^n.
1090 return x
.to_vector().inner_product(y
.to_vector())
1092 # The inner product used for the real symmetric simple EJA.
1093 # We keep it as a separate function because e.g. the complex
1094 # algebra uses the same inner product, except divided by 2.
1095 def _matrix_ip(X
,Y
):
1096 X_mat
= X
.natural_representation()
1097 Y_mat
= Y
.natural_representation()
1098 return (X_mat
*Y_mat
).trace()
1101 class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1103 The rank-n simple EJA consisting of real symmetric n-by-n
1104 matrices, the usual symmetric Jordan product, and the trace inner
1105 product. It has dimension `(n^2 + n)/2` over the reals.
1109 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
1113 sage: J = RealSymmetricEJA(2)
1114 sage: e0, e1, e2 = J.gens()
1124 The dimension of this algebra is `(n^2 + n) / 2`::
1126 sage: set_random_seed()
1127 sage: n = ZZ.random_element(1,5)
1128 sage: J = RealSymmetricEJA(n)
1129 sage: J.dimension() == (n^2 + n)/2
1132 The Jordan multiplication is what we think it is::
1134 sage: set_random_seed()
1135 sage: n = ZZ.random_element(1,5)
1136 sage: J = RealSymmetricEJA(n)
1137 sage: x = J.random_element()
1138 sage: y = J.random_element()
1139 sage: actual = (x*y).natural_representation()
1140 sage: X = x.natural_representation()
1141 sage: Y = y.natural_representation()
1142 sage: expected = (X*Y + Y*X)/2
1143 sage: actual == expected
1145 sage: J(expected) == x*y
1149 def __init__(self
, n
, field
=QQ
):
1150 S
= _real_symmetric_basis(n
, field
=field
)
1151 Qs
= _multiplication_table_from_matrix_basis(S
)
1153 fdeja
= super(RealSymmetricEJA
, self
)
1154 return fdeja
.__init
__(field
,
1159 def inner_product(self
, x
, y
):
1160 return _matrix_ip(x
,y
)
1163 class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1165 The rank-n simple EJA consisting of complex Hermitian n-by-n
1166 matrices over the real numbers, the usual symmetric Jordan product,
1167 and the real-part-of-trace inner product. It has dimension `n^2` over
1172 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
1176 The dimension of this algebra is `n^2`::
1178 sage: set_random_seed()
1179 sage: n = ZZ.random_element(1,5)
1180 sage: J = ComplexHermitianEJA(n)
1181 sage: J.dimension() == n^2
1184 The Jordan multiplication is what we think it is::
1186 sage: set_random_seed()
1187 sage: n = ZZ.random_element(1,5)
1188 sage: J = ComplexHermitianEJA(n)
1189 sage: x = J.random_element()
1190 sage: y = J.random_element()
1191 sage: actual = (x*y).natural_representation()
1192 sage: X = x.natural_representation()
1193 sage: Y = y.natural_representation()
1194 sage: expected = (X*Y + Y*X)/2
1195 sage: actual == expected
1197 sage: J(expected) == x*y
1201 def __init__(self
, n
, field
=QQ
):
1202 S
= _complex_hermitian_basis(n
)
1203 Qs
= _multiplication_table_from_matrix_basis(S
)
1205 fdeja
= super(ComplexHermitianEJA
, self
)
1206 return fdeja
.__init
__(field
,
1212 def inner_product(self
, x
, y
):
1213 # Since a+bi on the diagonal is represented as
1218 # we'll double-count the "a" entries if we take the trace of
1220 return _matrix_ip(x
,y
)/2
1223 class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1225 The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
1226 matrices, the usual symmetric Jordan product, and the
1227 real-part-of-trace inner product. It has dimension `2n^2 - n` over
1232 sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
1236 The dimension of this algebra is `n^2`::
1238 sage: set_random_seed()
1239 sage: n = ZZ.random_element(1,5)
1240 sage: J = QuaternionHermitianEJA(n)
1241 sage: J.dimension() == 2*(n^2) - n
1244 The Jordan multiplication is what we think it is::
1246 sage: set_random_seed()
1247 sage: n = ZZ.random_element(1,5)
1248 sage: J = QuaternionHermitianEJA(n)
1249 sage: x = J.random_element()
1250 sage: y = J.random_element()
1251 sage: actual = (x*y).natural_representation()
1252 sage: X = x.natural_representation()
1253 sage: Y = y.natural_representation()
1254 sage: expected = (X*Y + Y*X)/2
1255 sage: actual == expected
1257 sage: J(expected) == x*y
1261 def __init__(self
, n
, field
=QQ
):
1262 S
= _quaternion_hermitian_basis(n
)
1263 Qs
= _multiplication_table_from_matrix_basis(S
)
1265 fdeja
= super(QuaternionHermitianEJA
, self
)
1266 return fdeja
.__init
__(field
,
1271 def inner_product(self
, x
, y
):
1272 # Since a+bi+cj+dk on the diagonal is represented as
1274 # a + bi +cj + dk = [ a b c d]
1279 # we'll quadruple-count the "a" entries if we take the trace of
1281 return _matrix_ip(x
,y
)/4
1284 class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1286 The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
1287 with the usual inner product and jordan product ``x*y =
1288 (<x_bar,y_bar>, x0*y_bar + y0*x_bar)``. It has dimension `n` over
1293 sage: from mjo.eja.eja_algebra import JordanSpinEJA
1297 This multiplication table can be verified by hand::
1299 sage: J = JordanSpinEJA(4)
1300 sage: e0,e1,e2,e3 = J.gens()
1317 def __init__(self
, n
, field
=QQ
):
1318 V
= VectorSpace(field
, n
)
1319 mult_table
= [[V
.zero() for j
in range(n
)] for i
in range(n
)]
1329 z0
= x
.inner_product(y
)
1330 zbar
= y0
*xbar
+ x0
*ybar
1331 z
= V([z0
] + zbar
.list())
1332 mult_table
[i
][j
] = z
1334 # The rank of the spin algebra is two, unless we're in a
1335 # one-dimensional ambient space (because the rank is bounded by
1336 # the ambient dimension).
1337 fdeja
= super(JordanSpinEJA
, self
)
1338 return fdeja
.__init
__(field
, mult_table
, rank
=min(n
,2))
1340 def inner_product(self
, x
, y
):
1341 return _usual_ip(x
,y
)