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
)
84 class FiniteDimensionalEuclideanJordanAlgebraMorphism(FiniteDimensionalAlgebraMorphism
):
86 A linear map between two finite-dimensional EJAs.
88 This is a very thin wrapper around FiniteDimensionalAlgebraMorphism
89 that does only a few things:
91 1. Avoids the ``unitary`` and ``check`` arguments to the constructor
92 that will always be ``False``. This is necessary because these
93 are homomorphisms with respect to ADDITION, but the SageMath
94 machinery wants to check that they're homomorphisms with respect
95 to (Jordan) MULTIPLICATION. That obviously doesn't work.
97 2. Inputs and outputs the underlying matrix with respect to COLUMN
98 vectors, unlike the parent class.
100 3. Allows us to add, subtract, negate, multiply (compose), and
101 invert morphisms in the obvious way.
103 If this seems a bit heavyweight, it is. I would have been happy to
104 use a the ring morphism that underlies the finite-dimensional
105 algebra morphism, but they don't seem to be callable on elements of
106 our EJA, and you can't add/multiply/etc. them.
108 def _add_(self
, other
):
110 Add two EJA morphisms in the obvious way.
114 sage: J = RealSymmetricEJA(3)
117 sage: x.operator() + y.operator()
118 Morphism from Euclidean Jordan algebra of degree 6 over Rational
119 Field to Euclidean Jordan algebra of degree 6 over Rational Field
130 sage: set_random_seed()
131 sage: J = random_eja()
132 sage: x = J.random_element()
133 sage: y = J.random_element()
134 sage: (x.operator() + y.operator()) in J.Hom(J)
140 raise ValueError("summands must live in the same space")
142 return FiniteDimensionalEuclideanJordanAlgebraMorphism(
144 self
.matrix() + other
.matrix() )
147 def __init__(self
, parent
, f
):
148 FiniteDimensionalAlgebraMorphism
.__init
__(self
,
159 sage: J = RealSymmetricEJA(2)
160 sage: x = J.linear_combination(zip(range(len(J.gens())), J.gens()))
161 sage: x.is_invertible()
164 Morphism from Euclidean Jordan algebra of degree 3 over Rational
165 Field to Euclidean Jordan algebra of degree 3 over Rational Field
170 sage: x.operator_matrix().inverse()
177 sage: set_random_seed()
178 sage: J = random_eja()
179 sage: x = J.random_element()
180 sage: not x.is_invertible() or (
181 ....: (~x.operator()).matrix() == x.operator_matrix().inverse() )
186 if not A
.is_invertible():
187 raise ValueError("morphism is not invertible")
190 return FiniteDimensionalEuclideanJordanAlgebraMorphism(self
.parent(),
193 def _lmul_(self
, other
):
195 Compose two EJA morphisms using multiplicative notation.
199 sage: J = RealSymmetricEJA(3)
202 sage: x.operator() * y.operator()
203 Morphism from Euclidean Jordan algebra of degree 6 over Rational
204 Field to Euclidean Jordan algebra of degree 6 over Rational Field
215 sage: set_random_seed()
216 sage: J = random_eja()
217 sage: x = J.random_element()
218 sage: y = J.random_element()
219 sage: (x.operator() * y.operator()) in J.Hom(J)
223 if not other
.codomain() is self
.domain():
224 raise ValueError("(co)domains must agree for composition")
226 return FiniteDimensionalEuclideanJordanAlgebraMorphism(
228 self
.matrix()*other
.matrix() )
233 Negate this morphism.
237 sage: J = RealSymmetricEJA(2)
240 Morphism from Euclidean Jordan algebra of degree 3 over Rational
241 Field to Euclidean Jordan algebra of degree 3 over Rational Field
249 sage: set_random_seed()
250 sage: J = random_eja()
251 sage: x = J.random_element()
252 sage: -x.operator() in J.Hom(J)
256 return FiniteDimensionalEuclideanJordanAlgebraMorphism(
263 We override only the representation that is shown to the user,
264 because we want the matrix to be with respect to COLUMN vectors.
268 Ensure that we see the transpose of the underlying matrix object:
270 sage: J = RealSymmetricEJA(3)
271 sage: x = J.linear_combination(zip(range(len(J.gens())), J.gens()))
272 sage: L = x.operator()
274 Morphism from Euclidean Jordan algebra of degree 6 over Rational
275 Field to Euclidean Jordan algebra of degree 6 over Rational Field
292 return "Morphism from {} to {} given by matrix\n{}".format(
293 self
.domain(), self
.codomain(), self
.matrix())
296 def __sub__(self
, other
):
298 Subtract one morphism from another using addition and negation.
302 sage: J = RealSymmetricEJA(2)
303 sage: L1 = J.one().operator()
305 Morphism from Euclidean Jordan algebra of degree 3 over Rational
306 Field to Euclidean Jordan algebra of degree 3 over Rational
307 Field given by matrix
314 sage: set_random_seed()
315 sage: J = random_eja()
316 sage: x = J.random_element()
317 sage: y = J.random_element()
318 sage: x.operator() - y.operator() in J.Hom(J)
322 return self
+ (-other
)
327 Return the matrix of this morphism with respect to a left-action
330 return FiniteDimensionalAlgebraMorphism
.matrix(self
).transpose()
333 class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra
):
335 def __classcall_private__(cls
,
339 assume_associative
=False,
344 mult_table
= [b
.base_extend(field
) for b
in mult_table
]
347 if not (is_Matrix(b
) and b
.dimensions() == (n
, n
)):
348 raise ValueError("input is not a multiplication table")
349 mult_table
= tuple(mult_table
)
351 cat
= FiniteDimensionalAlgebrasWithBasis(field
)
352 cat
.or_subcategory(category
)
353 if assume_associative
:
354 cat
= cat
.Associative()
356 names
= normalize_names(n
, names
)
358 fda
= super(FiniteDimensionalEuclideanJordanAlgebra
, cls
)
359 return fda
.__classcall
__(cls
,
362 assume_associative
=assume_associative
,
366 natural_basis
=natural_basis
)
369 def _Hom_(self
, B
, cat
):
371 Construct a homset of ``self`` and ``B``.
373 return FiniteDimensionalEuclideanJordanAlgebraHomset(self
,
382 assume_associative
=False,
389 By definition, Jordan multiplication commutes::
391 sage: set_random_seed()
392 sage: J = random_eja()
393 sage: x = J.random_element()
394 sage: y = J.random_element()
400 self
._natural
_basis
= natural_basis
401 self
._multiplication
_table
= mult_table
402 fda
= super(FiniteDimensionalEuclideanJordanAlgebra
, self
)
411 Return a string representation of ``self``.
413 fmt
= "Euclidean Jordan algebra of degree {} over {}"
414 return fmt
.format(self
.degree(), self
.base_ring())
417 def _a_regular_element(self
):
419 Guess a regular element. Needed to compute the basis for our
420 characteristic polynomial coefficients.
423 z
= self
.sum( (i
+1)*gs
[i
] for i
in range(len(gs
)) )
424 if not z
.is_regular():
425 raise ValueError("don't know a regular element")
430 def _charpoly_basis_space(self
):
432 Return the vector space spanned by the basis used in our
433 characteristic polynomial coefficients. This is used not only to
434 compute those coefficients, but also any time we need to
435 evaluate the coefficients (like when we compute the trace or
438 z
= self
._a
_regular
_element
()
439 V
= z
.vector().parent().ambient_vector_space()
440 V1
= V
.span_of_basis( (z
**k
).vector() for k
in range(self
.rank()) )
441 b
= (V1
.basis() + V1
.complement().basis())
442 return V
.span_of_basis(b
)
446 def _charpoly_coeff(self
, i
):
448 Return the coefficient polynomial "a_{i}" of this algebra's
449 general characteristic polynomial.
451 Having this be a separate cached method lets us compute and
452 store the trace/determinant (a_{r-1} and a_{0} respectively)
453 separate from the entire characteristic polynomial.
455 (A_of_x
, x
, xr
, detA
) = self
._charpoly
_matrix
_system
()
456 R
= A_of_x
.base_ring()
458 # Guaranteed by theory
461 # Danger: the in-place modification is done for performance
462 # reasons (reconstructing a matrix with huge polynomial
463 # entries is slow), but I don't know how cached_method works,
464 # so it's highly possible that we're modifying some global
465 # list variable by reference, here. In other words, you
466 # probably shouldn't call this method twice on the same
467 # algebra, at the same time, in two threads
468 Ai_orig
= A_of_x
.column(i
)
469 A_of_x
.set_column(i
,xr
)
470 numerator
= A_of_x
.det()
471 A_of_x
.set_column(i
,Ai_orig
)
473 # We're relying on the theory here to ensure that each a_i is
474 # indeed back in R, and the added negative signs are to make
475 # the whole charpoly expression sum to zero.
476 return R(-numerator
/detA
)
480 def _charpoly_matrix_system(self
):
482 Compute the matrix whose entries A_ij are polynomials in
483 X1,...,XN, the vector ``x`` of variables X1,...,XN, the vector
484 corresponding to `x^r` and the determinent of the matrix A =
485 [A_ij]. In other words, all of the fixed (cachable) data needed
486 to compute the coefficients of the characteristic polynomial.
491 # Construct a new algebra over a multivariate polynomial ring...
492 names
= ['X' + str(i
) for i
in range(1,n
+1)]
493 R
= PolynomialRing(self
.base_ring(), names
)
494 J
= FiniteDimensionalEuclideanJordanAlgebra(R
,
495 self
._multiplication
_table
,
498 idmat
= identity_matrix(J
.base_ring(), n
)
500 W
= self
._charpoly
_basis
_space
()
501 W
= W
.change_ring(R
.fraction_field())
503 # Starting with the standard coordinates x = (X1,X2,...,Xn)
504 # and then converting the entries to W-coordinates allows us
505 # to pass in the standard coordinates to the charpoly and get
506 # back the right answer. Specifically, with x = (X1,X2,...,Xn),
509 # W.coordinates(x^2) eval'd at (standard z-coords)
513 # W-coords of (standard coords of x^2 eval'd at std-coords of z)
515 # We want the middle equivalent thing in our matrix, but use
516 # the first equivalent thing instead so that we can pass in
517 # standard coordinates.
518 x
= J(vector(R
, R
.gens()))
519 l1
= [column_matrix(W
.coordinates((x
**k
).vector())) for k
in range(r
)]
520 l2
= [idmat
.column(k
-1).column() for k
in range(r
+1, n
+1)]
521 A_of_x
= block_matrix(R
, 1, n
, (l1
+ l2
))
522 xr
= W
.coordinates((x
**r
).vector())
523 return (A_of_x
, x
, xr
, A_of_x
.det())
527 def characteristic_polynomial(self
):
532 This implementation doesn't guarantee that the polynomial
533 denominator in the coefficients is not identically zero, so
534 theoretically it could crash. The way that this is handled
535 in e.g. Faraut and Koranyi is to use a basis that guarantees
536 the denominator is non-zero. But, doing so requires knowledge
537 of at least one regular element, and we don't even know how
538 to do that. The trade-off is that, if we use the standard basis,
539 the resulting polynomial will accept the "usual" coordinates. In
540 other words, we don't have to do a change of basis before e.g.
541 computing the trace or determinant.
545 The characteristic polynomial in the spin algebra is given in
546 Alizadeh, Example 11.11::
548 sage: J = JordanSpinEJA(3)
549 sage: p = J.characteristic_polynomial(); p
550 X1^2 - X2^2 - X3^2 + (-2*t)*X1 + t^2
551 sage: xvec = J.one().vector()
559 # The list of coefficient polynomials a_1, a_2, ..., a_n.
560 a
= [ self
._charpoly
_coeff
(i
) for i
in range(n
) ]
562 # We go to a bit of trouble here to reorder the
563 # indeterminates, so that it's easier to evaluate the
564 # characteristic polynomial at x's coordinates and get back
565 # something in terms of t, which is what we want.
567 S
= PolynomialRing(self
.base_ring(),'t')
569 S
= PolynomialRing(S
, R
.variable_names())
572 # Note: all entries past the rth should be zero. The
573 # coefficient of the highest power (x^r) is 1, but it doesn't
574 # appear in the solution vector which contains coefficients
575 # for the other powers (to make them sum to x^r).
577 a
[r
] = 1 # corresponds to x^r
579 # When the rank is equal to the dimension, trying to
580 # assign a[r] goes out-of-bounds.
581 a
.append(1) # corresponds to x^r
583 return sum( a
[k
]*(t
**k
) for k
in range(len(a
)) )
586 def inner_product(self
, x
, y
):
588 The inner product associated with this Euclidean Jordan algebra.
590 Defaults to the trace inner product, but can be overridden by
591 subclasses if they are sure that the necessary properties are
596 The inner product must satisfy its axiom for this algebra to truly
597 be a Euclidean Jordan Algebra::
599 sage: set_random_seed()
600 sage: J = random_eja()
601 sage: x = J.random_element()
602 sage: y = J.random_element()
603 sage: z = J.random_element()
604 sage: (x*y).inner_product(z) == y.inner_product(x*z)
608 if (not x
in self
) or (not y
in self
):
609 raise TypeError("arguments must live in this algebra")
610 return x
.trace_inner_product(y
)
613 def natural_basis(self
):
615 Return a more-natural representation of this algebra's basis.
617 Every finite-dimensional Euclidean Jordan Algebra is a direct
618 sum of five simple algebras, four of which comprise Hermitian
619 matrices. This method returns the original "natural" basis
620 for our underlying vector space. (Typically, the natural basis
621 is used to construct the multiplication table in the first place.)
623 Note that this will always return a matrix. The standard basis
624 in `R^n` will be returned as `n`-by-`1` column matrices.
628 sage: J = RealSymmetricEJA(2)
631 sage: J.natural_basis()
639 sage: J = JordanSpinEJA(2)
642 sage: J.natural_basis()
649 if self
._natural
_basis
is None:
650 return tuple( b
.vector().column() for b
in self
.basis() )
652 return self
._natural
_basis
657 Return the rank of this EJA.
659 if self
._rank
is None:
660 raise ValueError("no rank specified at genesis")
665 class Element(FiniteDimensionalAlgebraElement
):
667 An element of a Euclidean Jordan algebra.
672 Oh man, I should not be doing this. This hides the "disabled"
673 methods ``left_matrix`` and ``matrix`` from introspection;
674 in particular it removes them from tab-completion.
676 return filter(lambda s
: s
not in ['left_matrix', 'matrix'],
677 dir(self
.__class
__) )
680 def __init__(self
, A
, elt
=None):
684 The identity in `S^n` is converted to the identity in the EJA::
686 sage: J = RealSymmetricEJA(3)
687 sage: I = identity_matrix(QQ,3)
688 sage: J(I) == J.one()
691 This skew-symmetric matrix can't be represented in the EJA::
693 sage: J = RealSymmetricEJA(3)
694 sage: A = matrix(QQ,3, lambda i,j: i-j)
696 Traceback (most recent call last):
698 ArithmeticError: vector is not in free module
701 # Goal: if we're given a matrix, and if it lives in our
702 # parent algebra's "natural ambient space," convert it
703 # into an algebra element.
705 # The catch is, we make a recursive call after converting
706 # the given matrix into a vector that lives in the algebra.
707 # This we need to try the parent class initializer first,
708 # to avoid recursing forever if we're given something that
709 # already fits into the algebra, but also happens to live
710 # in the parent's "natural ambient space" (this happens with
713 FiniteDimensionalAlgebraElement
.__init
__(self
, A
, elt
)
715 natural_basis
= A
.natural_basis()
716 if elt
in natural_basis
[0].matrix_space():
717 # Thanks for nothing! Matrix spaces aren't vector
718 # spaces in Sage, so we have to figure out its
719 # natural-basis coordinates ourselves.
720 V
= VectorSpace(elt
.base_ring(), elt
.nrows()**2)
721 W
= V
.span( _mat2vec(s
) for s
in natural_basis
)
722 coords
= W
.coordinates(_mat2vec(elt
))
723 FiniteDimensionalAlgebraElement
.__init
__(self
, A
, coords
)
725 def __pow__(self
, n
):
727 Return ``self`` raised to the power ``n``.
729 Jordan algebras are always power-associative; see for
730 example Faraut and Koranyi, Proposition II.1.2 (ii).
734 We have to override this because our superclass uses row vectors
735 instead of column vectors! We, on the other hand, assume column
740 sage: set_random_seed()
741 sage: x = random_eja().random_element()
742 sage: x.operator_matrix()*x.vector() == (x^2).vector()
745 A few examples of power-associativity::
747 sage: set_random_seed()
748 sage: x = random_eja().random_element()
749 sage: x*(x*x)*(x*x) == x^5
751 sage: (x*x)*(x*x*x) == x^5
754 We also know that powers operator-commute (Koecher, Chapter
757 sage: set_random_seed()
758 sage: x = random_eja().random_element()
759 sage: m = ZZ.random_element(0,10)
760 sage: n = ZZ.random_element(0,10)
761 sage: Lxm = (x^m).operator_matrix()
762 sage: Lxn = (x^n).operator_matrix()
763 sage: Lxm*Lxn == Lxn*Lxm
773 return A( (self
.operator_matrix()**(n
-1))*self
.vector() )
776 def apply_univariate_polynomial(self
, p
):
778 Apply the univariate polynomial ``p`` to this element.
780 A priori, SageMath won't allow us to apply a univariate
781 polynomial to an element of an EJA, because we don't know
782 that EJAs are rings (they are usually not associative). Of
783 course, we know that EJAs are power-associative, so the
784 operation is ultimately kosher. This function sidesteps
785 the CAS to get the answer we want and expect.
789 sage: R = PolynomialRing(QQ, 't')
791 sage: p = t^4 - t^3 + 5*t - 2
792 sage: J = RealCartesianProductEJA(5)
793 sage: J.one().apply_univariate_polynomial(p) == 3*J.one()
798 We should always get back an element of the algebra::
800 sage: set_random_seed()
801 sage: p = PolynomialRing(QQ, 't').random_element()
802 sage: J = random_eja()
803 sage: x = J.random_element()
804 sage: x.apply_univariate_polynomial(p) in J
808 if len(p
.variables()) > 1:
809 raise ValueError("not a univariate polynomial")
812 # Convert the coeficcients to the parent's base ring,
813 # because a priori they might live in an (unnecessarily)
814 # larger ring for which P.sum() would fail below.
815 cs
= [ R(c
) for c
in p
.coefficients(sparse
=False) ]
816 return P
.sum( cs
[k
]*(self
**k
) for k
in range(len(cs
)) )
819 def characteristic_polynomial(self
):
821 Return the characteristic polynomial of this element.
825 The rank of `R^3` is three, and the minimal polynomial of
826 the identity element is `(t-1)` from which it follows that
827 the characteristic polynomial should be `(t-1)^3`::
829 sage: J = RealCartesianProductEJA(3)
830 sage: J.one().characteristic_polynomial()
831 t^3 - 3*t^2 + 3*t - 1
833 Likewise, the characteristic of the zero element in the
834 rank-three algebra `R^{n}` should be `t^{3}`::
836 sage: J = RealCartesianProductEJA(3)
837 sage: J.zero().characteristic_polynomial()
840 The characteristic polynomial of an element should evaluate
841 to zero on that element::
843 sage: set_random_seed()
844 sage: x = RealCartesianProductEJA(3).random_element()
845 sage: p = x.characteristic_polynomial()
846 sage: x.apply_univariate_polynomial(p)
850 p
= self
.parent().characteristic_polynomial()
851 return p(*self
.vector())
854 def inner_product(self
, other
):
856 Return the parent algebra's inner product of myself and ``other``.
860 The inner product in the Jordan spin algebra is the usual
861 inner product on `R^n` (this example only works because the
862 basis for the Jordan algebra is the standard basis in `R^n`)::
864 sage: J = JordanSpinEJA(3)
865 sage: x = vector(QQ,[1,2,3])
866 sage: y = vector(QQ,[4,5,6])
867 sage: x.inner_product(y)
869 sage: J(x).inner_product(J(y))
872 The inner product on `S^n` is `<X,Y> = trace(X*Y)`, where
873 multiplication is the usual matrix multiplication in `S^n`,
874 so the inner product of the identity matrix with itself
877 sage: J = RealSymmetricEJA(3)
878 sage: J.one().inner_product(J.one())
881 Likewise, the inner product on `C^n` is `<X,Y> =
882 Re(trace(X*Y))`, where we must necessarily take the real
883 part because the product of Hermitian matrices may not be
886 sage: J = ComplexHermitianEJA(3)
887 sage: J.one().inner_product(J.one())
890 Ditto for the quaternions::
892 sage: J = QuaternionHermitianEJA(3)
893 sage: J.one().inner_product(J.one())
898 Ensure that we can always compute an inner product, and that
899 it gives us back a real number::
901 sage: set_random_seed()
902 sage: J = random_eja()
903 sage: x = J.random_element()
904 sage: y = J.random_element()
905 sage: x.inner_product(y) in RR
911 raise TypeError("'other' must live in the same algebra")
913 return P
.inner_product(self
, other
)
916 def operator_commutes_with(self
, other
):
918 Return whether or not this element operator-commutes
923 The definition of a Jordan algebra says that any element
924 operator-commutes with its square::
926 sage: set_random_seed()
927 sage: x = random_eja().random_element()
928 sage: x.operator_commutes_with(x^2)
933 Test Lemma 1 from Chapter III of Koecher::
935 sage: set_random_seed()
936 sage: J = random_eja()
937 sage: u = J.random_element()
938 sage: v = J.random_element()
939 sage: lhs = u.operator_commutes_with(u*v)
940 sage: rhs = v.operator_commutes_with(u^2)
945 if not other
in self
.parent():
946 raise TypeError("'other' must live in the same algebra")
948 A
= self
.operator_matrix()
949 B
= other
.operator_matrix()
955 Return my determinant, the product of my eigenvalues.
959 sage: J = JordanSpinEJA(2)
960 sage: e0,e1 = J.gens()
961 sage: x = sum( J.gens() )
967 sage: J = JordanSpinEJA(3)
968 sage: e0,e1,e2 = J.gens()
969 sage: x = sum( J.gens() )
975 An element is invertible if and only if its determinant is
978 sage: set_random_seed()
979 sage: x = random_eja().random_element()
980 sage: x.is_invertible() == (x.det() != 0)
986 p
= P
._charpoly
_coeff
(0)
987 # The _charpoly_coeff function already adds the factor of
988 # -1 to ensure that _charpoly_coeff(0) is really what
989 # appears in front of t^{0} in the charpoly. However,
990 # we want (-1)^r times THAT for the determinant.
991 return ((-1)**r
)*p(*self
.vector())
996 Return the Jordan-multiplicative inverse of this element.
1000 We appeal to the quadratic representation as in Koecher's
1001 Theorem 12 in Chapter III, Section 5.
1005 The inverse in the spin factor algebra is given in Alizadeh's
1008 sage: set_random_seed()
1009 sage: n = ZZ.random_element(1,10)
1010 sage: J = JordanSpinEJA(n)
1011 sage: x = J.random_element()
1012 sage: while x.is_zero():
1013 ....: x = J.random_element()
1014 sage: x_vec = x.vector()
1016 sage: x_bar = x_vec[1:]
1017 sage: coeff = 1/(x0^2 - x_bar.inner_product(x_bar))
1018 sage: inv_vec = x_vec.parent()([x0] + (-x_bar).list())
1019 sage: x_inverse = coeff*inv_vec
1020 sage: x.inverse() == J(x_inverse)
1025 The identity element is its own inverse::
1027 sage: set_random_seed()
1028 sage: J = random_eja()
1029 sage: J.one().inverse() == J.one()
1032 If an element has an inverse, it acts like one::
1034 sage: set_random_seed()
1035 sage: J = random_eja()
1036 sage: x = J.random_element()
1037 sage: (not x.is_invertible()) or (x.inverse()*x == J.one())
1040 The inverse of the inverse is what we started with::
1042 sage: set_random_seed()
1043 sage: J = random_eja()
1044 sage: x = J.random_element()
1045 sage: (not x.is_invertible()) or (x.inverse().inverse() == x)
1048 The zero element is never invertible::
1050 sage: set_random_seed()
1051 sage: J = random_eja().zero().inverse()
1052 Traceback (most recent call last):
1054 ValueError: element is not invertible
1057 if not self
.is_invertible():
1058 raise ValueError("element is not invertible")
1061 return P(self
.quadratic_representation().inverse()*self
.vector())
1064 def is_invertible(self
):
1066 Return whether or not this element is invertible.
1068 We can't use the superclass method because it relies on
1069 the algebra being associative.
1073 The usual way to do this is to check if the determinant is
1074 zero, but we need the characteristic polynomial for the
1075 determinant. The minimal polynomial is a lot easier to get,
1076 so we use Corollary 2 in Chapter V of Koecher to check
1077 whether or not the paren't algebra's zero element is a root
1078 of this element's minimal polynomial.
1082 The identity element is always invertible::
1084 sage: set_random_seed()
1085 sage: J = random_eja()
1086 sage: J.one().is_invertible()
1089 The zero element is never invertible::
1091 sage: set_random_seed()
1092 sage: J = random_eja()
1093 sage: J.zero().is_invertible()
1097 zero
= self
.parent().zero()
1098 p
= self
.minimal_polynomial()
1099 return not (p(zero
) == zero
)
1102 def is_nilpotent(self
):
1104 Return whether or not some power of this element is zero.
1106 The superclass method won't work unless we're in an
1107 associative algebra, and we aren't. However, we generate
1108 an assocoative subalgebra and we're nilpotent there if and
1109 only if we're nilpotent here (probably).
1113 The identity element is never nilpotent::
1115 sage: set_random_seed()
1116 sage: random_eja().one().is_nilpotent()
1119 The additive identity is always nilpotent::
1121 sage: set_random_seed()
1122 sage: random_eja().zero().is_nilpotent()
1126 # The element we're going to call "is_nilpotent()" on.
1127 # Either myself, interpreted as an element of a finite-
1128 # dimensional algebra, or an element of an associative
1132 if self
.parent().is_associative():
1133 elt
= FiniteDimensionalAlgebraElement(self
.parent(), self
)
1135 V
= self
.span_of_powers()
1136 assoc_subalg
= self
.subalgebra_generated_by()
1137 # Mis-design warning: the basis used for span_of_powers()
1138 # and subalgebra_generated_by() must be the same, and in
1140 elt
= assoc_subalg(V
.coordinates(self
.vector()))
1142 # Recursive call, but should work since elt lives in an
1143 # associative algebra.
1144 return elt
.is_nilpotent()
1147 def is_regular(self
):
1149 Return whether or not this is a regular element.
1153 The identity element always has degree one, but any element
1154 linearly-independent from it is regular::
1156 sage: J = JordanSpinEJA(5)
1157 sage: J.one().is_regular()
1159 sage: e0, e1, e2, e3, e4 = J.gens() # e0 is the identity
1160 sage: for x in J.gens():
1161 ....: (J.one() + x).is_regular()
1169 return self
.degree() == self
.parent().rank()
1174 Compute the degree of this element the straightforward way
1175 according to the definition; by appending powers to a list
1176 and figuring out its dimension (that is, whether or not
1177 they're linearly dependent).
1181 sage: J = JordanSpinEJA(4)
1182 sage: J.one().degree()
1184 sage: e0,e1,e2,e3 = J.gens()
1185 sage: (e0 - e1).degree()
1188 In the spin factor algebra (of rank two), all elements that
1189 aren't multiples of the identity are regular::
1191 sage: set_random_seed()
1192 sage: n = ZZ.random_element(1,10)
1193 sage: J = JordanSpinEJA(n)
1194 sage: x = J.random_element()
1195 sage: x == x.coefficient(0)*J.one() or x.degree() == 2
1199 return self
.span_of_powers().dimension()
1202 def left_matrix(self
):
1204 Our parent class defines ``left_matrix`` and ``matrix``
1205 methods whose names are misleading. We don't want them.
1207 raise NotImplementedError("use operator_matrix() instead")
1209 matrix
= left_matrix
1212 def minimal_polynomial(self
):
1214 Return the minimal polynomial of this element,
1215 as a function of the variable `t`.
1219 We restrict ourselves to the associative subalgebra
1220 generated by this element, and then return the minimal
1221 polynomial of this element's operator matrix (in that
1222 subalgebra). This works by Baes Proposition 2.3.16.
1226 The minimal polynomial of the identity and zero elements are
1229 sage: set_random_seed()
1230 sage: J = random_eja()
1231 sage: J.one().minimal_polynomial()
1233 sage: J.zero().minimal_polynomial()
1236 The degree of an element is (by one definition) the degree
1237 of its minimal polynomial::
1239 sage: set_random_seed()
1240 sage: x = random_eja().random_element()
1241 sage: x.degree() == x.minimal_polynomial().degree()
1244 The minimal polynomial and the characteristic polynomial coincide
1245 and are known (see Alizadeh, Example 11.11) for all elements of
1246 the spin factor algebra that aren't scalar multiples of the
1249 sage: set_random_seed()
1250 sage: n = ZZ.random_element(2,10)
1251 sage: J = JordanSpinEJA(n)
1252 sage: y = J.random_element()
1253 sage: while y == y.coefficient(0)*J.one():
1254 ....: y = J.random_element()
1255 sage: y0 = y.vector()[0]
1256 sage: y_bar = y.vector()[1:]
1257 sage: actual = y.minimal_polynomial()
1258 sage: t = PolynomialRing(J.base_ring(),'t').gen(0)
1259 sage: expected = t^2 - 2*y0*t + (y0^2 - norm(y_bar)^2)
1260 sage: bool(actual == expected)
1263 The minimal polynomial should always kill its element::
1265 sage: set_random_seed()
1266 sage: x = random_eja().random_element()
1267 sage: p = x.minimal_polynomial()
1268 sage: x.apply_univariate_polynomial(p)
1272 V
= self
.span_of_powers()
1273 assoc_subalg
= self
.subalgebra_generated_by()
1274 # Mis-design warning: the basis used for span_of_powers()
1275 # and subalgebra_generated_by() must be the same, and in
1277 elt
= assoc_subalg(V
.coordinates(self
.vector()))
1279 # We get back a symbolic polynomial in 'x' but want a real
1280 # polynomial in 't'.
1281 p_of_x
= elt
.operator_matrix().minimal_polynomial()
1282 return p_of_x
.change_variable_name('t')
1285 def natural_representation(self
):
1287 Return a more-natural representation of this element.
1289 Every finite-dimensional Euclidean Jordan Algebra is a
1290 direct sum of five simple algebras, four of which comprise
1291 Hermitian matrices. This method returns the original
1292 "natural" representation of this element as a Hermitian
1293 matrix, if it has one. If not, you get the usual representation.
1297 sage: J = ComplexHermitianEJA(3)
1300 sage: J.one().natural_representation()
1310 sage: J = QuaternionHermitianEJA(3)
1313 sage: J.one().natural_representation()
1314 [1 0 0 0 0 0 0 0 0 0 0 0]
1315 [0 1 0 0 0 0 0 0 0 0 0 0]
1316 [0 0 1 0 0 0 0 0 0 0 0 0]
1317 [0 0 0 1 0 0 0 0 0 0 0 0]
1318 [0 0 0 0 1 0 0 0 0 0 0 0]
1319 [0 0 0 0 0 1 0 0 0 0 0 0]
1320 [0 0 0 0 0 0 1 0 0 0 0 0]
1321 [0 0 0 0 0 0 0 1 0 0 0 0]
1322 [0 0 0 0 0 0 0 0 1 0 0 0]
1323 [0 0 0 0 0 0 0 0 0 1 0 0]
1324 [0 0 0 0 0 0 0 0 0 0 1 0]
1325 [0 0 0 0 0 0 0 0 0 0 0 1]
1328 B
= self
.parent().natural_basis()
1329 W
= B
[0].matrix_space()
1330 return W
.linear_combination(zip(self
.vector(), B
))
1335 Return the left-multiplication-by-this-element
1336 operator on the ambient algebra.
1340 sage: set_random_seed()
1341 sage: J = random_eja()
1342 sage: x = J.random_element()
1343 sage: y = J.random_element()
1344 sage: x.operator()(y) == x*y
1346 sage: y.operator()(x) == x*y
1351 return FiniteDimensionalEuclideanJordanAlgebraMorphism(
1353 self
.operator_matrix() )
1357 def operator_matrix(self
):
1359 Return the matrix that represents left- (or right-)
1360 multiplication by this element in the parent algebra.
1362 We implement this ourselves to work around the fact that
1363 our parent class represents everything with row vectors.
1367 Test the first polarization identity from my notes, Koecher Chapter
1368 III, or from Baes (2.3)::
1370 sage: set_random_seed()
1371 sage: J = random_eja()
1372 sage: x = J.random_element()
1373 sage: y = J.random_element()
1374 sage: Lx = x.operator_matrix()
1375 sage: Ly = y.operator_matrix()
1376 sage: Lxx = (x*x).operator_matrix()
1377 sage: Lxy = (x*y).operator_matrix()
1378 sage: bool(2*Lx*Lxy + Ly*Lxx == 2*Lxy*Lx + Lxx*Ly)
1381 Test the second polarization identity from my notes or from
1384 sage: set_random_seed()
1385 sage: J = random_eja()
1386 sage: x = J.random_element()
1387 sage: y = J.random_element()
1388 sage: z = J.random_element()
1389 sage: Lx = x.operator_matrix()
1390 sage: Ly = y.operator_matrix()
1391 sage: Lz = z.operator_matrix()
1392 sage: Lzy = (z*y).operator_matrix()
1393 sage: Lxy = (x*y).operator_matrix()
1394 sage: Lxz = (x*z).operator_matrix()
1395 sage: bool(Lx*Lzy + Lz*Lxy + Ly*Lxz == Lzy*Lx + Lxy*Lz + Lxz*Ly)
1398 Test the third polarization identity from my notes or from
1401 sage: set_random_seed()
1402 sage: J = random_eja()
1403 sage: u = J.random_element()
1404 sage: y = J.random_element()
1405 sage: z = J.random_element()
1406 sage: Lu = u.operator_matrix()
1407 sage: Ly = y.operator_matrix()
1408 sage: Lz = z.operator_matrix()
1409 sage: Lzy = (z*y).operator_matrix()
1410 sage: Luy = (u*y).operator_matrix()
1411 sage: Luz = (u*z).operator_matrix()
1412 sage: Luyz = (u*(y*z)).operator_matrix()
1413 sage: lhs = Lu*Lzy + Lz*Luy + Ly*Luz
1414 sage: rhs = Luyz + Ly*Lu*Lz + Lz*Lu*Ly
1415 sage: bool(lhs == rhs)
1419 fda_elt
= FiniteDimensionalAlgebraElement(self
.parent(), self
)
1420 return fda_elt
.matrix().transpose()
1423 def quadratic_representation(self
, other
=None):
1425 Return the quadratic representation of this element.
1429 The explicit form in the spin factor algebra is given by
1430 Alizadeh's Example 11.12::
1432 sage: set_random_seed()
1433 sage: n = ZZ.random_element(1,10)
1434 sage: J = JordanSpinEJA(n)
1435 sage: x = J.random_element()
1436 sage: x_vec = x.vector()
1438 sage: x_bar = x_vec[1:]
1439 sage: A = matrix(QQ, 1, [x_vec.inner_product(x_vec)])
1440 sage: B = 2*x0*x_bar.row()
1441 sage: C = 2*x0*x_bar.column()
1442 sage: D = identity_matrix(QQ, n-1)
1443 sage: D = (x0^2 - x_bar.inner_product(x_bar))*D
1444 sage: D = D + 2*x_bar.tensor_product(x_bar)
1445 sage: Q = block_matrix(2,2,[A,B,C,D])
1446 sage: Q == x.quadratic_representation().operator_matrix()
1449 Test all of the properties from Theorem 11.2 in Alizadeh::
1451 sage: set_random_seed()
1452 sage: J = random_eja()
1453 sage: x = J.random_element()
1454 sage: y = J.random_element()
1455 sage: Lx = x.operator()
1456 sage: Lxx = (x*x).operator()
1457 sage: Qx = x.quadratic_representation()
1458 sage: Qy = y.quadratic_representation()
1459 sage: Qxy = x.quadratic_representation(y)
1460 sage: Qex = J.one().quadratic_representation(x)
1461 sage: n = ZZ.random_element(10)
1462 sage: Qxn = (x^n).quadratic_representation()
1466 sage: 2*Qxy == (x+y).quadratic_representation() - Qx - Qy
1471 sage: alpha = QQ.random_element()
1472 sage: (alpha*x).quadratic_representation() == (alpha^2)*Qx
1477 sage: not x.is_invertible() or ( Qx(x.inverse()) == x )
1480 sage: not x.is_invertible() or (
1483 ....: x.inverse().quadratic_representation() )
1486 sage: Qxy(J.one()) == x*y
1491 sage: not x.is_invertible() or (
1492 ....: x.quadratic_representation(x.inverse())*Qx
1493 ....: == Qx*x.quadratic_representation(x.inverse()) )
1496 sage: not x.is_invertible() or (
1497 ....: x.quadratic_representation(x.inverse())*Qx
1499 ....: 2*x.operator()*Qex - Qx )
1502 sage: 2*x.operator()*Qex - Qx == Lxx
1507 sage: Qy(x).quadratic_representation() == Qy*Qx*Qy
1517 sage: not x.is_invertible() or (
1518 ....: Qx*x.inverse().operator() == Lx )
1523 sage: not x.operator_commutes_with(y) or (
1524 ....: Qx(y)^n == Qxn(y^n) )
1530 elif not other
in self
.parent():
1531 raise TypeError("'other' must live in the same algebra")
1534 M
= other
.operator()
1535 return ( L
*M
+ M
*L
- (self
*other
).operator() )
1538 def span_of_powers(self
):
1540 Return the vector space spanned by successive powers of
1543 # The dimension of the subalgebra can't be greater than
1544 # the big algebra, so just put everything into a list
1545 # and let span() get rid of the excess.
1547 # We do the extra ambient_vector_space() in case we're messing
1548 # with polynomials and the direct parent is a module.
1549 V
= self
.vector().parent().ambient_vector_space()
1550 return V
.span( (self
**d
).vector() for d
in xrange(V
.dimension()) )
1553 def subalgebra_generated_by(self
):
1555 Return the associative subalgebra of the parent EJA generated
1560 sage: set_random_seed()
1561 sage: x = random_eja().random_element()
1562 sage: x.subalgebra_generated_by().is_associative()
1565 Squaring in the subalgebra should be the same thing as
1566 squaring in the superalgebra::
1568 sage: set_random_seed()
1569 sage: x = random_eja().random_element()
1570 sage: u = x.subalgebra_generated_by().random_element()
1571 sage: u.operator_matrix()*u.vector() == (u**2).vector()
1575 # First get the subspace spanned by the powers of myself...
1576 V
= self
.span_of_powers()
1577 F
= self
.base_ring()
1579 # Now figure out the entries of the right-multiplication
1580 # matrix for the successive basis elements b0, b1,... of
1583 for b_right
in V
.basis():
1584 eja_b_right
= self
.parent()(b_right
)
1586 # The first row of the right-multiplication matrix by
1587 # b1 is what we get if we apply that matrix to b1. The
1588 # second row of the right multiplication matrix by b1
1589 # is what we get when we apply that matrix to b2...
1591 # IMPORTANT: this assumes that all vectors are COLUMN
1592 # vectors, unlike our superclass (which uses row vectors).
1593 for b_left
in V
.basis():
1594 eja_b_left
= self
.parent()(b_left
)
1595 # Multiply in the original EJA, but then get the
1596 # coordinates from the subalgebra in terms of its
1598 this_row
= V
.coordinates((eja_b_left
*eja_b_right
).vector())
1599 b_right_rows
.append(this_row
)
1600 b_right_matrix
= matrix(F
, b_right_rows
)
1601 mats
.append(b_right_matrix
)
1603 # It's an algebra of polynomials in one element, and EJAs
1604 # are power-associative.
1606 # TODO: choose generator names intelligently.
1607 return FiniteDimensionalEuclideanJordanAlgebra(F
, mats
, assume_associative
=True, names
='f')
1610 def subalgebra_idempotent(self
):
1612 Find an idempotent in the associative subalgebra I generate
1613 using Proposition 2.3.5 in Baes.
1617 sage: set_random_seed()
1618 sage: J = random_eja()
1619 sage: x = J.random_element()
1620 sage: while x.is_nilpotent():
1621 ....: x = J.random_element()
1622 sage: c = x.subalgebra_idempotent()
1627 if self
.is_nilpotent():
1628 raise ValueError("this only works with non-nilpotent elements!")
1630 V
= self
.span_of_powers()
1631 J
= self
.subalgebra_generated_by()
1632 # Mis-design warning: the basis used for span_of_powers()
1633 # and subalgebra_generated_by() must be the same, and in
1635 u
= J(V
.coordinates(self
.vector()))
1637 # The image of the matrix of left-u^m-multiplication
1638 # will be minimal for some natural number s...
1640 minimal_dim
= V
.dimension()
1641 for i
in xrange(1, V
.dimension()):
1642 this_dim
= (u
**i
).operator_matrix().image().dimension()
1643 if this_dim
< minimal_dim
:
1644 minimal_dim
= this_dim
1647 # Now minimal_matrix should correspond to the smallest
1648 # non-zero subspace in Baes's (or really, Koecher's)
1651 # However, we need to restrict the matrix to work on the
1652 # subspace... or do we? Can't we just solve, knowing that
1653 # A(c) = u^(s+1) should have a solution in the big space,
1656 # Beware, solve_right() means that we're using COLUMN vectors.
1657 # Our FiniteDimensionalAlgebraElement superclass uses rows.
1659 A
= u_next
.operator_matrix()
1660 c_coordinates
= A
.solve_right(u_next
.vector())
1662 # Now c_coordinates is the idempotent we want, but it's in
1663 # the coordinate system of the subalgebra.
1665 # We need the basis for J, but as elements of the parent algebra.
1667 basis
= [self
.parent(v
) for v
in V
.basis()]
1668 return self
.parent().linear_combination(zip(c_coordinates
, basis
))
1673 Return my trace, the sum of my eigenvalues.
1677 sage: J = JordanSpinEJA(3)
1678 sage: x = sum(J.gens())
1684 sage: J = RealCartesianProductEJA(5)
1685 sage: J.one().trace()
1690 The trace of an element is a real number::
1692 sage: set_random_seed()
1693 sage: J = random_eja()
1694 sage: J.random_element().trace() in J.base_ring()
1700 p
= P
._charpoly
_coeff
(r
-1)
1701 # The _charpoly_coeff function already adds the factor of
1702 # -1 to ensure that _charpoly_coeff(r-1) is really what
1703 # appears in front of t^{r-1} in the charpoly. However,
1704 # we want the negative of THAT for the trace.
1705 return -p(*self
.vector())
1708 def trace_inner_product(self
, other
):
1710 Return the trace inner product of myself and ``other``.
1714 The trace inner product is commutative::
1716 sage: set_random_seed()
1717 sage: J = random_eja()
1718 sage: x = J.random_element(); y = J.random_element()
1719 sage: x.trace_inner_product(y) == y.trace_inner_product(x)
1722 The trace inner product is bilinear::
1724 sage: set_random_seed()
1725 sage: J = random_eja()
1726 sage: x = J.random_element()
1727 sage: y = J.random_element()
1728 sage: z = J.random_element()
1729 sage: a = QQ.random_element();
1730 sage: actual = (a*(x+z)).trace_inner_product(y)
1731 sage: expected = ( a*x.trace_inner_product(y) +
1732 ....: a*z.trace_inner_product(y) )
1733 sage: actual == expected
1735 sage: actual = x.trace_inner_product(a*(y+z))
1736 sage: expected = ( a*x.trace_inner_product(y) +
1737 ....: a*x.trace_inner_product(z) )
1738 sage: actual == expected
1741 The trace inner product satisfies the compatibility
1742 condition in the definition of a Euclidean Jordan algebra::
1744 sage: set_random_seed()
1745 sage: J = random_eja()
1746 sage: x = J.random_element()
1747 sage: y = J.random_element()
1748 sage: z = J.random_element()
1749 sage: (x*y).trace_inner_product(z) == y.trace_inner_product(x*z)
1753 if not other
in self
.parent():
1754 raise TypeError("'other' must live in the same algebra")
1756 return (self
*other
).trace()
1759 class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1761 Return the Euclidean Jordan Algebra corresponding to the set
1762 `R^n` under the Hadamard product.
1764 Note: this is nothing more than the Cartesian product of ``n``
1765 copies of the spin algebra. Once Cartesian product algebras
1766 are implemented, this can go.
1770 This multiplication table can be verified by hand::
1772 sage: J = RealCartesianProductEJA(3)
1773 sage: e0,e1,e2 = J.gens()
1789 def __classcall_private__(cls
, n
, field
=QQ
):
1790 # The FiniteDimensionalAlgebra constructor takes a list of
1791 # matrices, the ith representing right multiplication by the ith
1792 # basis element in the vector space. So if e_1 = (1,0,0), then
1793 # right (Hadamard) multiplication of x by e_1 picks out the first
1794 # component of x; and likewise for the ith basis element e_i.
1795 Qs
= [ matrix(field
, n
, n
, lambda k
,j
: 1*(k
== j
== i
))
1796 for i
in xrange(n
) ]
1798 fdeja
= super(RealCartesianProductEJA
, cls
)
1799 return fdeja
.__classcall
_private
__(cls
, field
, Qs
, rank
=n
)
1801 def inner_product(self
, x
, y
):
1802 return _usual_ip(x
,y
)
1807 Return a "random" finite-dimensional Euclidean Jordan Algebra.
1811 For now, we choose a random natural number ``n`` (greater than zero)
1812 and then give you back one of the following:
1814 * The cartesian product of the rational numbers ``n`` times; this is
1815 ``QQ^n`` with the Hadamard product.
1817 * The Jordan spin algebra on ``QQ^n``.
1819 * The ``n``-by-``n`` rational symmetric matrices with the symmetric
1822 * The ``n``-by-``n`` complex-rational Hermitian matrices embedded
1823 in the space of ``2n``-by-``2n`` real symmetric matrices.
1825 * The ``n``-by-``n`` quaternion-rational Hermitian matrices embedded
1826 in the space of ``4n``-by-``4n`` real symmetric matrices.
1828 Later this might be extended to return Cartesian products of the
1834 Euclidean Jordan algebra of degree...
1838 # The max_n component lets us choose different upper bounds on the
1839 # value "n" that gets passed to the constructor. This is needed
1840 # because e.g. R^{10} is reasonable to test, while the Hermitian
1841 # 10-by-10 quaternion matrices are not.
1842 (constructor
, max_n
) = choice([(RealCartesianProductEJA
, 6),
1844 (RealSymmetricEJA
, 5),
1845 (ComplexHermitianEJA
, 4),
1846 (QuaternionHermitianEJA
, 3)])
1847 n
= ZZ
.random_element(1, max_n
)
1848 return constructor(n
, field
=QQ
)
1852 def _real_symmetric_basis(n
, field
=QQ
):
1854 Return a basis for the space of real symmetric n-by-n matrices.
1856 # The basis of symmetric matrices, as matrices, in their R^(n-by-n)
1860 for j
in xrange(i
+1):
1861 Eij
= matrix(field
, n
, lambda k
,l
: k
==i
and l
==j
)
1865 # Beware, orthogonal but not normalized!
1866 Sij
= Eij
+ Eij
.transpose()
1871 def _complex_hermitian_basis(n
, field
=QQ
):
1873 Returns a basis for the space of complex Hermitian n-by-n matrices.
1877 sage: set_random_seed()
1878 sage: n = ZZ.random_element(1,5)
1879 sage: all( M.is_symmetric() for M in _complex_hermitian_basis(n) )
1883 F
= QuadraticField(-1, 'I')
1886 # This is like the symmetric case, but we need to be careful:
1888 # * We want conjugate-symmetry, not just symmetry.
1889 # * The diagonal will (as a result) be real.
1893 for j
in xrange(i
+1):
1894 Eij
= matrix(field
, n
, lambda k
,l
: k
==i
and l
==j
)
1896 Sij
= _embed_complex_matrix(Eij
)
1899 # Beware, orthogonal but not normalized! The second one
1900 # has a minus because it's conjugated.
1901 Sij_real
= _embed_complex_matrix(Eij
+ Eij
.transpose())
1903 Sij_imag
= _embed_complex_matrix(I
*Eij
- I
*Eij
.transpose())
1908 def _quaternion_hermitian_basis(n
, field
=QQ
):
1910 Returns a basis for the space of quaternion Hermitian n-by-n matrices.
1914 sage: set_random_seed()
1915 sage: n = ZZ.random_element(1,5)
1916 sage: all( M.is_symmetric() for M in _quaternion_hermitian_basis(n) )
1920 Q
= QuaternionAlgebra(QQ
,-1,-1)
1923 # This is like the symmetric case, but we need to be careful:
1925 # * We want conjugate-symmetry, not just symmetry.
1926 # * The diagonal will (as a result) be real.
1930 for j
in xrange(i
+1):
1931 Eij
= matrix(Q
, n
, lambda k
,l
: k
==i
and l
==j
)
1933 Sij
= _embed_quaternion_matrix(Eij
)
1936 # Beware, orthogonal but not normalized! The second,
1937 # third, and fourth ones have a minus because they're
1939 Sij_real
= _embed_quaternion_matrix(Eij
+ Eij
.transpose())
1941 Sij_I
= _embed_quaternion_matrix(I
*Eij
- I
*Eij
.transpose())
1943 Sij_J
= _embed_quaternion_matrix(J
*Eij
- J
*Eij
.transpose())
1945 Sij_K
= _embed_quaternion_matrix(K
*Eij
- K
*Eij
.transpose())
1951 return vector(m
.base_ring(), m
.list())
1954 return matrix(v
.base_ring(), sqrt(v
.degree()), v
.list())
1956 def _multiplication_table_from_matrix_basis(basis
):
1958 At least three of the five simple Euclidean Jordan algebras have the
1959 symmetric multiplication (A,B) |-> (AB + BA)/2, where the
1960 multiplication on the right is matrix multiplication. Given a basis
1961 for the underlying matrix space, this function returns a
1962 multiplication table (obtained by looping through the basis
1963 elements) for an algebra of those matrices. A reordered copy
1964 of the basis is also returned to work around the fact that
1965 the ``span()`` in this function will change the order of the basis
1966 from what we think it is, to... something else.
1968 # In S^2, for example, we nominally have four coordinates even
1969 # though the space is of dimension three only. The vector space V
1970 # is supposed to hold the entire long vector, and the subspace W
1971 # of V will be spanned by the vectors that arise from symmetric
1972 # matrices. Thus for S^2, dim(V) == 4 and dim(W) == 3.
1973 field
= basis
[0].base_ring()
1974 dimension
= basis
[0].nrows()
1976 V
= VectorSpace(field
, dimension
**2)
1977 W
= V
.span( _mat2vec(s
) for s
in basis
)
1979 # Taking the span above reorders our basis (thanks, jerk!) so we
1980 # need to put our "matrix basis" in the same order as the
1981 # (reordered) vector basis.
1982 S
= tuple( _vec2mat(b
) for b
in W
.basis() )
1986 # Brute force the multiplication-by-s matrix by looping
1987 # through all elements of the basis and doing the computation
1988 # to find out what the corresponding row should be. BEWARE:
1989 # these multiplication tables won't be symmetric! It therefore
1990 # becomes REALLY IMPORTANT that the underlying algebra
1991 # constructor uses ROW vectors and not COLUMN vectors. That's
1992 # why we're computing rows here and not columns.
1995 this_row
= _mat2vec((s
*t
+ t
*s
)/2)
1996 Q_rows
.append(W
.coordinates(this_row
))
1997 Q
= matrix(field
, W
.dimension(), Q_rows
)
2003 def _embed_complex_matrix(M
):
2005 Embed the n-by-n complex matrix ``M`` into the space of real
2006 matrices of size 2n-by-2n via the map the sends each entry `z = a +
2007 bi` to the block matrix ``[[a,b],[-b,a]]``.
2011 sage: F = QuadraticField(-1,'i')
2012 sage: x1 = F(4 - 2*i)
2013 sage: x2 = F(1 + 2*i)
2016 sage: M = matrix(F,2,[[x1,x2],[x3,x4]])
2017 sage: _embed_complex_matrix(M)
2026 Embedding is a homomorphism (isomorphism, in fact)::
2028 sage: set_random_seed()
2029 sage: n = ZZ.random_element(5)
2030 sage: F = QuadraticField(-1, 'i')
2031 sage: X = random_matrix(F, n)
2032 sage: Y = random_matrix(F, n)
2033 sage: actual = _embed_complex_matrix(X) * _embed_complex_matrix(Y)
2034 sage: expected = _embed_complex_matrix(X*Y)
2035 sage: actual == expected
2041 raise ValueError("the matrix 'M' must be square")
2042 field
= M
.base_ring()
2047 blocks
.append(matrix(field
, 2, [[a
,b
],[-b
,a
]]))
2049 # We can drop the imaginaries here.
2050 return block_matrix(field
.base_ring(), n
, blocks
)
2053 def _unembed_complex_matrix(M
):
2055 The inverse of _embed_complex_matrix().
2059 sage: A = matrix(QQ,[ [ 1, 2, 3, 4],
2060 ....: [-2, 1, -4, 3],
2061 ....: [ 9, 10, 11, 12],
2062 ....: [-10, 9, -12, 11] ])
2063 sage: _unembed_complex_matrix(A)
2065 [ 10*i + 9 12*i + 11]
2069 Unembedding is the inverse of embedding::
2071 sage: set_random_seed()
2072 sage: F = QuadraticField(-1, 'i')
2073 sage: M = random_matrix(F, 3)
2074 sage: _unembed_complex_matrix(_embed_complex_matrix(M)) == M
2080 raise ValueError("the matrix 'M' must be square")
2081 if not n
.mod(2).is_zero():
2082 raise ValueError("the matrix 'M' must be a complex embedding")
2084 F
= QuadraticField(-1, 'i')
2087 # Go top-left to bottom-right (reading order), converting every
2088 # 2-by-2 block we see to a single complex element.
2090 for k
in xrange(n
/2):
2091 for j
in xrange(n
/2):
2092 submat
= M
[2*k
:2*k
+2,2*j
:2*j
+2]
2093 if submat
[0,0] != submat
[1,1]:
2094 raise ValueError('bad on-diagonal submatrix')
2095 if submat
[0,1] != -submat
[1,0]:
2096 raise ValueError('bad off-diagonal submatrix')
2097 z
= submat
[0,0] + submat
[0,1]*i
2100 return matrix(F
, n
/2, elements
)
2103 def _embed_quaternion_matrix(M
):
2105 Embed the n-by-n quaternion matrix ``M`` into the space of real
2106 matrices of size 4n-by-4n by first sending each quaternion entry
2107 `z = a + bi + cj + dk` to the block-complex matrix
2108 ``[[a + bi, c+di],[-c + di, a-bi]]`, and then embedding those into
2113 sage: Q = QuaternionAlgebra(QQ,-1,-1)
2114 sage: i,j,k = Q.gens()
2115 sage: x = 1 + 2*i + 3*j + 4*k
2116 sage: M = matrix(Q, 1, [[x]])
2117 sage: _embed_quaternion_matrix(M)
2123 Embedding is a homomorphism (isomorphism, in fact)::
2125 sage: set_random_seed()
2126 sage: n = ZZ.random_element(5)
2127 sage: Q = QuaternionAlgebra(QQ,-1,-1)
2128 sage: X = random_matrix(Q, n)
2129 sage: Y = random_matrix(Q, n)
2130 sage: actual = _embed_quaternion_matrix(X)*_embed_quaternion_matrix(Y)
2131 sage: expected = _embed_quaternion_matrix(X*Y)
2132 sage: actual == expected
2136 quaternions
= M
.base_ring()
2139 raise ValueError("the matrix 'M' must be square")
2141 F
= QuadraticField(-1, 'i')
2146 t
= z
.coefficient_tuple()
2151 cplx_matrix
= matrix(F
, 2, [[ a
+ b
*i
, c
+ d
*i
],
2152 [-c
+ d
*i
, a
- b
*i
]])
2153 blocks
.append(_embed_complex_matrix(cplx_matrix
))
2155 # We should have real entries by now, so use the realest field
2156 # we've got for the return value.
2157 return block_matrix(quaternions
.base_ring(), n
, blocks
)
2160 def _unembed_quaternion_matrix(M
):
2162 The inverse of _embed_quaternion_matrix().
2166 sage: M = matrix(QQ, [[ 1, 2, 3, 4],
2167 ....: [-2, 1, -4, 3],
2168 ....: [-3, 4, 1, -2],
2169 ....: [-4, -3, 2, 1]])
2170 sage: _unembed_quaternion_matrix(M)
2171 [1 + 2*i + 3*j + 4*k]
2175 Unembedding is the inverse of embedding::
2177 sage: set_random_seed()
2178 sage: Q = QuaternionAlgebra(QQ, -1, -1)
2179 sage: M = random_matrix(Q, 3)
2180 sage: _unembed_quaternion_matrix(_embed_quaternion_matrix(M)) == M
2186 raise ValueError("the matrix 'M' must be square")
2187 if not n
.mod(4).is_zero():
2188 raise ValueError("the matrix 'M' must be a complex embedding")
2190 Q
= QuaternionAlgebra(QQ
,-1,-1)
2193 # Go top-left to bottom-right (reading order), converting every
2194 # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1
2197 for l
in xrange(n
/4):
2198 for m
in xrange(n
/4):
2199 submat
= _unembed_complex_matrix(M
[4*l
:4*l
+4,4*m
:4*m
+4])
2200 if submat
[0,0] != submat
[1,1].conjugate():
2201 raise ValueError('bad on-diagonal submatrix')
2202 if submat
[0,1] != -submat
[1,0].conjugate():
2203 raise ValueError('bad off-diagonal submatrix')
2204 z
= submat
[0,0].real() + submat
[0,0].imag()*i
2205 z
+= submat
[0,1].real()*j
+ submat
[0,1].imag()*k
2208 return matrix(Q
, n
/4, elements
)
2211 # The usual inner product on R^n.
2213 return x
.vector().inner_product(y
.vector())
2215 # The inner product used for the real symmetric simple EJA.
2216 # We keep it as a separate function because e.g. the complex
2217 # algebra uses the same inner product, except divided by 2.
2218 def _matrix_ip(X
,Y
):
2219 X_mat
= X
.natural_representation()
2220 Y_mat
= Y
.natural_representation()
2221 return (X_mat
*Y_mat
).trace()
2224 class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra
):
2226 The rank-n simple EJA consisting of real symmetric n-by-n
2227 matrices, the usual symmetric Jordan product, and the trace inner
2228 product. It has dimension `(n^2 + n)/2` over the reals.
2232 sage: J = RealSymmetricEJA(2)
2233 sage: e0, e1, e2 = J.gens()
2243 The degree of this algebra is `(n^2 + n) / 2`::
2245 sage: set_random_seed()
2246 sage: n = ZZ.random_element(1,5)
2247 sage: J = RealSymmetricEJA(n)
2248 sage: J.degree() == (n^2 + n)/2
2251 The Jordan multiplication is what we think it is::
2253 sage: set_random_seed()
2254 sage: n = ZZ.random_element(1,5)
2255 sage: J = RealSymmetricEJA(n)
2256 sage: x = J.random_element()
2257 sage: y = J.random_element()
2258 sage: actual = (x*y).natural_representation()
2259 sage: X = x.natural_representation()
2260 sage: Y = y.natural_representation()
2261 sage: expected = (X*Y + Y*X)/2
2262 sage: actual == expected
2264 sage: J(expected) == x*y
2269 def __classcall_private__(cls
, n
, field
=QQ
):
2270 S
= _real_symmetric_basis(n
, field
=field
)
2271 (Qs
, T
) = _multiplication_table_from_matrix_basis(S
)
2273 fdeja
= super(RealSymmetricEJA
, cls
)
2274 return fdeja
.__classcall
_private
__(cls
,
2280 def inner_product(self
, x
, y
):
2281 return _matrix_ip(x
,y
)
2284 class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra
):
2286 The rank-n simple EJA consisting of complex Hermitian n-by-n
2287 matrices over the real numbers, the usual symmetric Jordan product,
2288 and the real-part-of-trace inner product. It has dimension `n^2` over
2293 The degree of this algebra is `n^2`::
2295 sage: set_random_seed()
2296 sage: n = ZZ.random_element(1,5)
2297 sage: J = ComplexHermitianEJA(n)
2298 sage: J.degree() == n^2
2301 The Jordan multiplication is what we think it is::
2303 sage: set_random_seed()
2304 sage: n = ZZ.random_element(1,5)
2305 sage: J = ComplexHermitianEJA(n)
2306 sage: x = J.random_element()
2307 sage: y = J.random_element()
2308 sage: actual = (x*y).natural_representation()
2309 sage: X = x.natural_representation()
2310 sage: Y = y.natural_representation()
2311 sage: expected = (X*Y + Y*X)/2
2312 sage: actual == expected
2314 sage: J(expected) == x*y
2319 def __classcall_private__(cls
, n
, field
=QQ
):
2320 S
= _complex_hermitian_basis(n
)
2321 (Qs
, T
) = _multiplication_table_from_matrix_basis(S
)
2323 fdeja
= super(ComplexHermitianEJA
, cls
)
2324 return fdeja
.__classcall
_private
__(cls
,
2330 def inner_product(self
, x
, y
):
2331 # Since a+bi on the diagonal is represented as
2336 # we'll double-count the "a" entries if we take the trace of
2338 return _matrix_ip(x
,y
)/2
2341 class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra
):
2343 The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
2344 matrices, the usual symmetric Jordan product, and the
2345 real-part-of-trace inner product. It has dimension `2n^2 - n` over
2350 The degree of this algebra is `n^2`::
2352 sage: set_random_seed()
2353 sage: n = ZZ.random_element(1,5)
2354 sage: J = QuaternionHermitianEJA(n)
2355 sage: J.degree() == 2*(n^2) - n
2358 The Jordan multiplication is what we think it is::
2360 sage: set_random_seed()
2361 sage: n = ZZ.random_element(1,5)
2362 sage: J = QuaternionHermitianEJA(n)
2363 sage: x = J.random_element()
2364 sage: y = J.random_element()
2365 sage: actual = (x*y).natural_representation()
2366 sage: X = x.natural_representation()
2367 sage: Y = y.natural_representation()
2368 sage: expected = (X*Y + Y*X)/2
2369 sage: actual == expected
2371 sage: J(expected) == x*y
2376 def __classcall_private__(cls
, n
, field
=QQ
):
2377 S
= _quaternion_hermitian_basis(n
)
2378 (Qs
, T
) = _multiplication_table_from_matrix_basis(S
)
2380 fdeja
= super(QuaternionHermitianEJA
, cls
)
2381 return fdeja
.__classcall
_private
__(cls
,
2387 def inner_product(self
, x
, y
):
2388 # Since a+bi+cj+dk on the diagonal is represented as
2390 # a + bi +cj + dk = [ a b c d]
2395 # we'll quadruple-count the "a" entries if we take the trace of
2397 return _matrix_ip(x
,y
)/4
2400 class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra
):
2402 The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
2403 with the usual inner product and jordan product ``x*y =
2404 (<x_bar,y_bar>, x0*y_bar + y0*x_bar)``. It has dimension `n` over
2409 This multiplication table can be verified by hand::
2411 sage: J = JordanSpinEJA(4)
2412 sage: e0,e1,e2,e3 = J.gens()
2430 def __classcall_private__(cls
, n
, field
=QQ
):
2432 id_matrix
= identity_matrix(field
, n
)
2434 ei
= id_matrix
.column(i
)
2435 Qi
= zero_matrix(field
, n
)
2437 Qi
.set_column(0, ei
)
2438 Qi
+= diagonal_matrix(n
, [ei
[0]]*n
)
2439 # The addition of the diagonal matrix adds an extra ei[0] in the
2440 # upper-left corner of the matrix.
2441 Qi
[0,0] = Qi
[0,0] * ~
field(2)
2444 # The rank of the spin algebra is two, unless we're in a
2445 # one-dimensional ambient space (because the rank is bounded by
2446 # the ambient dimension).
2447 fdeja
= super(JordanSpinEJA
, cls
)
2448 return fdeja
.__classcall
_private
__(cls
, field
, Qs
, rank
=min(n
,2))
2450 def inner_product(self
, x
, y
):
2451 return _usual_ip(x
,y
)