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
.categories
.finite_dimensional_algebras_with_basis
import FiniteDimensionalAlgebrasWithBasis
9 from sage
.categories
.morphism
import SetMorphism
10 from sage
.structure
.element
import is_Matrix
11 from sage
.structure
.category_object
import normalize_names
13 from sage
.algebras
.finite_dimensional_algebras
.finite_dimensional_algebra
import FiniteDimensionalAlgebra
14 from sage
.algebras
.finite_dimensional_algebras
.finite_dimensional_algebra_element
import FiniteDimensionalAlgebraElement
15 from sage
.algebras
.finite_dimensional_algebras
.finite_dimensional_algebra_morphism
import FiniteDimensionalAlgebraMorphism
, FiniteDimensionalAlgebraHomset
18 class FiniteDimensionalEuclideanJordanAlgebraHomset(FiniteDimensionalAlgebraHomset
):
20 def has_coerce_map_from(self
, S
):
24 sage: J = RealSymmetricEJA(2)
26 sage: H.has_coerce_map_from(QQ)
31 # The Homset classes override has_coerce_map_from() with
32 # something that crashes when it's given e.g. QQ.
33 if S
.is_subring(self
.codomain().base_ring()):
36 pclass
= super(FiniteDimensionalEuclideanJordanAlgebraHomset
, self
)
37 return pclass
.has_coerce_map_from(S
)
40 def _coerce_map_from_(self
, S
):
44 sage: J = RealSymmetricEJA(2)
47 Morphism from Euclidean Jordan algebra of degree 3 over Rational
48 Field to Euclidean Jordan algebra of degree 3 over Rational Field
58 h
= S
.hom(self
.codomain())
59 return SetMorphism(Hom(S
,C
), lambda x
: h(x
).operator())
62 def __call__(self
, x
):
66 sage: J = RealSymmetricEJA(2)
69 Morphism from Euclidean Jordan algebra of degree 3 over Rational
70 Field to Euclidean Jordan algebra of degree 3 over Rational Field
77 if x
in self
.base_ring():
78 cols
= self
.domain().dimension()
79 rows
= self
.codomain().dimension()
80 x
= x
*identity_matrix(self
.codomain().base_ring(), rows
, cols
)
81 return FiniteDimensionalEuclideanJordanAlgebraMorphism(self
, x
)
86 Return the identity morphism, but as a member of the right
87 space (so that we can add it, multiply it, etc.)
89 cols
= self
.domain().dimension()
90 rows
= self
.codomain().dimension()
91 mat
= identity_matrix(self
.base_ring(), rows
, cols
)
92 return FiniteDimensionalEuclideanJordanAlgebraMorphism(self
, mat
)
96 class FiniteDimensionalEuclideanJordanAlgebraMorphism(FiniteDimensionalAlgebraMorphism
):
98 A linear map between two finite-dimensional EJAs.
100 This is a very thin wrapper around FiniteDimensionalAlgebraMorphism
101 that does only a few things:
103 1. Avoids the ``unitary`` and ``check`` arguments to the constructor
104 that will always be ``False``. This is necessary because these
105 are homomorphisms with respect to ADDITION, but the SageMath
106 machinery wants to check that they're homomorphisms with respect
107 to (Jordan) MULTIPLICATION. That obviously doesn't work.
109 2. Inputs and outputs the underlying matrix with respect to COLUMN
110 vectors, unlike the parent class.
112 3. Allows us to add, subtract, negate, multiply (compose), and
113 invert morphisms in the obvious way.
115 If this seems a bit heavyweight, it is. I would have been happy to
116 use a the ring morphism that underlies the finite-dimensional
117 algebra morphism, but they don't seem to be callable on elements of
118 our EJA, and you can't add/multiply/etc. them.
120 def _add_(self
, other
):
122 Add two EJA morphisms in the obvious way.
126 sage: J = RealSymmetricEJA(3)
129 sage: x.operator() + y.operator()
130 Morphism from Euclidean Jordan algebra of degree 6 over Rational
131 Field to Euclidean Jordan algebra of degree 6 over Rational Field
142 sage: set_random_seed()
143 sage: J = random_eja()
144 sage: x = J.random_element()
145 sage: y = J.random_element()
146 sage: (x.operator() + y.operator()) in J.Hom(J)
152 raise ValueError("summands must live in the same space")
154 return FiniteDimensionalEuclideanJordanAlgebraMorphism(
156 self
.matrix() + other
.matrix() )
159 def __init__(self
, parent
, f
):
160 FiniteDimensionalAlgebraMorphism
.__init
__(self
,
167 def __invert__(self
):
171 sage: J = RealSymmetricEJA(2)
172 sage: x = J.linear_combination(zip(range(len(J.gens())), J.gens()))
173 sage: x.is_invertible()
176 Morphism from Euclidean Jordan algebra of degree 3 over Rational
177 Field to Euclidean Jordan algebra of degree 3 over Rational Field
182 sage: x.operator_matrix().inverse()
189 Beware, ``x`` being invertible isn't sufficient for its operator
190 to be invertible below::
192 sage: set_random_seed()
193 sage: J = random_eja()
194 sage: x = J.random_element()
195 sage: id = J.Hom(J).one()
196 sage: not x.operator_matrix().is_invertible() or (
197 ....: ~x.operator()*x.operator() == id )
202 if not A
.is_invertible():
203 raise ValueError("morphism is not invertible")
206 return FiniteDimensionalEuclideanJordanAlgebraMorphism(self
.parent(),
209 def _lmul_(self
, right
):
211 Compose two EJA morphisms using multiplicative notation.
215 sage: J = RealSymmetricEJA(2)
218 sage: x.operator() * y.operator()
219 Morphism from Euclidean Jordan algebra of degree 3 over Rational
220 Field to Euclidean Jordan algebra of degree 3 over Rational Field
228 sage: J = RealSymmetricEJA(2)
229 sage: x = J.linear_combination(zip(range(len(J.gens())), J.gens()))
231 Morphism from Euclidean Jordan algebra of degree 3 over Rational
232 Field to Euclidean Jordan algebra of degree 3 over Rational Field
238 Morphism from Euclidean Jordan algebra of degree 3 over Rational
239 Field to Euclidean Jordan algebra of degree 3 over Rational Field
245 Morphism from Euclidean Jordan algebra of degree 3 over Rational
246 Field to Euclidean Jordan algebra of degree 3 over Rational Field
254 sage: set_random_seed()
255 sage: J = random_eja()
256 sage: x = J.random_element()
257 sage: y = J.random_element()
258 sage: (x.operator() * y.operator()) in J.Hom(J)
263 # I think the morphism classes break the coercion framework
264 # somewhere along the way, so we have to do this ourselves.
265 right
= self
.parent().coerce(right
)
269 if not right
.codomain() is self
.domain():
270 raise ValueError("(co)domains must agree for composition")
272 return FiniteDimensionalEuclideanJordanAlgebraMorphism(
274 self
.matrix()*right
.matrix() )
279 def __pow__(self
, n
):
284 sage: J = JordanSpinEJA(4)
285 sage: e0,e1,e2,e3 = J.gens()
286 sage: x = -5/2*e0 + 1/2*e2 + 20*e3
287 sage: Qx = x.quadratic_representation()
289 Morphism from Euclidean Jordan algebra of degree 4 over Rational
290 Field to Euclidean Jordan algebra of degree 4 over Rational Field
296 sage: (x^0).quadratic_representation() == Qx^0
301 # We get back the stupid identity morphism which doesn't
302 # live in the right space.
303 return self
.parent().one()
307 return FiniteDimensionalAlgebraMorphism
.__pow
__(self
,n
)
312 Negate this morphism.
316 sage: J = RealSymmetricEJA(2)
319 Morphism from Euclidean Jordan algebra of degree 3 over Rational
320 Field to Euclidean Jordan algebra of degree 3 over Rational Field
328 sage: set_random_seed()
329 sage: J = random_eja()
330 sage: x = J.random_element()
331 sage: -x.operator() in J.Hom(J)
335 return FiniteDimensionalEuclideanJordanAlgebraMorphism(
342 We override only the representation that is shown to the user,
343 because we want the matrix to be with respect to COLUMN vectors.
347 Ensure that we see the transpose of the underlying matrix object:
349 sage: J = RealSymmetricEJA(3)
350 sage: x = J.linear_combination(zip(range(len(J.gens())), J.gens()))
351 sage: L = x.operator()
353 Morphism from Euclidean Jordan algebra of degree 6 over Rational
354 Field to Euclidean Jordan algebra of degree 6 over Rational Field
371 return "Morphism from {} to {} given by matrix\n{}".format(
372 self
.domain(), self
.codomain(), self
.matrix())
375 def __sub__(self
, other
):
377 Subtract one morphism from another using addition and negation.
381 sage: J = RealSymmetricEJA(2)
382 sage: L1 = J.one().operator()
384 Morphism from Euclidean Jordan algebra of degree 3 over Rational
385 Field to Euclidean Jordan algebra of degree 3 over Rational
386 Field given by matrix
393 sage: set_random_seed()
394 sage: J = random_eja()
395 sage: x = J.random_element()
396 sage: y = J.random_element()
397 sage: x.operator() - y.operator() in J.Hom(J)
401 return self
+ (-other
)
406 Return the matrix of this morphism with respect to a left-action
409 return FiniteDimensionalAlgebraMorphism
.matrix(self
).transpose()
412 class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra
):
414 def __classcall_private__(cls
,
418 assume_associative
=False,
423 mult_table
= [b
.base_extend(field
) for b
in mult_table
]
426 if not (is_Matrix(b
) and b
.dimensions() == (n
, n
)):
427 raise ValueError("input is not a multiplication table")
428 mult_table
= tuple(mult_table
)
430 cat
= FiniteDimensionalAlgebrasWithBasis(field
)
431 cat
.or_subcategory(category
)
432 if assume_associative
:
433 cat
= cat
.Associative()
435 names
= normalize_names(n
, names
)
437 fda
= super(FiniteDimensionalEuclideanJordanAlgebra
, cls
)
438 return fda
.__classcall
__(cls
,
441 assume_associative
=assume_associative
,
445 natural_basis
=natural_basis
)
448 def _Hom_(self
, B
, cat
):
450 Construct a homset of ``self`` and ``B``.
452 return FiniteDimensionalEuclideanJordanAlgebraHomset(self
,
461 assume_associative
=False,
468 By definition, Jordan multiplication commutes::
470 sage: set_random_seed()
471 sage: J = random_eja()
472 sage: x = J.random_element()
473 sage: y = J.random_element()
479 self
._natural
_basis
= natural_basis
480 self
._multiplication
_table
= mult_table
481 fda
= super(FiniteDimensionalEuclideanJordanAlgebra
, self
)
490 Return a string representation of ``self``.
492 fmt
= "Euclidean Jordan algebra of degree {} over {}"
493 return fmt
.format(self
.degree(), self
.base_ring())
496 def _a_regular_element(self
):
498 Guess a regular element. Needed to compute the basis for our
499 characteristic polynomial coefficients.
502 z
= self
.sum( (i
+1)*gs
[i
] for i
in range(len(gs
)) )
503 if not z
.is_regular():
504 raise ValueError("don't know a regular element")
509 def _charpoly_basis_space(self
):
511 Return the vector space spanned by the basis used in our
512 characteristic polynomial coefficients. This is used not only to
513 compute those coefficients, but also any time we need to
514 evaluate the coefficients (like when we compute the trace or
517 z
= self
._a
_regular
_element
()
518 V
= self
.vector_space()
519 V1
= V
.span_of_basis( (z
**k
).vector() for k
in range(self
.rank()) )
520 b
= (V1
.basis() + V1
.complement().basis())
521 return V
.span_of_basis(b
)
525 def _charpoly_coeff(self
, i
):
527 Return the coefficient polynomial "a_{i}" of this algebra's
528 general characteristic polynomial.
530 Having this be a separate cached method lets us compute and
531 store the trace/determinant (a_{r-1} and a_{0} respectively)
532 separate from the entire characteristic polynomial.
534 (A_of_x
, x
, xr
, detA
) = self
._charpoly
_matrix
_system
()
535 R
= A_of_x
.base_ring()
537 # Guaranteed by theory
540 # Danger: the in-place modification is done for performance
541 # reasons (reconstructing a matrix with huge polynomial
542 # entries is slow), but I don't know how cached_method works,
543 # so it's highly possible that we're modifying some global
544 # list variable by reference, here. In other words, you
545 # probably shouldn't call this method twice on the same
546 # algebra, at the same time, in two threads
547 Ai_orig
= A_of_x
.column(i
)
548 A_of_x
.set_column(i
,xr
)
549 numerator
= A_of_x
.det()
550 A_of_x
.set_column(i
,Ai_orig
)
552 # We're relying on the theory here to ensure that each a_i is
553 # indeed back in R, and the added negative signs are to make
554 # the whole charpoly expression sum to zero.
555 return R(-numerator
/detA
)
559 def _charpoly_matrix_system(self
):
561 Compute the matrix whose entries A_ij are polynomials in
562 X1,...,XN, the vector ``x`` of variables X1,...,XN, the vector
563 corresponding to `x^r` and the determinent of the matrix A =
564 [A_ij]. In other words, all of the fixed (cachable) data needed
565 to compute the coefficients of the characteristic polynomial.
570 # Construct a new algebra over a multivariate polynomial ring...
571 names
= ['X' + str(i
) for i
in range(1,n
+1)]
572 R
= PolynomialRing(self
.base_ring(), names
)
573 J
= FiniteDimensionalEuclideanJordanAlgebra(R
,
574 self
._multiplication
_table
,
577 idmat
= identity_matrix(J
.base_ring(), n
)
579 W
= self
._charpoly
_basis
_space
()
580 W
= W
.change_ring(R
.fraction_field())
582 # Starting with the standard coordinates x = (X1,X2,...,Xn)
583 # and then converting the entries to W-coordinates allows us
584 # to pass in the standard coordinates to the charpoly and get
585 # back the right answer. Specifically, with x = (X1,X2,...,Xn),
588 # W.coordinates(x^2) eval'd at (standard z-coords)
592 # W-coords of (standard coords of x^2 eval'd at std-coords of z)
594 # We want the middle equivalent thing in our matrix, but use
595 # the first equivalent thing instead so that we can pass in
596 # standard coordinates.
597 x
= J(vector(R
, R
.gens()))
598 l1
= [column_matrix(W
.coordinates((x
**k
).vector())) for k
in range(r
)]
599 l2
= [idmat
.column(k
-1).column() for k
in range(r
+1, n
+1)]
600 A_of_x
= block_matrix(R
, 1, n
, (l1
+ l2
))
601 xr
= W
.coordinates((x
**r
).vector())
602 return (A_of_x
, x
, xr
, A_of_x
.det())
606 def characteristic_polynomial(self
):
611 This implementation doesn't guarantee that the polynomial
612 denominator in the coefficients is not identically zero, so
613 theoretically it could crash. The way that this is handled
614 in e.g. Faraut and Koranyi is to use a basis that guarantees
615 the denominator is non-zero. But, doing so requires knowledge
616 of at least one regular element, and we don't even know how
617 to do that. The trade-off is that, if we use the standard basis,
618 the resulting polynomial will accept the "usual" coordinates. In
619 other words, we don't have to do a change of basis before e.g.
620 computing the trace or determinant.
624 The characteristic polynomial in the spin algebra is given in
625 Alizadeh, Example 11.11::
627 sage: J = JordanSpinEJA(3)
628 sage: p = J.characteristic_polynomial(); p
629 X1^2 - X2^2 - X3^2 + (-2*t)*X1 + t^2
630 sage: xvec = J.one().vector()
638 # The list of coefficient polynomials a_1, a_2, ..., a_n.
639 a
= [ self
._charpoly
_coeff
(i
) for i
in range(n
) ]
641 # We go to a bit of trouble here to reorder the
642 # indeterminates, so that it's easier to evaluate the
643 # characteristic polynomial at x's coordinates and get back
644 # something in terms of t, which is what we want.
646 S
= PolynomialRing(self
.base_ring(),'t')
648 S
= PolynomialRing(S
, R
.variable_names())
651 # Note: all entries past the rth should be zero. The
652 # coefficient of the highest power (x^r) is 1, but it doesn't
653 # appear in the solution vector which contains coefficients
654 # for the other powers (to make them sum to x^r).
656 a
[r
] = 1 # corresponds to x^r
658 # When the rank is equal to the dimension, trying to
659 # assign a[r] goes out-of-bounds.
660 a
.append(1) # corresponds to x^r
662 return sum( a
[k
]*(t
**k
) for k
in range(len(a
)) )
665 def inner_product(self
, x
, y
):
667 The inner product associated with this Euclidean Jordan algebra.
669 Defaults to the trace inner product, but can be overridden by
670 subclasses if they are sure that the necessary properties are
675 The inner product must satisfy its axiom for this algebra to truly
676 be a Euclidean Jordan Algebra::
678 sage: set_random_seed()
679 sage: J = random_eja()
680 sage: x = J.random_element()
681 sage: y = J.random_element()
682 sage: z = J.random_element()
683 sage: (x*y).inner_product(z) == y.inner_product(x*z)
687 if (not x
in self
) or (not y
in self
):
688 raise TypeError("arguments must live in this algebra")
689 return x
.trace_inner_product(y
)
692 def natural_basis(self
):
694 Return a more-natural representation of this algebra's basis.
696 Every finite-dimensional Euclidean Jordan Algebra is a direct
697 sum of five simple algebras, four of which comprise Hermitian
698 matrices. This method returns the original "natural" basis
699 for our underlying vector space. (Typically, the natural basis
700 is used to construct the multiplication table in the first place.)
702 Note that this will always return a matrix. The standard basis
703 in `R^n` will be returned as `n`-by-`1` column matrices.
707 sage: J = RealSymmetricEJA(2)
710 sage: J.natural_basis()
718 sage: J = JordanSpinEJA(2)
721 sage: J.natural_basis()
728 if self
._natural
_basis
is None:
729 return tuple( b
.vector().column() for b
in self
.basis() )
731 return self
._natural
_basis
736 Return the rank of this EJA.
738 if self
._rank
is None:
739 raise ValueError("no rank specified at genesis")
743 def vector_space(self
):
745 Return the vector space that underlies this algebra.
749 sage: J = RealSymmetricEJA(2)
750 sage: J.vector_space()
751 Vector space of dimension 3 over Rational Field
754 return self
.zero().vector().parent().ambient_vector_space()
757 class Element(FiniteDimensionalAlgebraElement
):
759 An element of a Euclidean Jordan algebra.
764 Oh man, I should not be doing this. This hides the "disabled"
765 methods ``left_matrix`` and ``matrix`` from introspection;
766 in particular it removes them from tab-completion.
768 return filter(lambda s
: s
not in ['left_matrix', 'matrix'],
769 dir(self
.__class
__) )
772 def __init__(self
, A
, elt
=None):
776 The identity in `S^n` is converted to the identity in the EJA::
778 sage: J = RealSymmetricEJA(3)
779 sage: I = identity_matrix(QQ,3)
780 sage: J(I) == J.one()
783 This skew-symmetric matrix can't be represented in the EJA::
785 sage: J = RealSymmetricEJA(3)
786 sage: A = matrix(QQ,3, lambda i,j: i-j)
788 Traceback (most recent call last):
790 ArithmeticError: vector is not in free module
793 # Goal: if we're given a matrix, and if it lives in our
794 # parent algebra's "natural ambient space," convert it
795 # into an algebra element.
797 # The catch is, we make a recursive call after converting
798 # the given matrix into a vector that lives in the algebra.
799 # This we need to try the parent class initializer first,
800 # to avoid recursing forever if we're given something that
801 # already fits into the algebra, but also happens to live
802 # in the parent's "natural ambient space" (this happens with
805 FiniteDimensionalAlgebraElement
.__init
__(self
, A
, elt
)
807 natural_basis
= A
.natural_basis()
808 if elt
in natural_basis
[0].matrix_space():
809 # Thanks for nothing! Matrix spaces aren't vector
810 # spaces in Sage, so we have to figure out its
811 # natural-basis coordinates ourselves.
812 V
= VectorSpace(elt
.base_ring(), elt
.nrows()**2)
813 W
= V
.span( _mat2vec(s
) for s
in natural_basis
)
814 coords
= W
.coordinates(_mat2vec(elt
))
815 FiniteDimensionalAlgebraElement
.__init
__(self
, A
, coords
)
817 def __pow__(self
, n
):
819 Return ``self`` raised to the power ``n``.
821 Jordan algebras are always power-associative; see for
822 example Faraut and Koranyi, Proposition II.1.2 (ii).
826 We have to override this because our superclass uses row vectors
827 instead of column vectors! We, on the other hand, assume column
832 sage: set_random_seed()
833 sage: x = random_eja().random_element()
834 sage: x.operator_matrix()*x.vector() == (x^2).vector()
837 A few examples of power-associativity::
839 sage: set_random_seed()
840 sage: x = random_eja().random_element()
841 sage: x*(x*x)*(x*x) == x^5
843 sage: (x*x)*(x*x*x) == x^5
846 We also know that powers operator-commute (Koecher, Chapter
849 sage: set_random_seed()
850 sage: x = random_eja().random_element()
851 sage: m = ZZ.random_element(0,10)
852 sage: n = ZZ.random_element(0,10)
853 sage: Lxm = (x^m).operator_matrix()
854 sage: Lxn = (x^n).operator_matrix()
855 sage: Lxm*Lxn == Lxn*Lxm
865 return A( (self
.operator_matrix()**(n
-1))*self
.vector() )
868 def apply_univariate_polynomial(self
, p
):
870 Apply the univariate polynomial ``p`` to this element.
872 A priori, SageMath won't allow us to apply a univariate
873 polynomial to an element of an EJA, because we don't know
874 that EJAs are rings (they are usually not associative). Of
875 course, we know that EJAs are power-associative, so the
876 operation is ultimately kosher. This function sidesteps
877 the CAS to get the answer we want and expect.
881 sage: R = PolynomialRing(QQ, 't')
883 sage: p = t^4 - t^3 + 5*t - 2
884 sage: J = RealCartesianProductEJA(5)
885 sage: J.one().apply_univariate_polynomial(p) == 3*J.one()
890 We should always get back an element of the algebra::
892 sage: set_random_seed()
893 sage: p = PolynomialRing(QQ, 't').random_element()
894 sage: J = random_eja()
895 sage: x = J.random_element()
896 sage: x.apply_univariate_polynomial(p) in J
900 if len(p
.variables()) > 1:
901 raise ValueError("not a univariate polynomial")
904 # Convert the coeficcients to the parent's base ring,
905 # because a priori they might live in an (unnecessarily)
906 # larger ring for which P.sum() would fail below.
907 cs
= [ R(c
) for c
in p
.coefficients(sparse
=False) ]
908 return P
.sum( cs
[k
]*(self
**k
) for k
in range(len(cs
)) )
911 def characteristic_polynomial(self
):
913 Return the characteristic polynomial of this element.
917 The rank of `R^3` is three, and the minimal polynomial of
918 the identity element is `(t-1)` from which it follows that
919 the characteristic polynomial should be `(t-1)^3`::
921 sage: J = RealCartesianProductEJA(3)
922 sage: J.one().characteristic_polynomial()
923 t^3 - 3*t^2 + 3*t - 1
925 Likewise, the characteristic of the zero element in the
926 rank-three algebra `R^{n}` should be `t^{3}`::
928 sage: J = RealCartesianProductEJA(3)
929 sage: J.zero().characteristic_polynomial()
932 The characteristic polynomial of an element should evaluate
933 to zero on that element::
935 sage: set_random_seed()
936 sage: x = RealCartesianProductEJA(3).random_element()
937 sage: p = x.characteristic_polynomial()
938 sage: x.apply_univariate_polynomial(p)
942 p
= self
.parent().characteristic_polynomial()
943 return p(*self
.vector())
946 def inner_product(self
, other
):
948 Return the parent algebra's inner product of myself and ``other``.
952 The inner product in the Jordan spin algebra is the usual
953 inner product on `R^n` (this example only works because the
954 basis for the Jordan algebra is the standard basis in `R^n`)::
956 sage: J = JordanSpinEJA(3)
957 sage: x = vector(QQ,[1,2,3])
958 sage: y = vector(QQ,[4,5,6])
959 sage: x.inner_product(y)
961 sage: J(x).inner_product(J(y))
964 The inner product on `S^n` is `<X,Y> = trace(X*Y)`, where
965 multiplication is the usual matrix multiplication in `S^n`,
966 so the inner product of the identity matrix with itself
969 sage: J = RealSymmetricEJA(3)
970 sage: J.one().inner_product(J.one())
973 Likewise, the inner product on `C^n` is `<X,Y> =
974 Re(trace(X*Y))`, where we must necessarily take the real
975 part because the product of Hermitian matrices may not be
978 sage: J = ComplexHermitianEJA(3)
979 sage: J.one().inner_product(J.one())
982 Ditto for the quaternions::
984 sage: J = QuaternionHermitianEJA(3)
985 sage: J.one().inner_product(J.one())
990 Ensure that we can always compute an inner product, and that
991 it gives us back a real number::
993 sage: set_random_seed()
994 sage: J = random_eja()
995 sage: x = J.random_element()
996 sage: y = J.random_element()
997 sage: x.inner_product(y) in RR
1003 raise TypeError("'other' must live in the same algebra")
1005 return P
.inner_product(self
, other
)
1008 def operator_commutes_with(self
, other
):
1010 Return whether or not this element operator-commutes
1015 The definition of a Jordan algebra says that any element
1016 operator-commutes with its square::
1018 sage: set_random_seed()
1019 sage: x = random_eja().random_element()
1020 sage: x.operator_commutes_with(x^2)
1025 Test Lemma 1 from Chapter III of Koecher::
1027 sage: set_random_seed()
1028 sage: J = random_eja()
1029 sage: u = J.random_element()
1030 sage: v = J.random_element()
1031 sage: lhs = u.operator_commutes_with(u*v)
1032 sage: rhs = v.operator_commutes_with(u^2)
1037 if not other
in self
.parent():
1038 raise TypeError("'other' must live in the same algebra")
1040 A
= self
.operator_matrix()
1041 B
= other
.operator_matrix()
1047 Return my determinant, the product of my eigenvalues.
1051 sage: J = JordanSpinEJA(2)
1052 sage: e0,e1 = J.gens()
1053 sage: x = sum( J.gens() )
1059 sage: J = JordanSpinEJA(3)
1060 sage: e0,e1,e2 = J.gens()
1061 sage: x = sum( J.gens() )
1067 An element is invertible if and only if its determinant is
1070 sage: set_random_seed()
1071 sage: x = random_eja().random_element()
1072 sage: x.is_invertible() == (x.det() != 0)
1078 p
= P
._charpoly
_coeff
(0)
1079 # The _charpoly_coeff function already adds the factor of
1080 # -1 to ensure that _charpoly_coeff(0) is really what
1081 # appears in front of t^{0} in the charpoly. However,
1082 # we want (-1)^r times THAT for the determinant.
1083 return ((-1)**r
)*p(*self
.vector())
1088 Return the Jordan-multiplicative inverse of this element.
1092 We appeal to the quadratic representation as in Koecher's
1093 Theorem 12 in Chapter III, Section 5.
1097 The inverse in the spin factor algebra is given in Alizadeh's
1100 sage: set_random_seed()
1101 sage: n = ZZ.random_element(1,10)
1102 sage: J = JordanSpinEJA(n)
1103 sage: x = J.random_element()
1104 sage: while not x.is_invertible():
1105 ....: x = J.random_element()
1106 sage: x_vec = x.vector()
1108 sage: x_bar = x_vec[1:]
1109 sage: coeff = ~(x0^2 - x_bar.inner_product(x_bar))
1110 sage: inv_vec = x_vec.parent()([x0] + (-x_bar).list())
1111 sage: x_inverse = coeff*inv_vec
1112 sage: x.inverse() == J(x_inverse)
1117 The identity element is its own inverse::
1119 sage: set_random_seed()
1120 sage: J = random_eja()
1121 sage: J.one().inverse() == J.one()
1124 If an element has an inverse, it acts like one::
1126 sage: set_random_seed()
1127 sage: J = random_eja()
1128 sage: x = J.random_element()
1129 sage: (not x.is_invertible()) or (x.inverse()*x == J.one())
1132 The inverse of the inverse is what we started with::
1134 sage: set_random_seed()
1135 sage: J = random_eja()
1136 sage: x = J.random_element()
1137 sage: (not x.is_invertible()) or (x.inverse().inverse() == x)
1140 The zero element is never invertible::
1142 sage: set_random_seed()
1143 sage: J = random_eja().zero().inverse()
1144 Traceback (most recent call last):
1146 ValueError: element is not invertible
1149 if not self
.is_invertible():
1150 raise ValueError("element is not invertible")
1152 return (~self
.quadratic_representation())(self
)
1155 def is_invertible(self
):
1157 Return whether or not this element is invertible.
1159 We can't use the superclass method because it relies on
1160 the algebra being associative.
1164 The usual way to do this is to check if the determinant is
1165 zero, but we need the characteristic polynomial for the
1166 determinant. The minimal polynomial is a lot easier to get,
1167 so we use Corollary 2 in Chapter V of Koecher to check
1168 whether or not the paren't algebra's zero element is a root
1169 of this element's minimal polynomial.
1173 The identity element is always invertible::
1175 sage: set_random_seed()
1176 sage: J = random_eja()
1177 sage: J.one().is_invertible()
1180 The zero element is never invertible::
1182 sage: set_random_seed()
1183 sage: J = random_eja()
1184 sage: J.zero().is_invertible()
1188 zero
= self
.parent().zero()
1189 p
= self
.minimal_polynomial()
1190 return not (p(zero
) == zero
)
1193 def is_nilpotent(self
):
1195 Return whether or not some power of this element is zero.
1197 The superclass method won't work unless we're in an
1198 associative algebra, and we aren't. However, we generate
1199 an assocoative subalgebra and we're nilpotent there if and
1200 only if we're nilpotent here (probably).
1204 The identity element is never nilpotent::
1206 sage: set_random_seed()
1207 sage: random_eja().one().is_nilpotent()
1210 The additive identity is always nilpotent::
1212 sage: set_random_seed()
1213 sage: random_eja().zero().is_nilpotent()
1217 # The element we're going to call "is_nilpotent()" on.
1218 # Either myself, interpreted as an element of a finite-
1219 # dimensional algebra, or an element of an associative
1223 if self
.parent().is_associative():
1224 elt
= FiniteDimensionalAlgebraElement(self
.parent(), self
)
1226 V
= self
.span_of_powers()
1227 assoc_subalg
= self
.subalgebra_generated_by()
1228 # Mis-design warning: the basis used for span_of_powers()
1229 # and subalgebra_generated_by() must be the same, and in
1231 elt
= assoc_subalg(V
.coordinates(self
.vector()))
1233 # Recursive call, but should work since elt lives in an
1234 # associative algebra.
1235 return elt
.is_nilpotent()
1238 def is_regular(self
):
1240 Return whether or not this is a regular element.
1244 The identity element always has degree one, but any element
1245 linearly-independent from it is regular::
1247 sage: J = JordanSpinEJA(5)
1248 sage: J.one().is_regular()
1250 sage: e0, e1, e2, e3, e4 = J.gens() # e0 is the identity
1251 sage: for x in J.gens():
1252 ....: (J.one() + x).is_regular()
1260 return self
.degree() == self
.parent().rank()
1265 Compute the degree of this element the straightforward way
1266 according to the definition; by appending powers to a list
1267 and figuring out its dimension (that is, whether or not
1268 they're linearly dependent).
1272 sage: J = JordanSpinEJA(4)
1273 sage: J.one().degree()
1275 sage: e0,e1,e2,e3 = J.gens()
1276 sage: (e0 - e1).degree()
1279 In the spin factor algebra (of rank two), all elements that
1280 aren't multiples of the identity are regular::
1282 sage: set_random_seed()
1283 sage: n = ZZ.random_element(1,10)
1284 sage: J = JordanSpinEJA(n)
1285 sage: x = J.random_element()
1286 sage: x == x.coefficient(0)*J.one() or x.degree() == 2
1290 return self
.span_of_powers().dimension()
1293 def left_matrix(self
):
1295 Our parent class defines ``left_matrix`` and ``matrix``
1296 methods whose names are misleading. We don't want them.
1298 raise NotImplementedError("use operator_matrix() instead")
1300 matrix
= left_matrix
1303 def minimal_polynomial(self
):
1305 Return the minimal polynomial of this element,
1306 as a function of the variable `t`.
1310 We restrict ourselves to the associative subalgebra
1311 generated by this element, and then return the minimal
1312 polynomial of this element's operator matrix (in that
1313 subalgebra). This works by Baes Proposition 2.3.16.
1317 The minimal polynomial of the identity and zero elements are
1320 sage: set_random_seed()
1321 sage: J = random_eja()
1322 sage: J.one().minimal_polynomial()
1324 sage: J.zero().minimal_polynomial()
1327 The degree of an element is (by one definition) the degree
1328 of its minimal polynomial::
1330 sage: set_random_seed()
1331 sage: x = random_eja().random_element()
1332 sage: x.degree() == x.minimal_polynomial().degree()
1335 The minimal polynomial and the characteristic polynomial coincide
1336 and are known (see Alizadeh, Example 11.11) for all elements of
1337 the spin factor algebra that aren't scalar multiples of the
1340 sage: set_random_seed()
1341 sage: n = ZZ.random_element(2,10)
1342 sage: J = JordanSpinEJA(n)
1343 sage: y = J.random_element()
1344 sage: while y == y.coefficient(0)*J.one():
1345 ....: y = J.random_element()
1346 sage: y0 = y.vector()[0]
1347 sage: y_bar = y.vector()[1:]
1348 sage: actual = y.minimal_polynomial()
1349 sage: t = PolynomialRing(J.base_ring(),'t').gen(0)
1350 sage: expected = t^2 - 2*y0*t + (y0^2 - norm(y_bar)^2)
1351 sage: bool(actual == expected)
1354 The minimal polynomial should always kill its element::
1356 sage: set_random_seed()
1357 sage: x = random_eja().random_element()
1358 sage: p = x.minimal_polynomial()
1359 sage: x.apply_univariate_polynomial(p)
1363 V
= self
.span_of_powers()
1364 assoc_subalg
= self
.subalgebra_generated_by()
1365 # Mis-design warning: the basis used for span_of_powers()
1366 # and subalgebra_generated_by() must be the same, and in
1368 elt
= assoc_subalg(V
.coordinates(self
.vector()))
1370 # We get back a symbolic polynomial in 'x' but want a real
1371 # polynomial in 't'.
1372 p_of_x
= elt
.operator_matrix().minimal_polynomial()
1373 return p_of_x
.change_variable_name('t')
1376 def natural_representation(self
):
1378 Return a more-natural representation of this element.
1380 Every finite-dimensional Euclidean Jordan Algebra is a
1381 direct sum of five simple algebras, four of which comprise
1382 Hermitian matrices. This method returns the original
1383 "natural" representation of this element as a Hermitian
1384 matrix, if it has one. If not, you get the usual representation.
1388 sage: J = ComplexHermitianEJA(3)
1391 sage: J.one().natural_representation()
1401 sage: J = QuaternionHermitianEJA(3)
1404 sage: J.one().natural_representation()
1405 [1 0 0 0 0 0 0 0 0 0 0 0]
1406 [0 1 0 0 0 0 0 0 0 0 0 0]
1407 [0 0 1 0 0 0 0 0 0 0 0 0]
1408 [0 0 0 1 0 0 0 0 0 0 0 0]
1409 [0 0 0 0 1 0 0 0 0 0 0 0]
1410 [0 0 0 0 0 1 0 0 0 0 0 0]
1411 [0 0 0 0 0 0 1 0 0 0 0 0]
1412 [0 0 0 0 0 0 0 1 0 0 0 0]
1413 [0 0 0 0 0 0 0 0 1 0 0 0]
1414 [0 0 0 0 0 0 0 0 0 1 0 0]
1415 [0 0 0 0 0 0 0 0 0 0 1 0]
1416 [0 0 0 0 0 0 0 0 0 0 0 1]
1419 B
= self
.parent().natural_basis()
1420 W
= B
[0].matrix_space()
1421 return W
.linear_combination(zip(self
.vector(), B
))
1426 Return the left-multiplication-by-this-element
1427 operator on the ambient algebra.
1431 sage: set_random_seed()
1432 sage: J = random_eja()
1433 sage: x = J.random_element()
1434 sage: y = J.random_element()
1435 sage: x.operator()(y) == x*y
1437 sage: y.operator()(x) == x*y
1442 return FiniteDimensionalEuclideanJordanAlgebraMorphism(
1444 self
.operator_matrix() )
1448 def operator_matrix(self
):
1450 Return the matrix that represents left- (or right-)
1451 multiplication by this element in the parent algebra.
1453 We implement this ourselves to work around the fact that
1454 our parent class represents everything with row vectors.
1458 Test the first polarization identity from my notes, Koecher Chapter
1459 III, or from Baes (2.3)::
1461 sage: set_random_seed()
1462 sage: J = random_eja()
1463 sage: x = J.random_element()
1464 sage: y = J.random_element()
1465 sage: Lx = x.operator_matrix()
1466 sage: Ly = y.operator_matrix()
1467 sage: Lxx = (x*x).operator_matrix()
1468 sage: Lxy = (x*y).operator_matrix()
1469 sage: bool(2*Lx*Lxy + Ly*Lxx == 2*Lxy*Lx + Lxx*Ly)
1472 Test the second polarization identity from my notes or from
1475 sage: set_random_seed()
1476 sage: J = random_eja()
1477 sage: x = J.random_element()
1478 sage: y = J.random_element()
1479 sage: z = J.random_element()
1480 sage: Lx = x.operator_matrix()
1481 sage: Ly = y.operator_matrix()
1482 sage: Lz = z.operator_matrix()
1483 sage: Lzy = (z*y).operator_matrix()
1484 sage: Lxy = (x*y).operator_matrix()
1485 sage: Lxz = (x*z).operator_matrix()
1486 sage: bool(Lx*Lzy + Lz*Lxy + Ly*Lxz == Lzy*Lx + Lxy*Lz + Lxz*Ly)
1489 Test the third polarization identity from my notes or from
1492 sage: set_random_seed()
1493 sage: J = random_eja()
1494 sage: u = J.random_element()
1495 sage: y = J.random_element()
1496 sage: z = J.random_element()
1497 sage: Lu = u.operator_matrix()
1498 sage: Ly = y.operator_matrix()
1499 sage: Lz = z.operator_matrix()
1500 sage: Lzy = (z*y).operator_matrix()
1501 sage: Luy = (u*y).operator_matrix()
1502 sage: Luz = (u*z).operator_matrix()
1503 sage: Luyz = (u*(y*z)).operator_matrix()
1504 sage: lhs = Lu*Lzy + Lz*Luy + Ly*Luz
1505 sage: rhs = Luyz + Ly*Lu*Lz + Lz*Lu*Ly
1506 sage: bool(lhs == rhs)
1510 fda_elt
= FiniteDimensionalAlgebraElement(self
.parent(), self
)
1511 return fda_elt
.matrix().transpose()
1514 def quadratic_representation(self
, other
=None):
1516 Return the quadratic representation of this element.
1520 The explicit form in the spin factor algebra is given by
1521 Alizadeh's Example 11.12::
1523 sage: set_random_seed()
1524 sage: n = ZZ.random_element(1,10)
1525 sage: J = JordanSpinEJA(n)
1526 sage: x = J.random_element()
1527 sage: x_vec = x.vector()
1529 sage: x_bar = x_vec[1:]
1530 sage: A = matrix(QQ, 1, [x_vec.inner_product(x_vec)])
1531 sage: B = 2*x0*x_bar.row()
1532 sage: C = 2*x0*x_bar.column()
1533 sage: D = identity_matrix(QQ, n-1)
1534 sage: D = (x0^2 - x_bar.inner_product(x_bar))*D
1535 sage: D = D + 2*x_bar.tensor_product(x_bar)
1536 sage: Q = block_matrix(2,2,[A,B,C,D])
1537 sage: Q == x.quadratic_representation().matrix()
1540 Test all of the properties from Theorem 11.2 in Alizadeh::
1542 sage: set_random_seed()
1543 sage: J = random_eja()
1544 sage: x = J.random_element()
1545 sage: y = J.random_element()
1546 sage: Lx = x.operator()
1547 sage: Lxx = (x*x).operator()
1548 sage: Qx = x.quadratic_representation()
1549 sage: Qy = y.quadratic_representation()
1550 sage: Qxy = x.quadratic_representation(y)
1551 sage: Qex = J.one().quadratic_representation(x)
1552 sage: n = ZZ.random_element(10)
1553 sage: Qxn = (x^n).quadratic_representation()
1557 sage: 2*Qxy == (x+y).quadratic_representation() - Qx - Qy
1562 sage: alpha = QQ.random_element()
1563 sage: (alpha*x).quadratic_representation() == (alpha^2)*Qx
1568 sage: not x.is_invertible() or ( Qx(x.inverse()) == x )
1571 sage: not x.is_invertible() or (
1574 ....: x.inverse().quadratic_representation() )
1577 sage: Qxy(J.one()) == x*y
1582 sage: not x.is_invertible() or (
1583 ....: x.quadratic_representation(x.inverse())*Qx
1584 ....: == Qx*x.quadratic_representation(x.inverse()) )
1587 sage: not x.is_invertible() or (
1588 ....: x.quadratic_representation(x.inverse())*Qx
1590 ....: 2*x.operator()*Qex - Qx )
1593 sage: 2*x.operator()*Qex - Qx == Lxx
1598 sage: Qy(x).quadratic_representation() == Qy*Qx*Qy
1608 sage: not x.is_invertible() or (
1609 ....: Qx*x.inverse().operator() == Lx )
1614 sage: not x.operator_commutes_with(y) or (
1615 ....: Qx(y)^n == Qxn(y^n) )
1621 elif not other
in self
.parent():
1622 raise TypeError("'other' must live in the same algebra")
1625 M
= other
.operator()
1626 return ( L
*M
+ M
*L
- (self
*other
).operator() )
1629 def span_of_powers(self
):
1631 Return the vector space spanned by successive powers of
1634 # The dimension of the subalgebra can't be greater than
1635 # the big algebra, so just put everything into a list
1636 # and let span() get rid of the excess.
1638 # We do the extra ambient_vector_space() in case we're messing
1639 # with polynomials and the direct parent is a module.
1640 V
= self
.parent().vector_space()
1641 return V
.span( (self
**d
).vector() for d
in xrange(V
.dimension()) )
1644 def subalgebra_generated_by(self
):
1646 Return the associative subalgebra of the parent EJA generated
1651 sage: set_random_seed()
1652 sage: x = random_eja().random_element()
1653 sage: x.subalgebra_generated_by().is_associative()
1656 Squaring in the subalgebra should be the same thing as
1657 squaring in the superalgebra::
1659 sage: set_random_seed()
1660 sage: x = random_eja().random_element()
1661 sage: u = x.subalgebra_generated_by().random_element()
1662 sage: u.operator_matrix()*u.vector() == (u**2).vector()
1666 # First get the subspace spanned by the powers of myself...
1667 V
= self
.span_of_powers()
1668 F
= self
.base_ring()
1670 # Now figure out the entries of the right-multiplication
1671 # matrix for the successive basis elements b0, b1,... of
1674 for b_right
in V
.basis():
1675 eja_b_right
= self
.parent()(b_right
)
1677 # The first row of the right-multiplication matrix by
1678 # b1 is what we get if we apply that matrix to b1. The
1679 # second row of the right multiplication matrix by b1
1680 # is what we get when we apply that matrix to b2...
1682 # IMPORTANT: this assumes that all vectors are COLUMN
1683 # vectors, unlike our superclass (which uses row vectors).
1684 for b_left
in V
.basis():
1685 eja_b_left
= self
.parent()(b_left
)
1686 # Multiply in the original EJA, but then get the
1687 # coordinates from the subalgebra in terms of its
1689 this_row
= V
.coordinates((eja_b_left
*eja_b_right
).vector())
1690 b_right_rows
.append(this_row
)
1691 b_right_matrix
= matrix(F
, b_right_rows
)
1692 mats
.append(b_right_matrix
)
1694 # It's an algebra of polynomials in one element, and EJAs
1695 # are power-associative.
1697 # TODO: choose generator names intelligently.
1698 return FiniteDimensionalEuclideanJordanAlgebra(F
, mats
, assume_associative
=True, names
='f')
1701 def subalgebra_idempotent(self
):
1703 Find an idempotent in the associative subalgebra I generate
1704 using Proposition 2.3.5 in Baes.
1708 sage: set_random_seed()
1709 sage: J = random_eja()
1710 sage: x = J.random_element()
1711 sage: while x.is_nilpotent():
1712 ....: x = J.random_element()
1713 sage: c = x.subalgebra_idempotent()
1718 if self
.is_nilpotent():
1719 raise ValueError("this only works with non-nilpotent elements!")
1721 V
= self
.span_of_powers()
1722 J
= self
.subalgebra_generated_by()
1723 # Mis-design warning: the basis used for span_of_powers()
1724 # and subalgebra_generated_by() must be the same, and in
1726 u
= J(V
.coordinates(self
.vector()))
1728 # The image of the matrix of left-u^m-multiplication
1729 # will be minimal for some natural number s...
1731 minimal_dim
= V
.dimension()
1732 for i
in xrange(1, V
.dimension()):
1733 this_dim
= (u
**i
).operator_matrix().image().dimension()
1734 if this_dim
< minimal_dim
:
1735 minimal_dim
= this_dim
1738 # Now minimal_matrix should correspond to the smallest
1739 # non-zero subspace in Baes's (or really, Koecher's)
1742 # However, we need to restrict the matrix to work on the
1743 # subspace... or do we? Can't we just solve, knowing that
1744 # A(c) = u^(s+1) should have a solution in the big space,
1747 # Beware, solve_right() means that we're using COLUMN vectors.
1748 # Our FiniteDimensionalAlgebraElement superclass uses rows.
1750 A
= u_next
.operator_matrix()
1751 c_coordinates
= A
.solve_right(u_next
.vector())
1753 # Now c_coordinates is the idempotent we want, but it's in
1754 # the coordinate system of the subalgebra.
1756 # We need the basis for J, but as elements of the parent algebra.
1758 basis
= [self
.parent(v
) for v
in V
.basis()]
1759 return self
.parent().linear_combination(zip(c_coordinates
, basis
))
1764 Return my trace, the sum of my eigenvalues.
1768 sage: J = JordanSpinEJA(3)
1769 sage: x = sum(J.gens())
1775 sage: J = RealCartesianProductEJA(5)
1776 sage: J.one().trace()
1781 The trace of an element is a real number::
1783 sage: set_random_seed()
1784 sage: J = random_eja()
1785 sage: J.random_element().trace() in J.base_ring()
1791 p
= P
._charpoly
_coeff
(r
-1)
1792 # The _charpoly_coeff function already adds the factor of
1793 # -1 to ensure that _charpoly_coeff(r-1) is really what
1794 # appears in front of t^{r-1} in the charpoly. However,
1795 # we want the negative of THAT for the trace.
1796 return -p(*self
.vector())
1799 def trace_inner_product(self
, other
):
1801 Return the trace inner product of myself and ``other``.
1805 The trace inner product is commutative::
1807 sage: set_random_seed()
1808 sage: J = random_eja()
1809 sage: x = J.random_element(); y = J.random_element()
1810 sage: x.trace_inner_product(y) == y.trace_inner_product(x)
1813 The trace inner product is bilinear::
1815 sage: set_random_seed()
1816 sage: J = random_eja()
1817 sage: x = J.random_element()
1818 sage: y = J.random_element()
1819 sage: z = J.random_element()
1820 sage: a = QQ.random_element();
1821 sage: actual = (a*(x+z)).trace_inner_product(y)
1822 sage: expected = ( a*x.trace_inner_product(y) +
1823 ....: a*z.trace_inner_product(y) )
1824 sage: actual == expected
1826 sage: actual = x.trace_inner_product(a*(y+z))
1827 sage: expected = ( a*x.trace_inner_product(y) +
1828 ....: a*x.trace_inner_product(z) )
1829 sage: actual == expected
1832 The trace inner product satisfies the compatibility
1833 condition in the definition of a Euclidean Jordan algebra::
1835 sage: set_random_seed()
1836 sage: J = random_eja()
1837 sage: x = J.random_element()
1838 sage: y = J.random_element()
1839 sage: z = J.random_element()
1840 sage: (x*y).trace_inner_product(z) == y.trace_inner_product(x*z)
1844 if not other
in self
.parent():
1845 raise TypeError("'other' must live in the same algebra")
1847 return (self
*other
).trace()
1850 class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1852 Return the Euclidean Jordan Algebra corresponding to the set
1853 `R^n` under the Hadamard product.
1855 Note: this is nothing more than the Cartesian product of ``n``
1856 copies of the spin algebra. Once Cartesian product algebras
1857 are implemented, this can go.
1861 This multiplication table can be verified by hand::
1863 sage: J = RealCartesianProductEJA(3)
1864 sage: e0,e1,e2 = J.gens()
1880 def __classcall_private__(cls
, n
, field
=QQ
):
1881 # The FiniteDimensionalAlgebra constructor takes a list of
1882 # matrices, the ith representing right multiplication by the ith
1883 # basis element in the vector space. So if e_1 = (1,0,0), then
1884 # right (Hadamard) multiplication of x by e_1 picks out the first
1885 # component of x; and likewise for the ith basis element e_i.
1886 Qs
= [ matrix(field
, n
, n
, lambda k
,j
: 1*(k
== j
== i
))
1887 for i
in xrange(n
) ]
1889 fdeja
= super(RealCartesianProductEJA
, cls
)
1890 return fdeja
.__classcall
_private
__(cls
, field
, Qs
, rank
=n
)
1892 def inner_product(self
, x
, y
):
1893 return _usual_ip(x
,y
)
1898 Return a "random" finite-dimensional Euclidean Jordan Algebra.
1902 For now, we choose a random natural number ``n`` (greater than zero)
1903 and then give you back one of the following:
1905 * The cartesian product of the rational numbers ``n`` times; this is
1906 ``QQ^n`` with the Hadamard product.
1908 * The Jordan spin algebra on ``QQ^n``.
1910 * The ``n``-by-``n`` rational symmetric matrices with the symmetric
1913 * The ``n``-by-``n`` complex-rational Hermitian matrices embedded
1914 in the space of ``2n``-by-``2n`` real symmetric matrices.
1916 * The ``n``-by-``n`` quaternion-rational Hermitian matrices embedded
1917 in the space of ``4n``-by-``4n`` real symmetric matrices.
1919 Later this might be extended to return Cartesian products of the
1925 Euclidean Jordan algebra of degree...
1929 # The max_n component lets us choose different upper bounds on the
1930 # value "n" that gets passed to the constructor. This is needed
1931 # because e.g. R^{10} is reasonable to test, while the Hermitian
1932 # 10-by-10 quaternion matrices are not.
1933 (constructor
, max_n
) = choice([(RealCartesianProductEJA
, 6),
1935 (RealSymmetricEJA
, 5),
1936 (ComplexHermitianEJA
, 4),
1937 (QuaternionHermitianEJA
, 3)])
1938 n
= ZZ
.random_element(1, max_n
)
1939 return constructor(n
, field
=QQ
)
1943 def _real_symmetric_basis(n
, field
=QQ
):
1945 Return a basis for the space of real symmetric n-by-n matrices.
1947 # The basis of symmetric matrices, as matrices, in their R^(n-by-n)
1951 for j
in xrange(i
+1):
1952 Eij
= matrix(field
, n
, lambda k
,l
: k
==i
and l
==j
)
1956 # Beware, orthogonal but not normalized!
1957 Sij
= Eij
+ Eij
.transpose()
1962 def _complex_hermitian_basis(n
, field
=QQ
):
1964 Returns a basis for the space of complex Hermitian n-by-n matrices.
1968 sage: set_random_seed()
1969 sage: n = ZZ.random_element(1,5)
1970 sage: all( M.is_symmetric() for M in _complex_hermitian_basis(n) )
1974 F
= QuadraticField(-1, 'I')
1977 # This is like the symmetric case, but we need to be careful:
1979 # * We want conjugate-symmetry, not just symmetry.
1980 # * The diagonal will (as a result) be real.
1984 for j
in xrange(i
+1):
1985 Eij
= matrix(field
, n
, lambda k
,l
: k
==i
and l
==j
)
1987 Sij
= _embed_complex_matrix(Eij
)
1990 # Beware, orthogonal but not normalized! The second one
1991 # has a minus because it's conjugated.
1992 Sij_real
= _embed_complex_matrix(Eij
+ Eij
.transpose())
1994 Sij_imag
= _embed_complex_matrix(I
*Eij
- I
*Eij
.transpose())
1999 def _quaternion_hermitian_basis(n
, field
=QQ
):
2001 Returns a basis for the space of quaternion Hermitian n-by-n matrices.
2005 sage: set_random_seed()
2006 sage: n = ZZ.random_element(1,5)
2007 sage: all( M.is_symmetric() for M in _quaternion_hermitian_basis(n) )
2011 Q
= QuaternionAlgebra(QQ
,-1,-1)
2014 # This is like the symmetric case, but we need to be careful:
2016 # * We want conjugate-symmetry, not just symmetry.
2017 # * The diagonal will (as a result) be real.
2021 for j
in xrange(i
+1):
2022 Eij
= matrix(Q
, n
, lambda k
,l
: k
==i
and l
==j
)
2024 Sij
= _embed_quaternion_matrix(Eij
)
2027 # Beware, orthogonal but not normalized! The second,
2028 # third, and fourth ones have a minus because they're
2030 Sij_real
= _embed_quaternion_matrix(Eij
+ Eij
.transpose())
2032 Sij_I
= _embed_quaternion_matrix(I
*Eij
- I
*Eij
.transpose())
2034 Sij_J
= _embed_quaternion_matrix(J
*Eij
- J
*Eij
.transpose())
2036 Sij_K
= _embed_quaternion_matrix(K
*Eij
- K
*Eij
.transpose())
2042 return vector(m
.base_ring(), m
.list())
2045 return matrix(v
.base_ring(), sqrt(v
.degree()), v
.list())
2047 def _multiplication_table_from_matrix_basis(basis
):
2049 At least three of the five simple Euclidean Jordan algebras have the
2050 symmetric multiplication (A,B) |-> (AB + BA)/2, where the
2051 multiplication on the right is matrix multiplication. Given a basis
2052 for the underlying matrix space, this function returns a
2053 multiplication table (obtained by looping through the basis
2054 elements) for an algebra of those matrices. A reordered copy
2055 of the basis is also returned to work around the fact that
2056 the ``span()`` in this function will change the order of the basis
2057 from what we think it is, to... something else.
2059 # In S^2, for example, we nominally have four coordinates even
2060 # though the space is of dimension three only. The vector space V
2061 # is supposed to hold the entire long vector, and the subspace W
2062 # of V will be spanned by the vectors that arise from symmetric
2063 # matrices. Thus for S^2, dim(V) == 4 and dim(W) == 3.
2064 field
= basis
[0].base_ring()
2065 dimension
= basis
[0].nrows()
2067 V
= VectorSpace(field
, dimension
**2)
2068 W
= V
.span( _mat2vec(s
) for s
in basis
)
2070 # Taking the span above reorders our basis (thanks, jerk!) so we
2071 # need to put our "matrix basis" in the same order as the
2072 # (reordered) vector basis.
2073 S
= tuple( _vec2mat(b
) for b
in W
.basis() )
2077 # Brute force the multiplication-by-s matrix by looping
2078 # through all elements of the basis and doing the computation
2079 # to find out what the corresponding row should be. BEWARE:
2080 # these multiplication tables won't be symmetric! It therefore
2081 # becomes REALLY IMPORTANT that the underlying algebra
2082 # constructor uses ROW vectors and not COLUMN vectors. That's
2083 # why we're computing rows here and not columns.
2086 this_row
= _mat2vec((s
*t
+ t
*s
)/2)
2087 Q_rows
.append(W
.coordinates(this_row
))
2088 Q
= matrix(field
, W
.dimension(), Q_rows
)
2094 def _embed_complex_matrix(M
):
2096 Embed the n-by-n complex matrix ``M`` into the space of real
2097 matrices of size 2n-by-2n via the map the sends each entry `z = a +
2098 bi` to the block matrix ``[[a,b],[-b,a]]``.
2102 sage: F = QuadraticField(-1,'i')
2103 sage: x1 = F(4 - 2*i)
2104 sage: x2 = F(1 + 2*i)
2107 sage: M = matrix(F,2,[[x1,x2],[x3,x4]])
2108 sage: _embed_complex_matrix(M)
2117 Embedding is a homomorphism (isomorphism, in fact)::
2119 sage: set_random_seed()
2120 sage: n = ZZ.random_element(5)
2121 sage: F = QuadraticField(-1, 'i')
2122 sage: X = random_matrix(F, n)
2123 sage: Y = random_matrix(F, n)
2124 sage: actual = _embed_complex_matrix(X) * _embed_complex_matrix(Y)
2125 sage: expected = _embed_complex_matrix(X*Y)
2126 sage: actual == expected
2132 raise ValueError("the matrix 'M' must be square")
2133 field
= M
.base_ring()
2138 blocks
.append(matrix(field
, 2, [[a
,b
],[-b
,a
]]))
2140 # We can drop the imaginaries here.
2141 return block_matrix(field
.base_ring(), n
, blocks
)
2144 def _unembed_complex_matrix(M
):
2146 The inverse of _embed_complex_matrix().
2150 sage: A = matrix(QQ,[ [ 1, 2, 3, 4],
2151 ....: [-2, 1, -4, 3],
2152 ....: [ 9, 10, 11, 12],
2153 ....: [-10, 9, -12, 11] ])
2154 sage: _unembed_complex_matrix(A)
2156 [ 10*i + 9 12*i + 11]
2160 Unembedding is the inverse of embedding::
2162 sage: set_random_seed()
2163 sage: F = QuadraticField(-1, 'i')
2164 sage: M = random_matrix(F, 3)
2165 sage: _unembed_complex_matrix(_embed_complex_matrix(M)) == M
2171 raise ValueError("the matrix 'M' must be square")
2172 if not n
.mod(2).is_zero():
2173 raise ValueError("the matrix 'M' must be a complex embedding")
2175 F
= QuadraticField(-1, 'i')
2178 # Go top-left to bottom-right (reading order), converting every
2179 # 2-by-2 block we see to a single complex element.
2181 for k
in xrange(n
/2):
2182 for j
in xrange(n
/2):
2183 submat
= M
[2*k
:2*k
+2,2*j
:2*j
+2]
2184 if submat
[0,0] != submat
[1,1]:
2185 raise ValueError('bad on-diagonal submatrix')
2186 if submat
[0,1] != -submat
[1,0]:
2187 raise ValueError('bad off-diagonal submatrix')
2188 z
= submat
[0,0] + submat
[0,1]*i
2191 return matrix(F
, n
/2, elements
)
2194 def _embed_quaternion_matrix(M
):
2196 Embed the n-by-n quaternion matrix ``M`` into the space of real
2197 matrices of size 4n-by-4n by first sending each quaternion entry
2198 `z = a + bi + cj + dk` to the block-complex matrix
2199 ``[[a + bi, c+di],[-c + di, a-bi]]`, and then embedding those into
2204 sage: Q = QuaternionAlgebra(QQ,-1,-1)
2205 sage: i,j,k = Q.gens()
2206 sage: x = 1 + 2*i + 3*j + 4*k
2207 sage: M = matrix(Q, 1, [[x]])
2208 sage: _embed_quaternion_matrix(M)
2214 Embedding is a homomorphism (isomorphism, in fact)::
2216 sage: set_random_seed()
2217 sage: n = ZZ.random_element(5)
2218 sage: Q = QuaternionAlgebra(QQ,-1,-1)
2219 sage: X = random_matrix(Q, n)
2220 sage: Y = random_matrix(Q, n)
2221 sage: actual = _embed_quaternion_matrix(X)*_embed_quaternion_matrix(Y)
2222 sage: expected = _embed_quaternion_matrix(X*Y)
2223 sage: actual == expected
2227 quaternions
= M
.base_ring()
2230 raise ValueError("the matrix 'M' must be square")
2232 F
= QuadraticField(-1, 'i')
2237 t
= z
.coefficient_tuple()
2242 cplx_matrix
= matrix(F
, 2, [[ a
+ b
*i
, c
+ d
*i
],
2243 [-c
+ d
*i
, a
- b
*i
]])
2244 blocks
.append(_embed_complex_matrix(cplx_matrix
))
2246 # We should have real entries by now, so use the realest field
2247 # we've got for the return value.
2248 return block_matrix(quaternions
.base_ring(), n
, blocks
)
2251 def _unembed_quaternion_matrix(M
):
2253 The inverse of _embed_quaternion_matrix().
2257 sage: M = matrix(QQ, [[ 1, 2, 3, 4],
2258 ....: [-2, 1, -4, 3],
2259 ....: [-3, 4, 1, -2],
2260 ....: [-4, -3, 2, 1]])
2261 sage: _unembed_quaternion_matrix(M)
2262 [1 + 2*i + 3*j + 4*k]
2266 Unembedding is the inverse of embedding::
2268 sage: set_random_seed()
2269 sage: Q = QuaternionAlgebra(QQ, -1, -1)
2270 sage: M = random_matrix(Q, 3)
2271 sage: _unembed_quaternion_matrix(_embed_quaternion_matrix(M)) == M
2277 raise ValueError("the matrix 'M' must be square")
2278 if not n
.mod(4).is_zero():
2279 raise ValueError("the matrix 'M' must be a complex embedding")
2281 Q
= QuaternionAlgebra(QQ
,-1,-1)
2284 # Go top-left to bottom-right (reading order), converting every
2285 # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1
2288 for l
in xrange(n
/4):
2289 for m
in xrange(n
/4):
2290 submat
= _unembed_complex_matrix(M
[4*l
:4*l
+4,4*m
:4*m
+4])
2291 if submat
[0,0] != submat
[1,1].conjugate():
2292 raise ValueError('bad on-diagonal submatrix')
2293 if submat
[0,1] != -submat
[1,0].conjugate():
2294 raise ValueError('bad off-diagonal submatrix')
2295 z
= submat
[0,0].real() + submat
[0,0].imag()*i
2296 z
+= submat
[0,1].real()*j
+ submat
[0,1].imag()*k
2299 return matrix(Q
, n
/4, elements
)
2302 # The usual inner product on R^n.
2304 return x
.vector().inner_product(y
.vector())
2306 # The inner product used for the real symmetric simple EJA.
2307 # We keep it as a separate function because e.g. the complex
2308 # algebra uses the same inner product, except divided by 2.
2309 def _matrix_ip(X
,Y
):
2310 X_mat
= X
.natural_representation()
2311 Y_mat
= Y
.natural_representation()
2312 return (X_mat
*Y_mat
).trace()
2315 class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra
):
2317 The rank-n simple EJA consisting of real symmetric n-by-n
2318 matrices, the usual symmetric Jordan product, and the trace inner
2319 product. It has dimension `(n^2 + n)/2` over the reals.
2323 sage: J = RealSymmetricEJA(2)
2324 sage: e0, e1, e2 = J.gens()
2334 The degree of this algebra is `(n^2 + n) / 2`::
2336 sage: set_random_seed()
2337 sage: n = ZZ.random_element(1,5)
2338 sage: J = RealSymmetricEJA(n)
2339 sage: J.degree() == (n^2 + n)/2
2342 The Jordan multiplication is what we think it is::
2344 sage: set_random_seed()
2345 sage: n = ZZ.random_element(1,5)
2346 sage: J = RealSymmetricEJA(n)
2347 sage: x = J.random_element()
2348 sage: y = J.random_element()
2349 sage: actual = (x*y).natural_representation()
2350 sage: X = x.natural_representation()
2351 sage: Y = y.natural_representation()
2352 sage: expected = (X*Y + Y*X)/2
2353 sage: actual == expected
2355 sage: J(expected) == x*y
2360 def __classcall_private__(cls
, n
, field
=QQ
):
2361 S
= _real_symmetric_basis(n
, field
=field
)
2362 (Qs
, T
) = _multiplication_table_from_matrix_basis(S
)
2364 fdeja
= super(RealSymmetricEJA
, cls
)
2365 return fdeja
.__classcall
_private
__(cls
,
2371 def inner_product(self
, x
, y
):
2372 return _matrix_ip(x
,y
)
2375 class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra
):
2377 The rank-n simple EJA consisting of complex Hermitian n-by-n
2378 matrices over the real numbers, the usual symmetric Jordan product,
2379 and the real-part-of-trace inner product. It has dimension `n^2` over
2384 The degree of this algebra is `n^2`::
2386 sage: set_random_seed()
2387 sage: n = ZZ.random_element(1,5)
2388 sage: J = ComplexHermitianEJA(n)
2389 sage: J.degree() == n^2
2392 The Jordan multiplication is what we think it is::
2394 sage: set_random_seed()
2395 sage: n = ZZ.random_element(1,5)
2396 sage: J = ComplexHermitianEJA(n)
2397 sage: x = J.random_element()
2398 sage: y = J.random_element()
2399 sage: actual = (x*y).natural_representation()
2400 sage: X = x.natural_representation()
2401 sage: Y = y.natural_representation()
2402 sage: expected = (X*Y + Y*X)/2
2403 sage: actual == expected
2405 sage: J(expected) == x*y
2410 def __classcall_private__(cls
, n
, field
=QQ
):
2411 S
= _complex_hermitian_basis(n
)
2412 (Qs
, T
) = _multiplication_table_from_matrix_basis(S
)
2414 fdeja
= super(ComplexHermitianEJA
, cls
)
2415 return fdeja
.__classcall
_private
__(cls
,
2421 def inner_product(self
, x
, y
):
2422 # Since a+bi on the diagonal is represented as
2427 # we'll double-count the "a" entries if we take the trace of
2429 return _matrix_ip(x
,y
)/2
2432 class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra
):
2434 The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
2435 matrices, the usual symmetric Jordan product, and the
2436 real-part-of-trace inner product. It has dimension `2n^2 - n` over
2441 The degree of this algebra is `n^2`::
2443 sage: set_random_seed()
2444 sage: n = ZZ.random_element(1,5)
2445 sage: J = QuaternionHermitianEJA(n)
2446 sage: J.degree() == 2*(n^2) - n
2449 The Jordan multiplication is what we think it is::
2451 sage: set_random_seed()
2452 sage: n = ZZ.random_element(1,5)
2453 sage: J = QuaternionHermitianEJA(n)
2454 sage: x = J.random_element()
2455 sage: y = J.random_element()
2456 sage: actual = (x*y).natural_representation()
2457 sage: X = x.natural_representation()
2458 sage: Y = y.natural_representation()
2459 sage: expected = (X*Y + Y*X)/2
2460 sage: actual == expected
2462 sage: J(expected) == x*y
2467 def __classcall_private__(cls
, n
, field
=QQ
):
2468 S
= _quaternion_hermitian_basis(n
)
2469 (Qs
, T
) = _multiplication_table_from_matrix_basis(S
)
2471 fdeja
= super(QuaternionHermitianEJA
, cls
)
2472 return fdeja
.__classcall
_private
__(cls
,
2478 def inner_product(self
, x
, y
):
2479 # Since a+bi+cj+dk on the diagonal is represented as
2481 # a + bi +cj + dk = [ a b c d]
2486 # we'll quadruple-count the "a" entries if we take the trace of
2488 return _matrix_ip(x
,y
)/4
2491 class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra
):
2493 The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
2494 with the usual inner product and jordan product ``x*y =
2495 (<x_bar,y_bar>, x0*y_bar + y0*x_bar)``. It has dimension `n` over
2500 This multiplication table can be verified by hand::
2502 sage: J = JordanSpinEJA(4)
2503 sage: e0,e1,e2,e3 = J.gens()
2521 def __classcall_private__(cls
, n
, field
=QQ
):
2523 id_matrix
= identity_matrix(field
, n
)
2525 ei
= id_matrix
.column(i
)
2526 Qi
= zero_matrix(field
, n
)
2528 Qi
.set_column(0, ei
)
2529 Qi
+= diagonal_matrix(n
, [ei
[0]]*n
)
2530 # The addition of the diagonal matrix adds an extra ei[0] in the
2531 # upper-left corner of the matrix.
2532 Qi
[0,0] = Qi
[0,0] * ~
field(2)
2535 # The rank of the spin algebra is two, unless we're in a
2536 # one-dimensional ambient space (because the rank is bounded by
2537 # the ambient dimension).
2538 fdeja
= super(JordanSpinEJA
, cls
)
2539 return fdeja
.__classcall
_private
__(cls
, field
, Qs
, rank
=min(n
,2))
2541 def inner_product(self
, x
, y
):
2542 return _usual_ip(x
,y
)