2 Euclidean Jordan Algebras. These are formally-real Jordan Algebras;
3 specifically those where u^2 + v^2 = 0 implies that u = v = 0. They
4 are used in optimization, and have some additional nice methods beyond
5 what can be supported in a general Jordan Algebra.
8 from itertools
import repeat
10 from sage
.algebras
.quatalg
.quaternion_algebra
import QuaternionAlgebra
11 from sage
.categories
.magmatic_algebras
import MagmaticAlgebras
12 from sage
.combinat
.free_module
import CombinatorialFreeModule
13 from sage
.matrix
.constructor
import matrix
14 from sage
.matrix
.matrix_space
import MatrixSpace
15 from sage
.misc
.cachefunc
import cached_method
16 from sage
.misc
.prandom
import choice
17 from sage
.misc
.table
import table
18 from sage
.modules
.free_module
import FreeModule
, VectorSpace
19 from sage
.rings
.integer_ring
import ZZ
20 from sage
.rings
.number_field
.number_field
import NumberField
, QuadraticField
21 from sage
.rings
.polynomial
.polynomial_ring_constructor
import PolynomialRing
22 from sage
.rings
.rational_field
import QQ
23 from sage
.rings
.real_lazy
import CLF
, RLF
24 from sage
.structure
.element
import is_Matrix
26 from mjo
.eja
.eja_element
import FiniteDimensionalEuclideanJordanAlgebraElement
27 from mjo
.eja
.eja_utils
import _mat2vec
29 class FiniteDimensionalEuclideanJordanAlgebra(CombinatorialFreeModule
):
30 # This is an ugly hack needed to prevent the category framework
31 # from implementing a coercion from our base ring (e.g. the
32 # rationals) into the algebra. First of all -- such a coercion is
33 # nonsense to begin with. But more importantly, it tries to do so
34 # in the category of rings, and since our algebras aren't
35 # associative they generally won't be rings.
36 _no_generic_basering_coercion
= True
48 sage: from mjo.eja.eja_algebra import random_eja
52 By definition, Jordan multiplication commutes::
54 sage: set_random_seed()
55 sage: J = random_eja()
56 sage: x,y = J.random_elements(2)
62 self
._natural
_basis
= natural_basis
65 category
= MagmaticAlgebras(field
).FiniteDimensional()
66 category
= category
.WithBasis().Unital()
68 fda
= super(FiniteDimensionalEuclideanJordanAlgebra
, self
)
70 range(len(mult_table
)),
73 self
.print_options(bracket
='')
75 # The multiplication table we're given is necessarily in terms
76 # of vectors, because we don't have an algebra yet for
77 # anything to be an element of. However, it's faster in the
78 # long run to have the multiplication table be in terms of
79 # algebra elements. We do this after calling the superclass
80 # constructor so that from_vector() knows what to do.
81 self
._multiplication
_table
= [ map(lambda x
: self
.from_vector(x
), ls
)
82 for ls
in mult_table
]
85 def _element_constructor_(self
, elt
):
87 Construct an element of this algebra from its natural
90 This gets called only after the parent element _call_ method
91 fails to find a coercion for the argument.
95 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
96 ....: RealCartesianProductEJA,
97 ....: RealSymmetricEJA)
101 The identity in `S^n` is converted to the identity in the EJA::
103 sage: J = RealSymmetricEJA(3)
104 sage: I = matrix.identity(QQ,3)
105 sage: J(I) == J.one()
108 This skew-symmetric matrix can't be represented in the EJA::
110 sage: J = RealSymmetricEJA(3)
111 sage: A = matrix(QQ,3, lambda i,j: i-j)
113 Traceback (most recent call last):
115 ArithmeticError: vector is not in free module
119 Ensure that we can convert any element of the two non-matrix
120 simple algebras (whose natural representations are their usual
121 vector representations) back and forth faithfully::
123 sage: set_random_seed()
124 sage: J = RealCartesianProductEJA.random_instance()
125 sage: x = J.random_element()
126 sage: J(x.to_vector().column()) == x
128 sage: J = JordanSpinEJA.random_instance()
129 sage: x = J.random_element()
130 sage: J(x.to_vector().column()) == x
135 # The superclass implementation of random_element()
136 # needs to be able to coerce "0" into the algebra.
139 natural_basis
= self
.natural_basis()
140 basis_space
= natural_basis
[0].matrix_space()
141 if elt
not in basis_space
:
142 raise ValueError("not a naturally-represented algebra element")
144 # Thanks for nothing! Matrix spaces aren't vector spaces in
145 # Sage, so we have to figure out its natural-basis coordinates
146 # ourselves. We use the basis space's ring instead of the
147 # element's ring because the basis space might be an algebraic
148 # closure whereas the base ring of the 3-by-3 identity matrix
149 # could be QQ instead of QQbar.
150 V
= VectorSpace(basis_space
.base_ring(), elt
.nrows()*elt
.ncols())
151 W
= V
.span_of_basis( _mat2vec(s
) for s
in natural_basis
)
152 coords
= W
.coordinate_vector(_mat2vec(elt
))
153 return self
.from_vector(coords
)
157 def _max_test_case_size():
159 Return an integer "size" that is an upper bound on the size of
160 this algebra when it is used in a random test
161 case. Unfortunately, the term "size" is quite vague -- when
162 dealing with `R^n` under either the Hadamard or Jordan spin
163 product, the "size" refers to the dimension `n`. When dealing
164 with a matrix algebra (real symmetric or complex/quaternion
165 Hermitian), it refers to the size of the matrix, which is
166 far less than the dimension of the underlying vector space.
168 We default to five in this class, which is safe in `R^n`. The
169 matrix algebra subclasses (or any class where the "size" is
170 interpreted to be far less than the dimension) should override
171 with a smaller number.
178 Return a string representation of ``self``.
182 sage: from mjo.eja.eja_algebra import JordanSpinEJA
186 Ensure that it says what we think it says::
188 sage: JordanSpinEJA(2, field=QQ)
189 Euclidean Jordan algebra of dimension 2 over Rational Field
190 sage: JordanSpinEJA(3, field=RDF)
191 Euclidean Jordan algebra of dimension 3 over Real Double Field
194 fmt
= "Euclidean Jordan algebra of dimension {} over {}"
195 return fmt
.format(self
.dimension(), self
.base_ring())
197 def product_on_basis(self
, i
, j
):
198 return self
._multiplication
_table
[i
][j
]
200 def _a_regular_element(self
):
202 Guess a regular element. Needed to compute the basis for our
203 characteristic polynomial coefficients.
207 sage: from mjo.eja.eja_algebra import random_eja
211 Ensure that this hacky method succeeds for every algebra that we
212 know how to construct::
214 sage: set_random_seed()
215 sage: J = random_eja()
216 sage: J._a_regular_element().is_regular()
221 z
= self
.sum( (i
+1)*gs
[i
] for i
in range(len(gs
)) )
222 if not z
.is_regular():
223 raise ValueError("don't know a regular element")
228 def _charpoly_basis_space(self
):
230 Return the vector space spanned by the basis used in our
231 characteristic polynomial coefficients. This is used not only to
232 compute those coefficients, but also any time we need to
233 evaluate the coefficients (like when we compute the trace or
236 z
= self
._a
_regular
_element
()
237 # Don't use the parent vector space directly here in case this
238 # happens to be a subalgebra. In that case, we would be e.g.
239 # two-dimensional but span_of_basis() would expect three
241 V
= VectorSpace(self
.base_ring(), self
.vector_space().dimension())
242 basis
= [ (z
**k
).to_vector() for k
in range(self
.rank()) ]
243 V1
= V
.span_of_basis( basis
)
244 b
= (V1
.basis() + V1
.complement().basis())
245 return V
.span_of_basis(b
)
250 def _charpoly_coeff(self
, i
):
252 Return the coefficient polynomial "a_{i}" of this algebra's
253 general characteristic polynomial.
255 Having this be a separate cached method lets us compute and
256 store the trace/determinant (a_{r-1} and a_{0} respectively)
257 separate from the entire characteristic polynomial.
259 (A_of_x
, x
, xr
, detA
) = self
._charpoly
_matrix
_system
()
260 R
= A_of_x
.base_ring()
262 # Guaranteed by theory
265 # Danger: the in-place modification is done for performance
266 # reasons (reconstructing a matrix with huge polynomial
267 # entries is slow), but I don't know how cached_method works,
268 # so it's highly possible that we're modifying some global
269 # list variable by reference, here. In other words, you
270 # probably shouldn't call this method twice on the same
271 # algebra, at the same time, in two threads
272 Ai_orig
= A_of_x
.column(i
)
273 A_of_x
.set_column(i
,xr
)
274 numerator
= A_of_x
.det()
275 A_of_x
.set_column(i
,Ai_orig
)
277 # We're relying on the theory here to ensure that each a_i is
278 # indeed back in R, and the added negative signs are to make
279 # the whole charpoly expression sum to zero.
280 return R(-numerator
/detA
)
284 def _charpoly_matrix_system(self
):
286 Compute the matrix whose entries A_ij are polynomials in
287 X1,...,XN, the vector ``x`` of variables X1,...,XN, the vector
288 corresponding to `x^r` and the determinent of the matrix A =
289 [A_ij]. In other words, all of the fixed (cachable) data needed
290 to compute the coefficients of the characteristic polynomial.
295 # Turn my vector space into a module so that "vectors" can
296 # have multivatiate polynomial entries.
297 names
= tuple('X' + str(i
) for i
in range(1,n
+1))
298 R
= PolynomialRing(self
.base_ring(), names
)
300 # Using change_ring() on the parent's vector space doesn't work
301 # here because, in a subalgebra, that vector space has a basis
302 # and change_ring() tries to bring the basis along with it. And
303 # that doesn't work unless the new ring is a PID, which it usually
307 # Now let x = (X1,X2,...,Xn) be the vector whose entries are
311 # And figure out the "left multiplication by x" matrix in
314 monomial_matrices
= [ self
.monomial(i
).operator().matrix()
315 for i
in range(n
) ] # don't recompute these!
317 ek
= self
.monomial(k
).to_vector()
319 sum( x
[i
]*(monomial_matrices
[i
]*ek
)
320 for i
in range(n
) ) )
321 Lx
= matrix
.column(R
, lmbx_cols
)
323 # Now we can compute powers of x "symbolically"
324 x_powers
= [self
.one().to_vector(), x
]
325 for d
in range(2, r
+1):
326 x_powers
.append( Lx
*(x_powers
[-1]) )
328 idmat
= matrix
.identity(R
, n
)
330 W
= self
._charpoly
_basis
_space
()
331 W
= W
.change_ring(R
.fraction_field())
333 # Starting with the standard coordinates x = (X1,X2,...,Xn)
334 # and then converting the entries to W-coordinates allows us
335 # to pass in the standard coordinates to the charpoly and get
336 # back the right answer. Specifically, with x = (X1,X2,...,Xn),
339 # W.coordinates(x^2) eval'd at (standard z-coords)
343 # W-coords of (standard coords of x^2 eval'd at std-coords of z)
345 # We want the middle equivalent thing in our matrix, but use
346 # the first equivalent thing instead so that we can pass in
347 # standard coordinates.
348 x_powers
= [ W
.coordinate_vector(xp
) for xp
in x_powers
]
349 l2
= [idmat
.column(k
-1) for k
in range(r
+1, n
+1)]
350 A_of_x
= matrix
.column(R
, n
, (x_powers
[:r
] + l2
))
351 return (A_of_x
, x
, x_powers
[r
], A_of_x
.det())
355 def characteristic_polynomial(self
):
357 Return a characteristic polynomial that works for all elements
360 The resulting polynomial has `n+1` variables, where `n` is the
361 dimension of this algebra. The first `n` variables correspond to
362 the coordinates of an algebra element: when evaluated at the
363 coordinates of an algebra element with respect to a certain
364 basis, the result is a univariate polynomial (in the one
365 remaining variable ``t``), namely the characteristic polynomial
370 sage: from mjo.eja.eja_algebra import JordanSpinEJA
374 The characteristic polynomial in the spin algebra is given in
375 Alizadeh, Example 11.11::
377 sage: J = JordanSpinEJA(3)
378 sage: p = J.characteristic_polynomial(); p
379 X1^2 - X2^2 - X3^2 + (-2*t)*X1 + t^2
380 sage: xvec = J.one().to_vector()
388 # The list of coefficient polynomials a_1, a_2, ..., a_n.
389 a
= [ self
._charpoly
_coeff
(i
) for i
in range(n
) ]
391 # We go to a bit of trouble here to reorder the
392 # indeterminates, so that it's easier to evaluate the
393 # characteristic polynomial at x's coordinates and get back
394 # something in terms of t, which is what we want.
396 S
= PolynomialRing(self
.base_ring(),'t')
398 S
= PolynomialRing(S
, R
.variable_names())
401 # Note: all entries past the rth should be zero. The
402 # coefficient of the highest power (x^r) is 1, but it doesn't
403 # appear in the solution vector which contains coefficients
404 # for the other powers (to make them sum to x^r).
406 a
[r
] = 1 # corresponds to x^r
408 # When the rank is equal to the dimension, trying to
409 # assign a[r] goes out-of-bounds.
410 a
.append(1) # corresponds to x^r
412 return sum( a
[k
]*(t
**k
) for k
in range(len(a
)) )
415 def inner_product(self
, x
, y
):
417 The inner product associated with this Euclidean Jordan algebra.
419 Defaults to the trace inner product, but can be overridden by
420 subclasses if they are sure that the necessary properties are
425 sage: from mjo.eja.eja_algebra import random_eja
429 Our inner product satisfies the Jordan axiom, which is also
430 referred to as "associativity" for a symmetric bilinear form::
432 sage: set_random_seed()
433 sage: J = random_eja()
434 sage: x,y,z = J.random_elements(3)
435 sage: (x*y).inner_product(z) == y.inner_product(x*z)
439 X
= x
.natural_representation()
440 Y
= y
.natural_representation()
441 return self
.natural_inner_product(X
,Y
)
444 def is_trivial(self
):
446 Return whether or not this algebra is trivial.
448 A trivial algebra contains only the zero element.
452 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
456 sage: J = ComplexHermitianEJA(3)
459 sage: A = J.zero().subalgebra_generated_by()
464 return self
.dimension() == 0
467 def multiplication_table(self
):
469 Return a visual representation of this algebra's multiplication
470 table (on basis elements).
474 sage: from mjo.eja.eja_algebra import JordanSpinEJA
478 sage: J = JordanSpinEJA(4)
479 sage: J.multiplication_table()
480 +----++----+----+----+----+
481 | * || e0 | e1 | e2 | e3 |
482 +====++====+====+====+====+
483 | e0 || e0 | e1 | e2 | e3 |
484 +----++----+----+----+----+
485 | e1 || e1 | e0 | 0 | 0 |
486 +----++----+----+----+----+
487 | e2 || e2 | 0 | e0 | 0 |
488 +----++----+----+----+----+
489 | e3 || e3 | 0 | 0 | e0 |
490 +----++----+----+----+----+
493 M
= list(self
._multiplication
_table
) # copy
494 for i
in range(len(M
)):
495 # M had better be "square"
496 M
[i
] = [self
.monomial(i
)] + M
[i
]
497 M
= [["*"] + list(self
.gens())] + M
498 return table(M
, header_row
=True, header_column
=True, frame
=True)
501 def natural_basis(self
):
503 Return a more-natural representation of this algebra's basis.
505 Every finite-dimensional Euclidean Jordan Algebra is a direct
506 sum of five simple algebras, four of which comprise Hermitian
507 matrices. This method returns the original "natural" basis
508 for our underlying vector space. (Typically, the natural basis
509 is used to construct the multiplication table in the first place.)
511 Note that this will always return a matrix. The standard basis
512 in `R^n` will be returned as `n`-by-`1` column matrices.
516 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
517 ....: RealSymmetricEJA)
521 sage: J = RealSymmetricEJA(2)
523 Finite family {0: e0, 1: e1, 2: e2}
524 sage: J.natural_basis()
526 [1 0] [ 0 1/2*sqrt2] [0 0]
527 [0 0], [1/2*sqrt2 0], [0 1]
532 sage: J = JordanSpinEJA(2)
534 Finite family {0: e0, 1: e1}
535 sage: J.natural_basis()
542 if self
._natural
_basis
is None:
543 M
= self
.natural_basis_space()
544 return tuple( M(b
.to_vector()) for b
in self
.basis() )
546 return self
._natural
_basis
549 def natural_basis_space(self
):
551 Return the matrix space in which this algebra's natural basis
554 if self
._natural
_basis
is None or len(self
._natural
_basis
) == 0:
555 return MatrixSpace(self
.base_ring(), self
.dimension(), 1)
557 return self
._natural
_basis
[0].matrix_space()
561 def natural_inner_product(X
,Y
):
563 Compute the inner product of two naturally-represented elements.
565 For example in the real symmetric matrix EJA, this will compute
566 the trace inner-product of two n-by-n symmetric matrices. The
567 default should work for the real cartesian product EJA, the
568 Jordan spin EJA, and the real symmetric matrices. The others
569 will have to be overridden.
571 return (X
.conjugate_transpose()*Y
).trace()
577 Return the unit element of this algebra.
581 sage: from mjo.eja.eja_algebra import (RealCartesianProductEJA,
586 sage: J = RealCartesianProductEJA(5)
588 e0 + e1 + e2 + e3 + e4
592 The identity element acts like the identity::
594 sage: set_random_seed()
595 sage: J = random_eja()
596 sage: x = J.random_element()
597 sage: J.one()*x == x and x*J.one() == x
600 The matrix of the unit element's operator is the identity::
602 sage: set_random_seed()
603 sage: J = random_eja()
604 sage: actual = J.one().operator().matrix()
605 sage: expected = matrix.identity(J.base_ring(), J.dimension())
606 sage: actual == expected
610 # We can brute-force compute the matrices of the operators
611 # that correspond to the basis elements of this algebra.
612 # If some linear combination of those basis elements is the
613 # algebra identity, then the same linear combination of
614 # their matrices has to be the identity matrix.
616 # Of course, matrices aren't vectors in sage, so we have to
617 # appeal to the "long vectors" isometry.
618 oper_vecs
= [ _mat2vec(g
.operator().matrix()) for g
in self
.gens() ]
620 # Now we use basis linear algebra to find the coefficients,
621 # of the matrices-as-vectors-linear-combination, which should
622 # work for the original algebra basis too.
623 A
= matrix
.column(self
.base_ring(), oper_vecs
)
625 # We used the isometry on the left-hand side already, but we
626 # still need to do it for the right-hand side. Recall that we
627 # wanted something that summed to the identity matrix.
628 b
= _mat2vec( matrix
.identity(self
.base_ring(), self
.dimension()) )
630 # Now if there's an identity element in the algebra, this should work.
631 coeffs
= A
.solve_right(b
)
632 return self
.linear_combination(zip(self
.gens(), coeffs
))
635 def random_element(self
):
636 # Temporary workaround for https://trac.sagemath.org/ticket/28327
637 if self
.is_trivial():
640 s
= super(FiniteDimensionalEuclideanJordanAlgebra
, self
)
641 return s
.random_element()
643 def random_elements(self
, count
):
645 Return ``count`` random elements as a tuple.
649 sage: from mjo.eja.eja_algebra import JordanSpinEJA
653 sage: J = JordanSpinEJA(3)
654 sage: x,y,z = J.random_elements(3)
655 sage: all( [ x in J, y in J, z in J ])
657 sage: len( J.random_elements(10) ) == 10
661 return tuple( self
.random_element() for idx
in xrange(count
) )
664 def random_instance(cls
, field
=QQ
, **kwargs
):
666 Return a random instance of this type of algebra.
668 In subclasses for algebras that we know how to construct, this
669 is a shortcut for constructing test cases and examples.
671 if cls
is FiniteDimensionalEuclideanJordanAlgebra
:
672 # Red flag! But in theory we could do this I guess. The
673 # only finite-dimensional exceptional EJA is the
674 # octononions. So, we could just create an EJA from an
675 # associative matrix algebra (generated by a subset of
676 # elements) with the symmetric product. Or, we could punt
677 # to random_eja() here, override it in our subclasses, and
678 # not worry about it.
679 raise NotImplementedError
681 n
= ZZ
.random_element(cls
._max
_test
_case
_size
()) + 1
682 return cls(n
, field
, **kwargs
)
687 Return the rank of this EJA.
691 The author knows of no algorithm to compute the rank of an EJA
692 where only the multiplication table is known. In lieu of one, we
693 require the rank to be specified when the algebra is created,
694 and simply pass along that number here.
698 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
699 ....: RealSymmetricEJA,
700 ....: ComplexHermitianEJA,
701 ....: QuaternionHermitianEJA,
706 The rank of the Jordan spin algebra is always two::
708 sage: JordanSpinEJA(2).rank()
710 sage: JordanSpinEJA(3).rank()
712 sage: JordanSpinEJA(4).rank()
715 The rank of the `n`-by-`n` Hermitian real, complex, or
716 quaternion matrices is `n`::
718 sage: RealSymmetricEJA(4).rank()
720 sage: ComplexHermitianEJA(3).rank()
722 sage: QuaternionHermitianEJA(2).rank()
727 Ensure that every EJA that we know how to construct has a
728 positive integer rank::
730 sage: set_random_seed()
731 sage: r = random_eja().rank()
732 sage: r in ZZ and r > 0
739 def vector_space(self
):
741 Return the vector space that underlies this algebra.
745 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
749 sage: J = RealSymmetricEJA(2)
750 sage: J.vector_space()
751 Vector space of dimension 3 over...
754 return self
.zero().to_vector().parent().ambient_vector_space()
757 Element
= FiniteDimensionalEuclideanJordanAlgebraElement
760 class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra
):
762 Return the Euclidean Jordan Algebra corresponding to the set
763 `R^n` under the Hadamard product.
765 Note: this is nothing more than the Cartesian product of ``n``
766 copies of the spin algebra. Once Cartesian product algebras
767 are implemented, this can go.
771 sage: from mjo.eja.eja_algebra import RealCartesianProductEJA
775 This multiplication table can be verified by hand::
777 sage: J = RealCartesianProductEJA(3)
778 sage: e0,e1,e2 = J.gens()
794 We can change the generator prefix::
796 sage: RealCartesianProductEJA(3, prefix='r').gens()
800 def __init__(self
, n
, field
=QQ
, **kwargs
):
801 V
= VectorSpace(field
, n
)
802 mult_table
= [ [ V
.gen(i
)*(i
== j
) for j
in range(n
) ]
805 fdeja
= super(RealCartesianProductEJA
, self
)
806 return fdeja
.__init
__(field
, mult_table
, rank
=n
, **kwargs
)
808 def inner_product(self
, x
, y
):
810 Faster to reimplement than to use natural representations.
814 sage: from mjo.eja.eja_algebra import RealCartesianProductEJA
818 Ensure that this is the usual inner product for the algebras
821 sage: set_random_seed()
822 sage: J = RealCartesianProductEJA.random_instance()
823 sage: x,y = J.random_elements(2)
824 sage: X = x.natural_representation()
825 sage: Y = y.natural_representation()
826 sage: x.inner_product(y) == J.natural_inner_product(X,Y)
830 return x
.to_vector().inner_product(y
.to_vector())
835 Return a "random" finite-dimensional Euclidean Jordan Algebra.
839 For now, we choose a random natural number ``n`` (greater than zero)
840 and then give you back one of the following:
842 * The cartesian product of the rational numbers ``n`` times; this is
843 ``QQ^n`` with the Hadamard product.
845 * The Jordan spin algebra on ``QQ^n``.
847 * The ``n``-by-``n`` rational symmetric matrices with the symmetric
850 * The ``n``-by-``n`` complex-rational Hermitian matrices embedded
851 in the space of ``2n``-by-``2n`` real symmetric matrices.
853 * The ``n``-by-``n`` quaternion-rational Hermitian matrices embedded
854 in the space of ``4n``-by-``4n`` real symmetric matrices.
856 Later this might be extended to return Cartesian products of the
861 sage: from mjo.eja.eja_algebra import random_eja
866 Euclidean Jordan algebra of dimension...
869 classname
= choice([RealCartesianProductEJA
,
873 QuaternionHermitianEJA
])
874 return classname
.random_instance()
881 class MatrixEuclideanJordanAlgebra(FiniteDimensionalEuclideanJordanAlgebra
):
883 def _max_test_case_size():
884 # Play it safe, since this will be squared and the underlying
885 # field can have dimension 4 (quaternions) too.
889 def _denormalized_basis(cls
, n
, field
):
890 raise NotImplementedError
892 def __init__(self
, n
, field
=QQ
, normalize_basis
=True, **kwargs
):
893 S
= self
._denormalized
_basis
(n
, field
)
895 # Used in this class's fast _charpoly_coeff() override.
896 self
._basis
_normalizers
= None
898 if n
> 1 and normalize_basis
:
899 # We'll need sqrt(2) to normalize the basis, and this
900 # winds up in the multiplication table, so the whole
901 # algebra needs to be over the field extension.
902 R
= PolynomialRing(field
, 'z')
905 if p
.is_irreducible():
906 field
= NumberField(p
, 'sqrt2', embedding
=RLF(2).sqrt())
907 S
= [ s
.change_ring(field
) for s
in S
]
908 self
._basis
_normalizers
= tuple(
909 ~
(self
.natural_inner_product(s
,s
).sqrt()) for s
in S
)
910 S
= tuple( s
*c
for (s
,c
) in zip(S
,self
._basis
_normalizers
) )
912 Qs
= self
.multiplication_table_from_matrix_basis(S
)
914 fdeja
= super(MatrixEuclideanJordanAlgebra
, self
)
915 return fdeja
.__init
__(field
,
923 def _charpoly_coeff(self
, i
):
925 Override the parent method with something that tries to compute
926 over a faster (non-extension) field.
928 if self
._basis
_normalizers
is None:
929 # We didn't normalize, so assume that the basis we started
930 # with had entries in a nice field.
931 return super(MatrixEuclideanJordanAlgebra
, self
)._charpoly
_coeff
(i
)
933 # If we didn't unembed first, this number would be wrong
934 # by a power-of-two factor for complex/quaternion matrices.
935 n
= self
.real_unembed(self
.natural_basis_space().zero()).nrows()
936 field
= self
.base_ring().base_ring() # yeeeeaaaahhh
937 J
= self
.__class
__(n
, field
, False)
938 (_
,x
,_
,_
) = J
._charpoly
_matrix
_system
()
939 p
= J
._charpoly
_coeff
(i
)
940 # p might be missing some vars, have to substitute "optionally"
941 pairs
= zip(x
.base_ring().gens(), self
._basis
_normalizers
)
942 substitutions
= { v: v*c for (v,c) in pairs }
943 return p
.subs(substitutions
)
947 def multiplication_table_from_matrix_basis(basis
):
949 At least three of the five simple Euclidean Jordan algebras have the
950 symmetric multiplication (A,B) |-> (AB + BA)/2, where the
951 multiplication on the right is matrix multiplication. Given a basis
952 for the underlying matrix space, this function returns a
953 multiplication table (obtained by looping through the basis
954 elements) for an algebra of those matrices.
956 # In S^2, for example, we nominally have four coordinates even
957 # though the space is of dimension three only. The vector space V
958 # is supposed to hold the entire long vector, and the subspace W
959 # of V will be spanned by the vectors that arise from symmetric
960 # matrices. Thus for S^2, dim(V) == 4 and dim(W) == 3.
961 field
= basis
[0].base_ring()
962 dimension
= basis
[0].nrows()
964 V
= VectorSpace(field
, dimension
**2)
965 W
= V
.span_of_basis( _mat2vec(s
) for s
in basis
)
967 mult_table
= [[W
.zero() for j
in range(n
)] for i
in range(n
)]
970 mat_entry
= (basis
[i
]*basis
[j
] + basis
[j
]*basis
[i
])/2
971 mult_table
[i
][j
] = W
.coordinate_vector(_mat2vec(mat_entry
))
979 Embed the matrix ``M`` into a space of real matrices.
981 The matrix ``M`` can have entries in any field at the moment:
982 the real numbers, complex numbers, or quaternions. And although
983 they are not a field, we can probably support octonions at some
984 point, too. This function returns a real matrix that "acts like"
985 the original with respect to matrix multiplication; i.e.
987 real_embed(M*N) = real_embed(M)*real_embed(N)
990 raise NotImplementedError
996 The inverse of :meth:`real_embed`.
998 raise NotImplementedError
1002 def natural_inner_product(cls
,X
,Y
):
1003 Xu
= cls
.real_unembed(X
)
1004 Yu
= cls
.real_unembed(Y
)
1005 tr
= (Xu
*Yu
).trace()
1007 # It's real already.
1010 # Otherwise, try the thing that works for complex numbers; and
1011 # if that doesn't work, the thing that works for quaternions.
1013 return tr
.vector()[0] # real part, imag part is index 1
1014 except AttributeError:
1015 # A quaternions doesn't have a vector() method, but does
1016 # have coefficient_tuple() method that returns the
1017 # coefficients of 1, i, j, and k -- in that order.
1018 return tr
.coefficient_tuple()[0]
1021 class RealMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra
):
1025 Embed the matrix ``M`` into a space of real matrices.
1027 The matrix ``M`` can have entries in any field at the moment:
1028 the real numbers, complex numbers, or quaternions. And although
1029 they are not a field, we can probably support octonions at some
1030 point, too. This function returns a real matrix that "acts like"
1031 the original with respect to matrix multiplication; i.e.
1033 real_embed(M*N) = real_embed(M)*real_embed(N)
1040 def real_unembed(M
):
1042 The inverse of :meth:`real_embed`.
1047 class RealSymmetricEJA(RealMatrixEuclideanJordanAlgebra
):
1049 The rank-n simple EJA consisting of real symmetric n-by-n
1050 matrices, the usual symmetric Jordan product, and the trace inner
1051 product. It has dimension `(n^2 + n)/2` over the reals.
1055 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
1059 sage: J = RealSymmetricEJA(2)
1060 sage: e0, e1, e2 = J.gens()
1070 The dimension of this algebra is `(n^2 + n) / 2`::
1072 sage: set_random_seed()
1073 sage: n_max = RealSymmetricEJA._max_test_case_size()
1074 sage: n = ZZ.random_element(1, n_max)
1075 sage: J = RealSymmetricEJA(n)
1076 sage: J.dimension() == (n^2 + n)/2
1079 The Jordan multiplication is what we think it is::
1081 sage: set_random_seed()
1082 sage: J = RealSymmetricEJA.random_instance()
1083 sage: x,y = J.random_elements(2)
1084 sage: actual = (x*y).natural_representation()
1085 sage: X = x.natural_representation()
1086 sage: Y = y.natural_representation()
1087 sage: expected = (X*Y + Y*X)/2
1088 sage: actual == expected
1090 sage: J(expected) == x*y
1093 We can change the generator prefix::
1095 sage: RealSymmetricEJA(3, prefix='q').gens()
1096 (q0, q1, q2, q3, q4, q5)
1098 Our natural basis is normalized with respect to the natural inner
1099 product unless we specify otherwise::
1101 sage: set_random_seed()
1102 sage: J = RealSymmetricEJA.random_instance()
1103 sage: all( b.norm() == 1 for b in J.gens() )
1106 Since our natural basis is normalized with respect to the natural
1107 inner product, and since we know that this algebra is an EJA, any
1108 left-multiplication operator's matrix will be symmetric because
1109 natural->EJA basis representation is an isometry and within the EJA
1110 the operator is self-adjoint by the Jordan axiom::
1112 sage: set_random_seed()
1113 sage: x = RealSymmetricEJA.random_instance().random_element()
1114 sage: x.operator().matrix().is_symmetric()
1119 def _denormalized_basis(cls
, n
, field
):
1121 Return a basis for the space of real symmetric n-by-n matrices.
1125 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
1129 sage: set_random_seed()
1130 sage: n = ZZ.random_element(1,5)
1131 sage: B = RealSymmetricEJA._denormalized_basis(n,QQ)
1132 sage: all( M.is_symmetric() for M in B)
1136 # The basis of symmetric matrices, as matrices, in their R^(n-by-n)
1140 for j
in xrange(i
+1):
1141 Eij
= matrix(field
, n
, lambda k
,l
: k
==i
and l
==j
)
1145 Sij
= Eij
+ Eij
.transpose()
1151 def _max_test_case_size():
1152 return 4 # Dimension 10
1156 class ComplexMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra
):
1160 Embed the n-by-n complex matrix ``M`` into the space of real
1161 matrices of size 2n-by-2n via the map the sends each entry `z = a +
1162 bi` to the block matrix ``[[a,b],[-b,a]]``.
1166 sage: from mjo.eja.eja_algebra import \
1167 ....: ComplexMatrixEuclideanJordanAlgebra
1171 sage: F = QuadraticField(-1, 'i')
1172 sage: x1 = F(4 - 2*i)
1173 sage: x2 = F(1 + 2*i)
1176 sage: M = matrix(F,2,[[x1,x2],[x3,x4]])
1177 sage: ComplexMatrixEuclideanJordanAlgebra.real_embed(M)
1186 Embedding is a homomorphism (isomorphism, in fact)::
1188 sage: set_random_seed()
1189 sage: n_max = ComplexMatrixEuclideanJordanAlgebra._max_test_case_size()
1190 sage: n = ZZ.random_element(n_max)
1191 sage: F = QuadraticField(-1, 'i')
1192 sage: X = random_matrix(F, n)
1193 sage: Y = random_matrix(F, n)
1194 sage: Xe = ComplexMatrixEuclideanJordanAlgebra.real_embed(X)
1195 sage: Ye = ComplexMatrixEuclideanJordanAlgebra.real_embed(Y)
1196 sage: XYe = ComplexMatrixEuclideanJordanAlgebra.real_embed(X*Y)
1203 raise ValueError("the matrix 'M' must be square")
1204 field
= M
.base_ring()
1207 a
= z
.vector()[0] # real part, I guess
1208 b
= z
.vector()[1] # imag part, I guess
1209 blocks
.append(matrix(field
, 2, [[a
,b
],[-b
,a
]]))
1211 # We can drop the imaginaries here.
1212 return matrix
.block(field
.base_ring(), n
, blocks
)
1216 def real_unembed(M
):
1218 The inverse of _embed_complex_matrix().
1222 sage: from mjo.eja.eja_algebra import \
1223 ....: ComplexMatrixEuclideanJordanAlgebra
1227 sage: A = matrix(QQ,[ [ 1, 2, 3, 4],
1228 ....: [-2, 1, -4, 3],
1229 ....: [ 9, 10, 11, 12],
1230 ....: [-10, 9, -12, 11] ])
1231 sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(A)
1233 [ 10*i + 9 12*i + 11]
1237 Unembedding is the inverse of embedding::
1239 sage: set_random_seed()
1240 sage: F = QuadraticField(-1, 'i')
1241 sage: M = random_matrix(F, 3)
1242 sage: Me = ComplexMatrixEuclideanJordanAlgebra.real_embed(M)
1243 sage: ComplexMatrixEuclideanJordanAlgebra.real_unembed(Me) == M
1249 raise ValueError("the matrix 'M' must be square")
1250 if not n
.mod(2).is_zero():
1251 raise ValueError("the matrix 'M' must be a complex embedding")
1253 field
= M
.base_ring() # This should already have sqrt2
1254 R
= PolynomialRing(field
, 'z')
1256 F
= NumberField(z
**2 + 1,'i', embedding
=CLF(-1).sqrt())
1259 # Go top-left to bottom-right (reading order), converting every
1260 # 2-by-2 block we see to a single complex element.
1262 for k
in xrange(n
/2):
1263 for j
in xrange(n
/2):
1264 submat
= M
[2*k
:2*k
+2,2*j
:2*j
+2]
1265 if submat
[0,0] != submat
[1,1]:
1266 raise ValueError('bad on-diagonal submatrix')
1267 if submat
[0,1] != -submat
[1,0]:
1268 raise ValueError('bad off-diagonal submatrix')
1269 z
= submat
[0,0] + submat
[0,1]*i
1272 return matrix(F
, n
/2, elements
)
1276 def natural_inner_product(cls
,X
,Y
):
1278 Compute a natural inner product in this algebra directly from
1283 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
1287 This gives the same answer as the slow, default method implemented
1288 in :class:`MatrixEuclideanJordanAlgebra`::
1290 sage: set_random_seed()
1291 sage: J = ComplexHermitianEJA.random_instance()
1292 sage: x,y = J.random_elements(2)
1293 sage: Xe = x.natural_representation()
1294 sage: Ye = y.natural_representation()
1295 sage: X = ComplexHermitianEJA.real_unembed(Xe)
1296 sage: Y = ComplexHermitianEJA.real_unembed(Ye)
1297 sage: expected = (X*Y).trace().vector()[0]
1298 sage: actual = ComplexHermitianEJA.natural_inner_product(Xe,Ye)
1299 sage: actual == expected
1303 return RealMatrixEuclideanJordanAlgebra
.natural_inner_product(X
,Y
)/2
1306 class ComplexHermitianEJA(ComplexMatrixEuclideanJordanAlgebra
):
1308 The rank-n simple EJA consisting of complex Hermitian n-by-n
1309 matrices over the real numbers, the usual symmetric Jordan product,
1310 and the real-part-of-trace inner product. It has dimension `n^2` over
1315 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
1319 The dimension of this algebra is `n^2`::
1321 sage: set_random_seed()
1322 sage: n_max = ComplexHermitianEJA._max_test_case_size()
1323 sage: n = ZZ.random_element(1, n_max)
1324 sage: J = ComplexHermitianEJA(n)
1325 sage: J.dimension() == n^2
1328 The Jordan multiplication is what we think it is::
1330 sage: set_random_seed()
1331 sage: J = ComplexHermitianEJA.random_instance()
1332 sage: x,y = J.random_elements(2)
1333 sage: actual = (x*y).natural_representation()
1334 sage: X = x.natural_representation()
1335 sage: Y = y.natural_representation()
1336 sage: expected = (X*Y + Y*X)/2
1337 sage: actual == expected
1339 sage: J(expected) == x*y
1342 We can change the generator prefix::
1344 sage: ComplexHermitianEJA(2, prefix='z').gens()
1347 Our natural basis is normalized with respect to the natural inner
1348 product unless we specify otherwise::
1350 sage: set_random_seed()
1351 sage: J = ComplexHermitianEJA.random_instance()
1352 sage: all( b.norm() == 1 for b in J.gens() )
1355 Since our natural basis is normalized with respect to the natural
1356 inner product, and since we know that this algebra is an EJA, any
1357 left-multiplication operator's matrix will be symmetric because
1358 natural->EJA basis representation is an isometry and within the EJA
1359 the operator is self-adjoint by the Jordan axiom::
1361 sage: set_random_seed()
1362 sage: x = ComplexHermitianEJA.random_instance().random_element()
1363 sage: x.operator().matrix().is_symmetric()
1368 def _denormalized_basis(cls
, n
, field
):
1370 Returns a basis for the space of complex Hermitian n-by-n matrices.
1372 Why do we embed these? Basically, because all of numerical linear
1373 algebra assumes that you're working with vectors consisting of `n`
1374 entries from a field and scalars from the same field. There's no way
1375 to tell SageMath that (for example) the vectors contain complex
1376 numbers, while the scalar field is real.
1380 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
1384 sage: set_random_seed()
1385 sage: n = ZZ.random_element(1,5)
1386 sage: field = QuadraticField(2, 'sqrt2')
1387 sage: B = ComplexHermitianEJA._denormalized_basis(n, field)
1388 sage: all( M.is_symmetric() for M in B)
1392 R
= PolynomialRing(field
, 'z')
1394 F
= NumberField(z
**2 + 1, 'I', embedding
=CLF(-1).sqrt())
1397 # This is like the symmetric case, but we need to be careful:
1399 # * We want conjugate-symmetry, not just symmetry.
1400 # * The diagonal will (as a result) be real.
1404 for j
in xrange(i
+1):
1405 Eij
= matrix(F
, n
, lambda k
,l
: k
==i
and l
==j
)
1407 Sij
= cls
.real_embed(Eij
)
1410 # The second one has a minus because it's conjugated.
1411 Sij_real
= cls
.real_embed(Eij
+ Eij
.transpose())
1413 Sij_imag
= cls
.real_embed(I
*Eij
- I
*Eij
.transpose())
1416 # Since we embedded these, we can drop back to the "field" that we
1417 # started with instead of the complex extension "F".
1418 return tuple( s
.change_ring(field
) for s
in S
)
1422 class QuaternionMatrixEuclideanJordanAlgebra(MatrixEuclideanJordanAlgebra
):
1426 Embed the n-by-n quaternion matrix ``M`` into the space of real
1427 matrices of size 4n-by-4n by first sending each quaternion entry `z
1428 = a + bi + cj + dk` to the block-complex matrix ``[[a + bi,
1429 c+di],[-c + di, a-bi]]`, and then embedding those into a real
1434 sage: from mjo.eja.eja_algebra import \
1435 ....: QuaternionMatrixEuclideanJordanAlgebra
1439 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1440 sage: i,j,k = Q.gens()
1441 sage: x = 1 + 2*i + 3*j + 4*k
1442 sage: M = matrix(Q, 1, [[x]])
1443 sage: QuaternionMatrixEuclideanJordanAlgebra.real_embed(M)
1449 Embedding is a homomorphism (isomorphism, in fact)::
1451 sage: set_random_seed()
1452 sage: n_max = QuaternionMatrixEuclideanJordanAlgebra._max_test_case_size()
1453 sage: n = ZZ.random_element(n_max)
1454 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1455 sage: X = random_matrix(Q, n)
1456 sage: Y = random_matrix(Q, n)
1457 sage: Xe = QuaternionMatrixEuclideanJordanAlgebra.real_embed(X)
1458 sage: Ye = QuaternionMatrixEuclideanJordanAlgebra.real_embed(Y)
1459 sage: XYe = QuaternionMatrixEuclideanJordanAlgebra.real_embed(X*Y)
1464 quaternions
= M
.base_ring()
1467 raise ValueError("the matrix 'M' must be square")
1469 F
= QuadraticField(-1, 'i')
1474 t
= z
.coefficient_tuple()
1479 cplxM
= matrix(F
, 2, [[ a
+ b
*i
, c
+ d
*i
],
1480 [-c
+ d
*i
, a
- b
*i
]])
1481 realM
= ComplexMatrixEuclideanJordanAlgebra
.real_embed(cplxM
)
1482 blocks
.append(realM
)
1484 # We should have real entries by now, so use the realest field
1485 # we've got for the return value.
1486 return matrix
.block(quaternions
.base_ring(), n
, blocks
)
1491 def real_unembed(M
):
1493 The inverse of _embed_quaternion_matrix().
1497 sage: from mjo.eja.eja_algebra import \
1498 ....: QuaternionMatrixEuclideanJordanAlgebra
1502 sage: M = matrix(QQ, [[ 1, 2, 3, 4],
1503 ....: [-2, 1, -4, 3],
1504 ....: [-3, 4, 1, -2],
1505 ....: [-4, -3, 2, 1]])
1506 sage: QuaternionMatrixEuclideanJordanAlgebra.real_unembed(M)
1507 [1 + 2*i + 3*j + 4*k]
1511 Unembedding is the inverse of embedding::
1513 sage: set_random_seed()
1514 sage: Q = QuaternionAlgebra(QQ, -1, -1)
1515 sage: M = random_matrix(Q, 3)
1516 sage: Me = QuaternionMatrixEuclideanJordanAlgebra.real_embed(M)
1517 sage: QuaternionMatrixEuclideanJordanAlgebra.real_unembed(Me) == M
1523 raise ValueError("the matrix 'M' must be square")
1524 if not n
.mod(4).is_zero():
1525 raise ValueError("the matrix 'M' must be a complex embedding")
1527 # Use the base ring of the matrix to ensure that its entries can be
1528 # multiplied by elements of the quaternion algebra.
1529 field
= M
.base_ring()
1530 Q
= QuaternionAlgebra(field
,-1,-1)
1533 # Go top-left to bottom-right (reading order), converting every
1534 # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1
1537 for l
in xrange(n
/4):
1538 for m
in xrange(n
/4):
1539 submat
= ComplexMatrixEuclideanJordanAlgebra
.real_unembed(
1540 M
[4*l
:4*l
+4,4*m
:4*m
+4] )
1541 if submat
[0,0] != submat
[1,1].conjugate():
1542 raise ValueError('bad on-diagonal submatrix')
1543 if submat
[0,1] != -submat
[1,0].conjugate():
1544 raise ValueError('bad off-diagonal submatrix')
1545 z
= submat
[0,0].vector()[0] # real part
1546 z
+= submat
[0,0].vector()[1]*i
# imag part
1547 z
+= submat
[0,1].vector()[0]*j
# real part
1548 z
+= submat
[0,1].vector()[1]*k
# imag part
1551 return matrix(Q
, n
/4, elements
)
1555 def natural_inner_product(cls
,X
,Y
):
1557 Compute a natural inner product in this algebra directly from
1562 sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
1566 This gives the same answer as the slow, default method implemented
1567 in :class:`MatrixEuclideanJordanAlgebra`::
1569 sage: set_random_seed()
1570 sage: J = QuaternionHermitianEJA.random_instance()
1571 sage: x,y = J.random_elements(2)
1572 sage: Xe = x.natural_representation()
1573 sage: Ye = y.natural_representation()
1574 sage: X = QuaternionHermitianEJA.real_unembed(Xe)
1575 sage: Y = QuaternionHermitianEJA.real_unembed(Ye)
1576 sage: expected = (X*Y).trace().coefficient_tuple()[0]
1577 sage: actual = QuaternionHermitianEJA.natural_inner_product(Xe,Ye)
1578 sage: actual == expected
1582 return RealMatrixEuclideanJordanAlgebra
.natural_inner_product(X
,Y
)/4
1585 class QuaternionHermitianEJA(QuaternionMatrixEuclideanJordanAlgebra
):
1587 The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
1588 matrices, the usual symmetric Jordan product, and the
1589 real-part-of-trace inner product. It has dimension `2n^2 - n` over
1594 sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
1598 The dimension of this algebra is `2*n^2 - n`::
1600 sage: set_random_seed()
1601 sage: n_max = QuaternionHermitianEJA._max_test_case_size()
1602 sage: n = ZZ.random_element(1, n_max)
1603 sage: J = QuaternionHermitianEJA(n)
1604 sage: J.dimension() == 2*(n^2) - n
1607 The Jordan multiplication is what we think it is::
1609 sage: set_random_seed()
1610 sage: J = QuaternionHermitianEJA.random_instance()
1611 sage: x,y = J.random_elements(2)
1612 sage: actual = (x*y).natural_representation()
1613 sage: X = x.natural_representation()
1614 sage: Y = y.natural_representation()
1615 sage: expected = (X*Y + Y*X)/2
1616 sage: actual == expected
1618 sage: J(expected) == x*y
1621 We can change the generator prefix::
1623 sage: QuaternionHermitianEJA(2, prefix='a').gens()
1624 (a0, a1, a2, a3, a4, a5)
1626 Our natural basis is normalized with respect to the natural inner
1627 product unless we specify otherwise::
1629 sage: set_random_seed()
1630 sage: J = QuaternionHermitianEJA.random_instance()
1631 sage: all( b.norm() == 1 for b in J.gens() )
1634 Since our natural basis is normalized with respect to the natural
1635 inner product, and since we know that this algebra is an EJA, any
1636 left-multiplication operator's matrix will be symmetric because
1637 natural->EJA basis representation is an isometry and within the EJA
1638 the operator is self-adjoint by the Jordan axiom::
1640 sage: set_random_seed()
1641 sage: x = QuaternionHermitianEJA.random_instance().random_element()
1642 sage: x.operator().matrix().is_symmetric()
1647 def _denormalized_basis(cls
, n
, field
):
1649 Returns a basis for the space of quaternion Hermitian n-by-n matrices.
1651 Why do we embed these? Basically, because all of numerical
1652 linear algebra assumes that you're working with vectors consisting
1653 of `n` entries from a field and scalars from the same field. There's
1654 no way to tell SageMath that (for example) the vectors contain
1655 complex numbers, while the scalar field is real.
1659 sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
1663 sage: set_random_seed()
1664 sage: n = ZZ.random_element(1,5)
1665 sage: B = QuaternionHermitianEJA._denormalized_basis(n,QQ)
1666 sage: all( M.is_symmetric() for M in B )
1670 Q
= QuaternionAlgebra(QQ
,-1,-1)
1673 # This is like the symmetric case, but we need to be careful:
1675 # * We want conjugate-symmetry, not just symmetry.
1676 # * The diagonal will (as a result) be real.
1680 for j
in xrange(i
+1):
1681 Eij
= matrix(Q
, n
, lambda k
,l
: k
==i
and l
==j
)
1683 Sij
= cls
.real_embed(Eij
)
1686 # The second, third, and fourth ones have a minus
1687 # because they're conjugated.
1688 Sij_real
= cls
.real_embed(Eij
+ Eij
.transpose())
1690 Sij_I
= cls
.real_embed(I
*Eij
- I
*Eij
.transpose())
1692 Sij_J
= cls
.real_embed(J
*Eij
- J
*Eij
.transpose())
1694 Sij_K
= cls
.real_embed(K
*Eij
- K
*Eij
.transpose())
1700 class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1702 The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
1703 with the usual inner product and jordan product ``x*y =
1704 (<x_bar,y_bar>, x0*y_bar + y0*x_bar)``. It has dimension `n` over
1709 sage: from mjo.eja.eja_algebra import JordanSpinEJA
1713 This multiplication table can be verified by hand::
1715 sage: J = JordanSpinEJA(4)
1716 sage: e0,e1,e2,e3 = J.gens()
1732 We can change the generator prefix::
1734 sage: JordanSpinEJA(2, prefix='B').gens()
1738 def __init__(self
, n
, field
=QQ
, **kwargs
):
1739 V
= VectorSpace(field
, n
)
1740 mult_table
= [[V
.zero() for j
in range(n
)] for i
in range(n
)]
1750 z0
= x
.inner_product(y
)
1751 zbar
= y0
*xbar
+ x0
*ybar
1752 z
= V([z0
] + zbar
.list())
1753 mult_table
[i
][j
] = z
1755 # The rank of the spin algebra is two, unless we're in a
1756 # one-dimensional ambient space (because the rank is bounded by
1757 # the ambient dimension).
1758 fdeja
= super(JordanSpinEJA
, self
)
1759 return fdeja
.__init
__(field
, mult_table
, rank
=min(n
,2), **kwargs
)
1761 def inner_product(self
, x
, y
):
1763 Faster to reimplement than to use natural representations.
1767 sage: from mjo.eja.eja_algebra import JordanSpinEJA
1771 Ensure that this is the usual inner product for the algebras
1774 sage: set_random_seed()
1775 sage: J = JordanSpinEJA.random_instance()
1776 sage: x,y = J.random_elements(2)
1777 sage: X = x.natural_representation()
1778 sage: Y = y.natural_representation()
1779 sage: x.inner_product(y) == J.natural_inner_product(X,Y)
1783 return x
.to_vector().inner_product(y
.to_vector())