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
.magmatic_algebras
import MagmaticAlgebras
9 from sage
.structure
.element
import is_Matrix
10 from sage
.structure
.category_object
import normalize_names
12 from sage
.algebras
.finite_dimensional_algebras
.finite_dimensional_algebra
import FiniteDimensionalAlgebra
13 from sage
.algebras
.finite_dimensional_algebras
.finite_dimensional_algebra_element
import FiniteDimensionalAlgebraElement
14 from sage
.algebras
.finite_dimensional_algebras
.finite_dimensional_algebra_morphism
import FiniteDimensionalAlgebraMorphism
17 class FiniteDimensionalEuclideanJordanAlgebraMorphism(FiniteDimensionalAlgebraMorphism
):
19 A linear map between two finite-dimensional EJAs.
21 This is a very thin wrapper around FiniteDimensionalAlgebraMorphism
22 that does only a few things:
24 1. Avoids the ``unitary`` and ``check`` arguments to the constructor
25 that will always be ``False``. This is necessary because these
26 are homomorphisms with respect to ADDITION, but the SageMath
27 machinery wants to check that they're homomorphisms with respect
28 to (Jordan) MULTIPLICATION. That obviously doesn't work.
30 2. Inputs and outputs the underlying matrix with respect to COLUMN
31 vectors, unlike the parent class.
33 3. Allows us to add morphisms in the obvious way.
35 4. Allows us to invert morphisms.
37 If this seems a bit heavyweight, it is. I would have been happy to
38 use a the ring morphism that underlies the finite-dimensional
39 algebra morphism, but they don't seem to be callable on elements of
40 our EJA, and you can't add/invert them.
43 def __add__(self
, other
):
45 Add two EJA morphisms in the obvious way.
49 sage: J = RealSymmetricEJA(3)
52 sage: x.operator() + y.operator()
53 Morphism from Euclidean Jordan algebra of degree 6 over Rational
54 Field to Euclidean Jordan algebra of degree 6 over Rational Field
65 sage: set_random_seed()
66 sage: J = random_eja()
67 sage: x = J.random_element()
68 sage: y = J.random_element()
69 sage: (x.operator() + y.operator()) in J.Hom(J)
75 raise ValueError("summands must live in the same space")
77 return FiniteDimensionalEuclideanJordanAlgebraMorphism(
79 self
.matrix() + other
.matrix() )
82 def __init__(self
, parent
, f
):
83 FiniteDimensionalAlgebraMorphism
.__init
__(self
,
94 sage: J = RealSymmetricEJA(2)
95 sage: x = J.linear_combination(zip(range(len(J.gens())), J.gens()))
96 sage: x.is_invertible()
99 Morphism from Euclidean Jordan algebra of degree 3 over Rational
100 Field to Euclidean Jordan algebra of degree 3 over Rational Field
105 sage: x.operator_matrix().inverse()
112 sage: set_random_seed()
113 sage: J = random_eja()
114 sage: x = J.random_element()
115 sage: not x.is_invertible() or (
116 ....: (~x.operator()).matrix() == x.operator_matrix().inverse() )
121 if not A
.is_invertible():
122 raise ValueError("morphism is not invertible")
125 return FiniteDimensionalEuclideanJordanAlgebraMorphism(self
.parent(),
130 We override only the representation that is shown to the user,
131 because we want the matrix to be with respect to COLUMN vectors.
135 Ensure that we see the transpose of the underlying matrix object:
137 sage: J = RealSymmetricEJA(3)
138 sage: x = J.linear_combination(zip(range(len(J.gens())), J.gens()))
139 sage: L = x.operator()
141 Morphism from Euclidean Jordan algebra of degree 6 over Rational
142 Field to Euclidean Jordan algebra of degree 6 over Rational Field
159 return "Morphism from {} to {} given by matrix\n{}".format(
160 self
.domain(), self
.codomain(), self
.matrix())
164 Return the matrix of this morphism with respect to a left-action
167 return FiniteDimensionalAlgebraMorphism
.matrix(self
).transpose()
170 class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra
):
172 def __classcall_private__(cls
,
176 assume_associative
=False,
181 mult_table
= [b
.base_extend(field
) for b
in mult_table
]
184 if not (is_Matrix(b
) and b
.dimensions() == (n
, n
)):
185 raise ValueError("input is not a multiplication table")
186 mult_table
= tuple(mult_table
)
188 cat
= MagmaticAlgebras(field
).FiniteDimensional().WithBasis()
189 cat
.or_subcategory(category
)
190 if assume_associative
:
191 cat
= cat
.Associative()
193 names
= normalize_names(n
, names
)
195 fda
= super(FiniteDimensionalEuclideanJordanAlgebra
, cls
)
196 return fda
.__classcall
__(cls
,
199 assume_associative
=assume_associative
,
203 natural_basis
=natural_basis
)
210 assume_associative
=False,
217 By definition, Jordan multiplication commutes::
219 sage: set_random_seed()
220 sage: J = random_eja()
221 sage: x = J.random_element()
222 sage: y = J.random_element()
228 self
._natural
_basis
= natural_basis
229 self
._multiplication
_table
= mult_table
230 fda
= super(FiniteDimensionalEuclideanJordanAlgebra
, self
)
239 Return a string representation of ``self``.
241 fmt
= "Euclidean Jordan algebra of degree {} over {}"
242 return fmt
.format(self
.degree(), self
.base_ring())
245 def _a_regular_element(self
):
247 Guess a regular element. Needed to compute the basis for our
248 characteristic polynomial coefficients.
251 z
= self
.sum( (i
+1)*gs
[i
] for i
in range(len(gs
)) )
252 if not z
.is_regular():
253 raise ValueError("don't know a regular element")
258 def _charpoly_basis_space(self
):
260 Return the vector space spanned by the basis used in our
261 characteristic polynomial coefficients. This is used not only to
262 compute those coefficients, but also any time we need to
263 evaluate the coefficients (like when we compute the trace or
266 z
= self
._a
_regular
_element
()
267 V
= z
.vector().parent().ambient_vector_space()
268 V1
= V
.span_of_basis( (z
**k
).vector() for k
in range(self
.rank()) )
269 b
= (V1
.basis() + V1
.complement().basis())
270 return V
.span_of_basis(b
)
274 def _charpoly_coeff(self
, i
):
276 Return the coefficient polynomial "a_{i}" of this algebra's
277 general characteristic polynomial.
279 Having this be a separate cached method lets us compute and
280 store the trace/determinant (a_{r-1} and a_{0} respectively)
281 separate from the entire characteristic polynomial.
283 (A_of_x
, x
, xr
, detA
) = self
._charpoly
_matrix
_system
()
284 R
= A_of_x
.base_ring()
286 # Guaranteed by theory
289 # Danger: the in-place modification is done for performance
290 # reasons (reconstructing a matrix with huge polynomial
291 # entries is slow), but I don't know how cached_method works,
292 # so it's highly possible that we're modifying some global
293 # list variable by reference, here. In other words, you
294 # probably shouldn't call this method twice on the same
295 # algebra, at the same time, in two threads
296 Ai_orig
= A_of_x
.column(i
)
297 A_of_x
.set_column(i
,xr
)
298 numerator
= A_of_x
.det()
299 A_of_x
.set_column(i
,Ai_orig
)
301 # We're relying on the theory here to ensure that each a_i is
302 # indeed back in R, and the added negative signs are to make
303 # the whole charpoly expression sum to zero.
304 return R(-numerator
/detA
)
308 def _charpoly_matrix_system(self
):
310 Compute the matrix whose entries A_ij are polynomials in
311 X1,...,XN, the vector ``x`` of variables X1,...,XN, the vector
312 corresponding to `x^r` and the determinent of the matrix A =
313 [A_ij]. In other words, all of the fixed (cachable) data needed
314 to compute the coefficients of the characteristic polynomial.
319 # Construct a new algebra over a multivariate polynomial ring...
320 names
= ['X' + str(i
) for i
in range(1,n
+1)]
321 R
= PolynomialRing(self
.base_ring(), names
)
322 J
= FiniteDimensionalEuclideanJordanAlgebra(R
,
323 self
._multiplication
_table
,
326 idmat
= identity_matrix(J
.base_ring(), n
)
328 W
= self
._charpoly
_basis
_space
()
329 W
= W
.change_ring(R
.fraction_field())
331 # Starting with the standard coordinates x = (X1,X2,...,Xn)
332 # and then converting the entries to W-coordinates allows us
333 # to pass in the standard coordinates to the charpoly and get
334 # back the right answer. Specifically, with x = (X1,X2,...,Xn),
337 # W.coordinates(x^2) eval'd at (standard z-coords)
341 # W-coords of (standard coords of x^2 eval'd at std-coords of z)
343 # We want the middle equivalent thing in our matrix, but use
344 # the first equivalent thing instead so that we can pass in
345 # standard coordinates.
346 x
= J(vector(R
, R
.gens()))
347 l1
= [column_matrix(W
.coordinates((x
**k
).vector())) for k
in range(r
)]
348 l2
= [idmat
.column(k
-1).column() for k
in range(r
+1, n
+1)]
349 A_of_x
= block_matrix(R
, 1, n
, (l1
+ l2
))
350 xr
= W
.coordinates((x
**r
).vector())
351 return (A_of_x
, x
, xr
, A_of_x
.det())
355 def characteristic_polynomial(self
):
360 This implementation doesn't guarantee that the polynomial
361 denominator in the coefficients is not identically zero, so
362 theoretically it could crash. The way that this is handled
363 in e.g. Faraut and Koranyi is to use a basis that guarantees
364 the denominator is non-zero. But, doing so requires knowledge
365 of at least one regular element, and we don't even know how
366 to do that. The trade-off is that, if we use the standard basis,
367 the resulting polynomial will accept the "usual" coordinates. In
368 other words, we don't have to do a change of basis before e.g.
369 computing the trace or determinant.
373 The characteristic polynomial in the spin algebra is given in
374 Alizadeh, Example 11.11::
376 sage: J = JordanSpinEJA(3)
377 sage: p = J.characteristic_polynomial(); p
378 X1^2 - X2^2 - X3^2 + (-2*t)*X1 + t^2
379 sage: xvec = J.one().vector()
387 # The list of coefficient polynomials a_1, a_2, ..., a_n.
388 a
= [ self
._charpoly
_coeff
(i
) for i
in range(n
) ]
390 # We go to a bit of trouble here to reorder the
391 # indeterminates, so that it's easier to evaluate the
392 # characteristic polynomial at x's coordinates and get back
393 # something in terms of t, which is what we want.
395 S
= PolynomialRing(self
.base_ring(),'t')
397 S
= PolynomialRing(S
, R
.variable_names())
400 # Note: all entries past the rth should be zero. The
401 # coefficient of the highest power (x^r) is 1, but it doesn't
402 # appear in the solution vector which contains coefficients
403 # for the other powers (to make them sum to x^r).
405 a
[r
] = 1 # corresponds to x^r
407 # When the rank is equal to the dimension, trying to
408 # assign a[r] goes out-of-bounds.
409 a
.append(1) # corresponds to x^r
411 return sum( a
[k
]*(t
**k
) for k
in range(len(a
)) )
414 def inner_product(self
, x
, y
):
416 The inner product associated with this Euclidean Jordan algebra.
418 Defaults to the trace inner product, but can be overridden by
419 subclasses if they are sure that the necessary properties are
424 The inner product must satisfy its axiom for this algebra to truly
425 be a Euclidean Jordan Algebra::
427 sage: set_random_seed()
428 sage: J = random_eja()
429 sage: x = J.random_element()
430 sage: y = J.random_element()
431 sage: z = J.random_element()
432 sage: (x*y).inner_product(z) == y.inner_product(x*z)
436 if (not x
in self
) or (not y
in self
):
437 raise TypeError("arguments must live in this algebra")
438 return x
.trace_inner_product(y
)
441 def natural_basis(self
):
443 Return a more-natural representation of this algebra's basis.
445 Every finite-dimensional Euclidean Jordan Algebra is a direct
446 sum of five simple algebras, four of which comprise Hermitian
447 matrices. This method returns the original "natural" basis
448 for our underlying vector space. (Typically, the natural basis
449 is used to construct the multiplication table in the first place.)
451 Note that this will always return a matrix. The standard basis
452 in `R^n` will be returned as `n`-by-`1` column matrices.
456 sage: J = RealSymmetricEJA(2)
459 sage: J.natural_basis()
467 sage: J = JordanSpinEJA(2)
470 sage: J.natural_basis()
477 if self
._natural
_basis
is None:
478 return tuple( b
.vector().column() for b
in self
.basis() )
480 return self
._natural
_basis
485 Return the rank of this EJA.
487 if self
._rank
is None:
488 raise ValueError("no rank specified at genesis")
493 class Element(FiniteDimensionalAlgebraElement
):
495 An element of a Euclidean Jordan algebra.
500 Oh man, I should not be doing this. This hides the "disabled"
501 methods ``left_matrix`` and ``matrix`` from introspection;
502 in particular it removes them from tab-completion.
504 return filter(lambda s
: s
not in ['left_matrix', 'matrix'],
505 dir(self
.__class
__) )
508 def __init__(self
, A
, elt
=None):
512 The identity in `S^n` is converted to the identity in the EJA::
514 sage: J = RealSymmetricEJA(3)
515 sage: I = identity_matrix(QQ,3)
516 sage: J(I) == J.one()
519 This skew-symmetric matrix can't be represented in the EJA::
521 sage: J = RealSymmetricEJA(3)
522 sage: A = matrix(QQ,3, lambda i,j: i-j)
524 Traceback (most recent call last):
526 ArithmeticError: vector is not in free module
529 # Goal: if we're given a matrix, and if it lives in our
530 # parent algebra's "natural ambient space," convert it
531 # into an algebra element.
533 # The catch is, we make a recursive call after converting
534 # the given matrix into a vector that lives in the algebra.
535 # This we need to try the parent class initializer first,
536 # to avoid recursing forever if we're given something that
537 # already fits into the algebra, but also happens to live
538 # in the parent's "natural ambient space" (this happens with
541 FiniteDimensionalAlgebraElement
.__init
__(self
, A
, elt
)
543 natural_basis
= A
.natural_basis()
544 if elt
in natural_basis
[0].matrix_space():
545 # Thanks for nothing! Matrix spaces aren't vector
546 # spaces in Sage, so we have to figure out its
547 # natural-basis coordinates ourselves.
548 V
= VectorSpace(elt
.base_ring(), elt
.nrows()**2)
549 W
= V
.span( _mat2vec(s
) for s
in natural_basis
)
550 coords
= W
.coordinates(_mat2vec(elt
))
551 FiniteDimensionalAlgebraElement
.__init
__(self
, A
, coords
)
553 def __pow__(self
, n
):
555 Return ``self`` raised to the power ``n``.
557 Jordan algebras are always power-associative; see for
558 example Faraut and Koranyi, Proposition II.1.2 (ii).
562 We have to override this because our superclass uses row vectors
563 instead of column vectors! We, on the other hand, assume column
568 sage: set_random_seed()
569 sage: x = random_eja().random_element()
570 sage: x.operator_matrix()*x.vector() == (x^2).vector()
573 A few examples of power-associativity::
575 sage: set_random_seed()
576 sage: x = random_eja().random_element()
577 sage: x*(x*x)*(x*x) == x^5
579 sage: (x*x)*(x*x*x) == x^5
582 We also know that powers operator-commute (Koecher, Chapter
585 sage: set_random_seed()
586 sage: x = random_eja().random_element()
587 sage: m = ZZ.random_element(0,10)
588 sage: n = ZZ.random_element(0,10)
589 sage: Lxm = (x^m).operator_matrix()
590 sage: Lxn = (x^n).operator_matrix()
591 sage: Lxm*Lxn == Lxn*Lxm
601 return A( (self
.operator_matrix()**(n
-1))*self
.vector() )
604 def apply_univariate_polynomial(self
, p
):
606 Apply the univariate polynomial ``p`` to this element.
608 A priori, SageMath won't allow us to apply a univariate
609 polynomial to an element of an EJA, because we don't know
610 that EJAs are rings (they are usually not associative). Of
611 course, we know that EJAs are power-associative, so the
612 operation is ultimately kosher. This function sidesteps
613 the CAS to get the answer we want and expect.
617 sage: R = PolynomialRing(QQ, 't')
619 sage: p = t^4 - t^3 + 5*t - 2
620 sage: J = RealCartesianProductEJA(5)
621 sage: J.one().apply_univariate_polynomial(p) == 3*J.one()
626 We should always get back an element of the algebra::
628 sage: set_random_seed()
629 sage: p = PolynomialRing(QQ, 't').random_element()
630 sage: J = random_eja()
631 sage: x = J.random_element()
632 sage: x.apply_univariate_polynomial(p) in J
636 if len(p
.variables()) > 1:
637 raise ValueError("not a univariate polynomial")
640 # Convert the coeficcients to the parent's base ring,
641 # because a priori they might live in an (unnecessarily)
642 # larger ring for which P.sum() would fail below.
643 cs
= [ R(c
) for c
in p
.coefficients(sparse
=False) ]
644 return P
.sum( cs
[k
]*(self
**k
) for k
in range(len(cs
)) )
647 def characteristic_polynomial(self
):
649 Return the characteristic polynomial of this element.
653 The rank of `R^3` is three, and the minimal polynomial of
654 the identity element is `(t-1)` from which it follows that
655 the characteristic polynomial should be `(t-1)^3`::
657 sage: J = RealCartesianProductEJA(3)
658 sage: J.one().characteristic_polynomial()
659 t^3 - 3*t^2 + 3*t - 1
661 Likewise, the characteristic of the zero element in the
662 rank-three algebra `R^{n}` should be `t^{3}`::
664 sage: J = RealCartesianProductEJA(3)
665 sage: J.zero().characteristic_polynomial()
668 The characteristic polynomial of an element should evaluate
669 to zero on that element::
671 sage: set_random_seed()
672 sage: x = RealCartesianProductEJA(3).random_element()
673 sage: p = x.characteristic_polynomial()
674 sage: x.apply_univariate_polynomial(p)
678 p
= self
.parent().characteristic_polynomial()
679 return p(*self
.vector())
682 def inner_product(self
, other
):
684 Return the parent algebra's inner product of myself and ``other``.
688 The inner product in the Jordan spin algebra is the usual
689 inner product on `R^n` (this example only works because the
690 basis for the Jordan algebra is the standard basis in `R^n`)::
692 sage: J = JordanSpinEJA(3)
693 sage: x = vector(QQ,[1,2,3])
694 sage: y = vector(QQ,[4,5,6])
695 sage: x.inner_product(y)
697 sage: J(x).inner_product(J(y))
700 The inner product on `S^n` is `<X,Y> = trace(X*Y)`, where
701 multiplication is the usual matrix multiplication in `S^n`,
702 so the inner product of the identity matrix with itself
705 sage: J = RealSymmetricEJA(3)
706 sage: J.one().inner_product(J.one())
709 Likewise, the inner product on `C^n` is `<X,Y> =
710 Re(trace(X*Y))`, where we must necessarily take the real
711 part because the product of Hermitian matrices may not be
714 sage: J = ComplexHermitianEJA(3)
715 sage: J.one().inner_product(J.one())
718 Ditto for the quaternions::
720 sage: J = QuaternionHermitianEJA(3)
721 sage: J.one().inner_product(J.one())
726 Ensure that we can always compute an inner product, and that
727 it gives us back a real number::
729 sage: set_random_seed()
730 sage: J = random_eja()
731 sage: x = J.random_element()
732 sage: y = J.random_element()
733 sage: x.inner_product(y) in RR
739 raise TypeError("'other' must live in the same algebra")
741 return P
.inner_product(self
, other
)
744 def operator_commutes_with(self
, other
):
746 Return whether or not this element operator-commutes
751 The definition of a Jordan algebra says that any element
752 operator-commutes with its square::
754 sage: set_random_seed()
755 sage: x = random_eja().random_element()
756 sage: x.operator_commutes_with(x^2)
761 Test Lemma 1 from Chapter III of Koecher::
763 sage: set_random_seed()
764 sage: J = random_eja()
765 sage: u = J.random_element()
766 sage: v = J.random_element()
767 sage: lhs = u.operator_commutes_with(u*v)
768 sage: rhs = v.operator_commutes_with(u^2)
773 if not other
in self
.parent():
774 raise TypeError("'other' must live in the same algebra")
776 A
= self
.operator_matrix()
777 B
= other
.operator_matrix()
783 Return my determinant, the product of my eigenvalues.
787 sage: J = JordanSpinEJA(2)
788 sage: e0,e1 = J.gens()
789 sage: x = sum( J.gens() )
795 sage: J = JordanSpinEJA(3)
796 sage: e0,e1,e2 = J.gens()
797 sage: x = sum( J.gens() )
803 An element is invertible if and only if its determinant is
806 sage: set_random_seed()
807 sage: x = random_eja().random_element()
808 sage: x.is_invertible() == (x.det() != 0)
814 p
= P
._charpoly
_coeff
(0)
815 # The _charpoly_coeff function already adds the factor of
816 # -1 to ensure that _charpoly_coeff(0) is really what
817 # appears in front of t^{0} in the charpoly. However,
818 # we want (-1)^r times THAT for the determinant.
819 return ((-1)**r
)*p(*self
.vector())
824 Return the Jordan-multiplicative inverse of this element.
828 We appeal to the quadratic representation as in Koecher's
829 Theorem 12 in Chapter III, Section 5.
833 The inverse in the spin factor algebra is given in Alizadeh's
836 sage: set_random_seed()
837 sage: n = ZZ.random_element(1,10)
838 sage: J = JordanSpinEJA(n)
839 sage: x = J.random_element()
840 sage: while x.is_zero():
841 ....: x = J.random_element()
842 sage: x_vec = x.vector()
844 sage: x_bar = x_vec[1:]
845 sage: coeff = 1/(x0^2 - x_bar.inner_product(x_bar))
846 sage: inv_vec = x_vec.parent()([x0] + (-x_bar).list())
847 sage: x_inverse = coeff*inv_vec
848 sage: x.inverse() == J(x_inverse)
853 The identity element is its own inverse::
855 sage: set_random_seed()
856 sage: J = random_eja()
857 sage: J.one().inverse() == J.one()
860 If an element has an inverse, it acts like one::
862 sage: set_random_seed()
863 sage: J = random_eja()
864 sage: x = J.random_element()
865 sage: (not x.is_invertible()) or (x.inverse()*x == J.one())
868 The inverse of the inverse is what we started with::
870 sage: set_random_seed()
871 sage: J = random_eja()
872 sage: x = J.random_element()
873 sage: (not x.is_invertible()) or (x.inverse().inverse() == x)
876 The zero element is never invertible::
878 sage: set_random_seed()
879 sage: J = random_eja().zero().inverse()
880 Traceback (most recent call last):
882 ValueError: element is not invertible
885 if not self
.is_invertible():
886 raise ValueError("element is not invertible")
889 return P(self
.quadratic_representation().inverse()*self
.vector())
892 def is_invertible(self
):
894 Return whether or not this element is invertible.
896 We can't use the superclass method because it relies on
897 the algebra being associative.
901 The usual way to do this is to check if the determinant is
902 zero, but we need the characteristic polynomial for the
903 determinant. The minimal polynomial is a lot easier to get,
904 so we use Corollary 2 in Chapter V of Koecher to check
905 whether or not the paren't algebra's zero element is a root
906 of this element's minimal polynomial.
910 The identity element is always invertible::
912 sage: set_random_seed()
913 sage: J = random_eja()
914 sage: J.one().is_invertible()
917 The zero element is never invertible::
919 sage: set_random_seed()
920 sage: J = random_eja()
921 sage: J.zero().is_invertible()
925 zero
= self
.parent().zero()
926 p
= self
.minimal_polynomial()
927 return not (p(zero
) == zero
)
930 def is_nilpotent(self
):
932 Return whether or not some power of this element is zero.
934 The superclass method won't work unless we're in an
935 associative algebra, and we aren't. However, we generate
936 an assocoative subalgebra and we're nilpotent there if and
937 only if we're nilpotent here (probably).
941 The identity element is never nilpotent::
943 sage: set_random_seed()
944 sage: random_eja().one().is_nilpotent()
947 The additive identity is always nilpotent::
949 sage: set_random_seed()
950 sage: random_eja().zero().is_nilpotent()
954 # The element we're going to call "is_nilpotent()" on.
955 # Either myself, interpreted as an element of a finite-
956 # dimensional algebra, or an element of an associative
960 if self
.parent().is_associative():
961 elt
= FiniteDimensionalAlgebraElement(self
.parent(), self
)
963 V
= self
.span_of_powers()
964 assoc_subalg
= self
.subalgebra_generated_by()
965 # Mis-design warning: the basis used for span_of_powers()
966 # and subalgebra_generated_by() must be the same, and in
968 elt
= assoc_subalg(V
.coordinates(self
.vector()))
970 # Recursive call, but should work since elt lives in an
971 # associative algebra.
972 return elt
.is_nilpotent()
975 def is_regular(self
):
977 Return whether or not this is a regular element.
981 The identity element always has degree one, but any element
982 linearly-independent from it is regular::
984 sage: J = JordanSpinEJA(5)
985 sage: J.one().is_regular()
987 sage: e0, e1, e2, e3, e4 = J.gens() # e0 is the identity
988 sage: for x in J.gens():
989 ....: (J.one() + x).is_regular()
997 return self
.degree() == self
.parent().rank()
1002 Compute the degree of this element the straightforward way
1003 according to the definition; by appending powers to a list
1004 and figuring out its dimension (that is, whether or not
1005 they're linearly dependent).
1009 sage: J = JordanSpinEJA(4)
1010 sage: J.one().degree()
1012 sage: e0,e1,e2,e3 = J.gens()
1013 sage: (e0 - e1).degree()
1016 In the spin factor algebra (of rank two), all elements that
1017 aren't multiples of the identity are regular::
1019 sage: set_random_seed()
1020 sage: n = ZZ.random_element(1,10)
1021 sage: J = JordanSpinEJA(n)
1022 sage: x = J.random_element()
1023 sage: x == x.coefficient(0)*J.one() or x.degree() == 2
1027 return self
.span_of_powers().dimension()
1030 def left_matrix(self
):
1032 Our parent class defines ``left_matrix`` and ``matrix``
1033 methods whose names are misleading. We don't want them.
1035 raise NotImplementedError("use operator_matrix() instead")
1037 matrix
= left_matrix
1040 def minimal_polynomial(self
):
1042 Return the minimal polynomial of this element,
1043 as a function of the variable `t`.
1047 We restrict ourselves to the associative subalgebra
1048 generated by this element, and then return the minimal
1049 polynomial of this element's operator matrix (in that
1050 subalgebra). This works by Baes Proposition 2.3.16.
1054 The minimal polynomial of the identity and zero elements are
1057 sage: set_random_seed()
1058 sage: J = random_eja()
1059 sage: J.one().minimal_polynomial()
1061 sage: J.zero().minimal_polynomial()
1064 The degree of an element is (by one definition) the degree
1065 of its minimal polynomial::
1067 sage: set_random_seed()
1068 sage: x = random_eja().random_element()
1069 sage: x.degree() == x.minimal_polynomial().degree()
1072 The minimal polynomial and the characteristic polynomial coincide
1073 and are known (see Alizadeh, Example 11.11) for all elements of
1074 the spin factor algebra that aren't scalar multiples of the
1077 sage: set_random_seed()
1078 sage: n = ZZ.random_element(2,10)
1079 sage: J = JordanSpinEJA(n)
1080 sage: y = J.random_element()
1081 sage: while y == y.coefficient(0)*J.one():
1082 ....: y = J.random_element()
1083 sage: y0 = y.vector()[0]
1084 sage: y_bar = y.vector()[1:]
1085 sage: actual = y.minimal_polynomial()
1086 sage: t = PolynomialRing(J.base_ring(),'t').gen(0)
1087 sage: expected = t^2 - 2*y0*t + (y0^2 - norm(y_bar)^2)
1088 sage: bool(actual == expected)
1091 The minimal polynomial should always kill its element::
1093 sage: set_random_seed()
1094 sage: x = random_eja().random_element()
1095 sage: p = x.minimal_polynomial()
1096 sage: x.apply_univariate_polynomial(p)
1100 V
= self
.span_of_powers()
1101 assoc_subalg
= self
.subalgebra_generated_by()
1102 # Mis-design warning: the basis used for span_of_powers()
1103 # and subalgebra_generated_by() must be the same, and in
1105 elt
= assoc_subalg(V
.coordinates(self
.vector()))
1107 # We get back a symbolic polynomial in 'x' but want a real
1108 # polynomial in 't'.
1109 p_of_x
= elt
.operator_matrix().minimal_polynomial()
1110 return p_of_x
.change_variable_name('t')
1113 def natural_representation(self
):
1115 Return a more-natural representation of this element.
1117 Every finite-dimensional Euclidean Jordan Algebra is a
1118 direct sum of five simple algebras, four of which comprise
1119 Hermitian matrices. This method returns the original
1120 "natural" representation of this element as a Hermitian
1121 matrix, if it has one. If not, you get the usual representation.
1125 sage: J = ComplexHermitianEJA(3)
1128 sage: J.one().natural_representation()
1138 sage: J = QuaternionHermitianEJA(3)
1141 sage: J.one().natural_representation()
1142 [1 0 0 0 0 0 0 0 0 0 0 0]
1143 [0 1 0 0 0 0 0 0 0 0 0 0]
1144 [0 0 1 0 0 0 0 0 0 0 0 0]
1145 [0 0 0 1 0 0 0 0 0 0 0 0]
1146 [0 0 0 0 1 0 0 0 0 0 0 0]
1147 [0 0 0 0 0 1 0 0 0 0 0 0]
1148 [0 0 0 0 0 0 1 0 0 0 0 0]
1149 [0 0 0 0 0 0 0 1 0 0 0 0]
1150 [0 0 0 0 0 0 0 0 1 0 0 0]
1151 [0 0 0 0 0 0 0 0 0 1 0 0]
1152 [0 0 0 0 0 0 0 0 0 0 1 0]
1153 [0 0 0 0 0 0 0 0 0 0 0 1]
1156 B
= self
.parent().natural_basis()
1157 W
= B
[0].matrix_space()
1158 return W
.linear_combination(zip(self
.vector(), B
))
1163 Return the left-multiplication-by-this-element
1164 operator on the ambient algebra.
1168 sage: set_random_seed()
1169 sage: J = random_eja()
1170 sage: x = J.random_element()
1171 sage: y = J.random_element()
1172 sage: x.operator()(y) == x*y
1174 sage: y.operator()(x) == x*y
1179 return FiniteDimensionalEuclideanJordanAlgebraMorphism(
1181 self
.operator_matrix() )
1185 def operator_matrix(self
):
1187 Return the matrix that represents left- (or right-)
1188 multiplication by this element in the parent algebra.
1190 We implement this ourselves to work around the fact that
1191 our parent class represents everything with row vectors.
1195 Test the first polarization identity from my notes, Koecher Chapter
1196 III, or from Baes (2.3)::
1198 sage: set_random_seed()
1199 sage: J = random_eja()
1200 sage: x = J.random_element()
1201 sage: y = J.random_element()
1202 sage: Lx = x.operator_matrix()
1203 sage: Ly = y.operator_matrix()
1204 sage: Lxx = (x*x).operator_matrix()
1205 sage: Lxy = (x*y).operator_matrix()
1206 sage: bool(2*Lx*Lxy + Ly*Lxx == 2*Lxy*Lx + Lxx*Ly)
1209 Test the second polarization identity from my notes or from
1212 sage: set_random_seed()
1213 sage: J = random_eja()
1214 sage: x = J.random_element()
1215 sage: y = J.random_element()
1216 sage: z = J.random_element()
1217 sage: Lx = x.operator_matrix()
1218 sage: Ly = y.operator_matrix()
1219 sage: Lz = z.operator_matrix()
1220 sage: Lzy = (z*y).operator_matrix()
1221 sage: Lxy = (x*y).operator_matrix()
1222 sage: Lxz = (x*z).operator_matrix()
1223 sage: bool(Lx*Lzy + Lz*Lxy + Ly*Lxz == Lzy*Lx + Lxy*Lz + Lxz*Ly)
1226 Test the third polarization identity from my notes or from
1229 sage: set_random_seed()
1230 sage: J = random_eja()
1231 sage: u = J.random_element()
1232 sage: y = J.random_element()
1233 sage: z = J.random_element()
1234 sage: Lu = u.operator_matrix()
1235 sage: Ly = y.operator_matrix()
1236 sage: Lz = z.operator_matrix()
1237 sage: Lzy = (z*y).operator_matrix()
1238 sage: Luy = (u*y).operator_matrix()
1239 sage: Luz = (u*z).operator_matrix()
1240 sage: Luyz = (u*(y*z)).operator_matrix()
1241 sage: lhs = Lu*Lzy + Lz*Luy + Ly*Luz
1242 sage: rhs = Luyz + Ly*Lu*Lz + Lz*Lu*Ly
1243 sage: bool(lhs == rhs)
1247 fda_elt
= FiniteDimensionalAlgebraElement(self
.parent(), self
)
1248 return fda_elt
.matrix().transpose()
1251 def quadratic_representation(self
, other
=None):
1253 Return the quadratic representation of this element.
1257 The explicit form in the spin factor algebra is given by
1258 Alizadeh's Example 11.12::
1260 sage: set_random_seed()
1261 sage: n = ZZ.random_element(1,10)
1262 sage: J = JordanSpinEJA(n)
1263 sage: x = J.random_element()
1264 sage: x_vec = x.vector()
1266 sage: x_bar = x_vec[1:]
1267 sage: A = matrix(QQ, 1, [x_vec.inner_product(x_vec)])
1268 sage: B = 2*x0*x_bar.row()
1269 sage: C = 2*x0*x_bar.column()
1270 sage: D = identity_matrix(QQ, n-1)
1271 sage: D = (x0^2 - x_bar.inner_product(x_bar))*D
1272 sage: D = D + 2*x_bar.tensor_product(x_bar)
1273 sage: Q = block_matrix(2,2,[A,B,C,D])
1274 sage: Q == x.quadratic_representation()
1277 Test all of the properties from Theorem 11.2 in Alizadeh::
1279 sage: set_random_seed()
1280 sage: J = random_eja()
1281 sage: x = J.random_element()
1282 sage: y = J.random_element()
1283 sage: Lx = x.operator_matrix()
1284 sage: Lxx = (x*x).operator_matrix()
1285 sage: Qx = x.quadratic_representation()
1286 sage: Qy = y.quadratic_representation()
1287 sage: Qxy = x.quadratic_representation(y)
1288 sage: Qex = J.one().quadratic_representation(x)
1289 sage: n = ZZ.random_element(10)
1290 sage: Qxn = (x^n).quadratic_representation()
1294 sage: 2*Qxy == (x+y).quadratic_representation() - Qx - Qy
1299 sage: alpha = QQ.random_element()
1300 sage: (alpha*x).quadratic_representation() == (alpha^2)*Qx
1305 sage: not x.is_invertible() or (
1306 ....: Qx*x.inverse().vector() == x.vector() )
1309 sage: not x.is_invertible() or (
1312 ....: x.inverse().quadratic_representation() )
1315 sage: Qxy*(J.one().vector()) == (x*y).vector()
1320 sage: not x.is_invertible() or (
1321 ....: x.quadratic_representation(x.inverse())*Qx
1322 ....: == Qx*x.quadratic_representation(x.inverse()) )
1325 sage: not x.is_invertible() or (
1326 ....: x.quadratic_representation(x.inverse())*Qx
1328 ....: 2*x.operator_matrix()*Qex - Qx )
1331 sage: 2*x.operator_matrix()*Qex - Qx == Lxx
1336 sage: J(Qy*x.vector()).quadratic_representation() == Qy*Qx*Qy
1346 sage: not x.is_invertible() or (
1347 ....: Qx*x.inverse().operator_matrix() == Lx )
1352 sage: not x.operator_commutes_with(y) or (
1353 ....: J(Qx*y.vector())^n == J(Qxn*(y^n).vector()) )
1359 elif not other
in self
.parent():
1360 raise TypeError("'other' must live in the same algebra")
1362 L
= self
.operator_matrix()
1363 M
= other
.operator_matrix()
1364 return ( L
*M
+ M
*L
- (self
*other
).operator_matrix() )
1367 def span_of_powers(self
):
1369 Return the vector space spanned by successive powers of
1372 # The dimension of the subalgebra can't be greater than
1373 # the big algebra, so just put everything into a list
1374 # and let span() get rid of the excess.
1376 # We do the extra ambient_vector_space() in case we're messing
1377 # with polynomials and the direct parent is a module.
1378 V
= self
.vector().parent().ambient_vector_space()
1379 return V
.span( (self
**d
).vector() for d
in xrange(V
.dimension()) )
1382 def subalgebra_generated_by(self
):
1384 Return the associative subalgebra of the parent EJA generated
1389 sage: set_random_seed()
1390 sage: x = random_eja().random_element()
1391 sage: x.subalgebra_generated_by().is_associative()
1394 Squaring in the subalgebra should be the same thing as
1395 squaring in the superalgebra::
1397 sage: set_random_seed()
1398 sage: x = random_eja().random_element()
1399 sage: u = x.subalgebra_generated_by().random_element()
1400 sage: u.operator_matrix()*u.vector() == (u**2).vector()
1404 # First get the subspace spanned by the powers of myself...
1405 V
= self
.span_of_powers()
1406 F
= self
.base_ring()
1408 # Now figure out the entries of the right-multiplication
1409 # matrix for the successive basis elements b0, b1,... of
1412 for b_right
in V
.basis():
1413 eja_b_right
= self
.parent()(b_right
)
1415 # The first row of the right-multiplication matrix by
1416 # b1 is what we get if we apply that matrix to b1. The
1417 # second row of the right multiplication matrix by b1
1418 # is what we get when we apply that matrix to b2...
1420 # IMPORTANT: this assumes that all vectors are COLUMN
1421 # vectors, unlike our superclass (which uses row vectors).
1422 for b_left
in V
.basis():
1423 eja_b_left
= self
.parent()(b_left
)
1424 # Multiply in the original EJA, but then get the
1425 # coordinates from the subalgebra in terms of its
1427 this_row
= V
.coordinates((eja_b_left
*eja_b_right
).vector())
1428 b_right_rows
.append(this_row
)
1429 b_right_matrix
= matrix(F
, b_right_rows
)
1430 mats
.append(b_right_matrix
)
1432 # It's an algebra of polynomials in one element, and EJAs
1433 # are power-associative.
1435 # TODO: choose generator names intelligently.
1436 return FiniteDimensionalEuclideanJordanAlgebra(F
, mats
, assume_associative
=True, names
='f')
1439 def subalgebra_idempotent(self
):
1441 Find an idempotent in the associative subalgebra I generate
1442 using Proposition 2.3.5 in Baes.
1446 sage: set_random_seed()
1447 sage: J = random_eja()
1448 sage: x = J.random_element()
1449 sage: while x.is_nilpotent():
1450 ....: x = J.random_element()
1451 sage: c = x.subalgebra_idempotent()
1456 if self
.is_nilpotent():
1457 raise ValueError("this only works with non-nilpotent elements!")
1459 V
= self
.span_of_powers()
1460 J
= self
.subalgebra_generated_by()
1461 # Mis-design warning: the basis used for span_of_powers()
1462 # and subalgebra_generated_by() must be the same, and in
1464 u
= J(V
.coordinates(self
.vector()))
1466 # The image of the matrix of left-u^m-multiplication
1467 # will be minimal for some natural number s...
1469 minimal_dim
= V
.dimension()
1470 for i
in xrange(1, V
.dimension()):
1471 this_dim
= (u
**i
).operator_matrix().image().dimension()
1472 if this_dim
< minimal_dim
:
1473 minimal_dim
= this_dim
1476 # Now minimal_matrix should correspond to the smallest
1477 # non-zero subspace in Baes's (or really, Koecher's)
1480 # However, we need to restrict the matrix to work on the
1481 # subspace... or do we? Can't we just solve, knowing that
1482 # A(c) = u^(s+1) should have a solution in the big space,
1485 # Beware, solve_right() means that we're using COLUMN vectors.
1486 # Our FiniteDimensionalAlgebraElement superclass uses rows.
1488 A
= u_next
.operator_matrix()
1489 c_coordinates
= A
.solve_right(u_next
.vector())
1491 # Now c_coordinates is the idempotent we want, but it's in
1492 # the coordinate system of the subalgebra.
1494 # We need the basis for J, but as elements of the parent algebra.
1496 basis
= [self
.parent(v
) for v
in V
.basis()]
1497 return self
.parent().linear_combination(zip(c_coordinates
, basis
))
1502 Return my trace, the sum of my eigenvalues.
1506 sage: J = JordanSpinEJA(3)
1507 sage: x = sum(J.gens())
1513 sage: J = RealCartesianProductEJA(5)
1514 sage: J.one().trace()
1519 The trace of an element is a real number::
1521 sage: set_random_seed()
1522 sage: J = random_eja()
1523 sage: J.random_element().trace() in J.base_ring()
1529 p
= P
._charpoly
_coeff
(r
-1)
1530 # The _charpoly_coeff function already adds the factor of
1531 # -1 to ensure that _charpoly_coeff(r-1) is really what
1532 # appears in front of t^{r-1} in the charpoly. However,
1533 # we want the negative of THAT for the trace.
1534 return -p(*self
.vector())
1537 def trace_inner_product(self
, other
):
1539 Return the trace inner product of myself and ``other``.
1543 The trace inner product is commutative::
1545 sage: set_random_seed()
1546 sage: J = random_eja()
1547 sage: x = J.random_element(); y = J.random_element()
1548 sage: x.trace_inner_product(y) == y.trace_inner_product(x)
1551 The trace inner product is bilinear::
1553 sage: set_random_seed()
1554 sage: J = random_eja()
1555 sage: x = J.random_element()
1556 sage: y = J.random_element()
1557 sage: z = J.random_element()
1558 sage: a = QQ.random_element();
1559 sage: actual = (a*(x+z)).trace_inner_product(y)
1560 sage: expected = ( a*x.trace_inner_product(y) +
1561 ....: a*z.trace_inner_product(y) )
1562 sage: actual == expected
1564 sage: actual = x.trace_inner_product(a*(y+z))
1565 sage: expected = ( a*x.trace_inner_product(y) +
1566 ....: a*x.trace_inner_product(z) )
1567 sage: actual == expected
1570 The trace inner product satisfies the compatibility
1571 condition in the definition of a Euclidean Jordan algebra::
1573 sage: set_random_seed()
1574 sage: J = random_eja()
1575 sage: x = J.random_element()
1576 sage: y = J.random_element()
1577 sage: z = J.random_element()
1578 sage: (x*y).trace_inner_product(z) == y.trace_inner_product(x*z)
1582 if not other
in self
.parent():
1583 raise TypeError("'other' must live in the same algebra")
1585 return (self
*other
).trace()
1588 class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1590 Return the Euclidean Jordan Algebra corresponding to the set
1591 `R^n` under the Hadamard product.
1593 Note: this is nothing more than the Cartesian product of ``n``
1594 copies of the spin algebra. Once Cartesian product algebras
1595 are implemented, this can go.
1599 This multiplication table can be verified by hand::
1601 sage: J = RealCartesianProductEJA(3)
1602 sage: e0,e1,e2 = J.gens()
1618 def __classcall_private__(cls
, n
, field
=QQ
):
1619 # The FiniteDimensionalAlgebra constructor takes a list of
1620 # matrices, the ith representing right multiplication by the ith
1621 # basis element in the vector space. So if e_1 = (1,0,0), then
1622 # right (Hadamard) multiplication of x by e_1 picks out the first
1623 # component of x; and likewise for the ith basis element e_i.
1624 Qs
= [ matrix(field
, n
, n
, lambda k
,j
: 1*(k
== j
== i
))
1625 for i
in xrange(n
) ]
1627 fdeja
= super(RealCartesianProductEJA
, cls
)
1628 return fdeja
.__classcall
_private
__(cls
, field
, Qs
, rank
=n
)
1630 def inner_product(self
, x
, y
):
1631 return _usual_ip(x
,y
)
1636 Return a "random" finite-dimensional Euclidean Jordan Algebra.
1640 For now, we choose a random natural number ``n`` (greater than zero)
1641 and then give you back one of the following:
1643 * The cartesian product of the rational numbers ``n`` times; this is
1644 ``QQ^n`` with the Hadamard product.
1646 * The Jordan spin algebra on ``QQ^n``.
1648 * The ``n``-by-``n`` rational symmetric matrices with the symmetric
1651 * The ``n``-by-``n`` complex-rational Hermitian matrices embedded
1652 in the space of ``2n``-by-``2n`` real symmetric matrices.
1654 * The ``n``-by-``n`` quaternion-rational Hermitian matrices embedded
1655 in the space of ``4n``-by-``4n`` real symmetric matrices.
1657 Later this might be extended to return Cartesian products of the
1663 Euclidean Jordan algebra of degree...
1667 # The max_n component lets us choose different upper bounds on the
1668 # value "n" that gets passed to the constructor. This is needed
1669 # because e.g. R^{10} is reasonable to test, while the Hermitian
1670 # 10-by-10 quaternion matrices are not.
1671 (constructor
, max_n
) = choice([(RealCartesianProductEJA
, 6),
1673 (RealSymmetricEJA
, 5),
1674 (ComplexHermitianEJA
, 4),
1675 (QuaternionHermitianEJA
, 3)])
1676 n
= ZZ
.random_element(1, max_n
)
1677 return constructor(n
, field
=QQ
)
1681 def _real_symmetric_basis(n
, field
=QQ
):
1683 Return a basis for the space of real symmetric n-by-n matrices.
1685 # The basis of symmetric matrices, as matrices, in their R^(n-by-n)
1689 for j
in xrange(i
+1):
1690 Eij
= matrix(field
, n
, lambda k
,l
: k
==i
and l
==j
)
1694 # Beware, orthogonal but not normalized!
1695 Sij
= Eij
+ Eij
.transpose()
1700 def _complex_hermitian_basis(n
, field
=QQ
):
1702 Returns a basis for the space of complex Hermitian n-by-n matrices.
1706 sage: set_random_seed()
1707 sage: n = ZZ.random_element(1,5)
1708 sage: all( M.is_symmetric() for M in _complex_hermitian_basis(n) )
1712 F
= QuadraticField(-1, 'I')
1715 # This is like the symmetric case, but we need to be careful:
1717 # * We want conjugate-symmetry, not just symmetry.
1718 # * The diagonal will (as a result) be real.
1722 for j
in xrange(i
+1):
1723 Eij
= matrix(field
, n
, lambda k
,l
: k
==i
and l
==j
)
1725 Sij
= _embed_complex_matrix(Eij
)
1728 # Beware, orthogonal but not normalized! The second one
1729 # has a minus because it's conjugated.
1730 Sij_real
= _embed_complex_matrix(Eij
+ Eij
.transpose())
1732 Sij_imag
= _embed_complex_matrix(I
*Eij
- I
*Eij
.transpose())
1737 def _quaternion_hermitian_basis(n
, field
=QQ
):
1739 Returns a basis for the space of quaternion Hermitian n-by-n matrices.
1743 sage: set_random_seed()
1744 sage: n = ZZ.random_element(1,5)
1745 sage: all( M.is_symmetric() for M in _quaternion_hermitian_basis(n) )
1749 Q
= QuaternionAlgebra(QQ
,-1,-1)
1752 # This is like the symmetric case, but we need to be careful:
1754 # * We want conjugate-symmetry, not just symmetry.
1755 # * The diagonal will (as a result) be real.
1759 for j
in xrange(i
+1):
1760 Eij
= matrix(Q
, n
, lambda k
,l
: k
==i
and l
==j
)
1762 Sij
= _embed_quaternion_matrix(Eij
)
1765 # Beware, orthogonal but not normalized! The second,
1766 # third, and fourth ones have a minus because they're
1768 Sij_real
= _embed_quaternion_matrix(Eij
+ Eij
.transpose())
1770 Sij_I
= _embed_quaternion_matrix(I
*Eij
- I
*Eij
.transpose())
1772 Sij_J
= _embed_quaternion_matrix(J
*Eij
- J
*Eij
.transpose())
1774 Sij_K
= _embed_quaternion_matrix(K
*Eij
- K
*Eij
.transpose())
1780 return vector(m
.base_ring(), m
.list())
1783 return matrix(v
.base_ring(), sqrt(v
.degree()), v
.list())
1785 def _multiplication_table_from_matrix_basis(basis
):
1787 At least three of the five simple Euclidean Jordan algebras have the
1788 symmetric multiplication (A,B) |-> (AB + BA)/2, where the
1789 multiplication on the right is matrix multiplication. Given a basis
1790 for the underlying matrix space, this function returns a
1791 multiplication table (obtained by looping through the basis
1792 elements) for an algebra of those matrices. A reordered copy
1793 of the basis is also returned to work around the fact that
1794 the ``span()`` in this function will change the order of the basis
1795 from what we think it is, to... something else.
1797 # In S^2, for example, we nominally have four coordinates even
1798 # though the space is of dimension three only. The vector space V
1799 # is supposed to hold the entire long vector, and the subspace W
1800 # of V will be spanned by the vectors that arise from symmetric
1801 # matrices. Thus for S^2, dim(V) == 4 and dim(W) == 3.
1802 field
= basis
[0].base_ring()
1803 dimension
= basis
[0].nrows()
1805 V
= VectorSpace(field
, dimension
**2)
1806 W
= V
.span( _mat2vec(s
) for s
in basis
)
1808 # Taking the span above reorders our basis (thanks, jerk!) so we
1809 # need to put our "matrix basis" in the same order as the
1810 # (reordered) vector basis.
1811 S
= tuple( _vec2mat(b
) for b
in W
.basis() )
1815 # Brute force the multiplication-by-s matrix by looping
1816 # through all elements of the basis and doing the computation
1817 # to find out what the corresponding row should be. BEWARE:
1818 # these multiplication tables won't be symmetric! It therefore
1819 # becomes REALLY IMPORTANT that the underlying algebra
1820 # constructor uses ROW vectors and not COLUMN vectors. That's
1821 # why we're computing rows here and not columns.
1824 this_row
= _mat2vec((s
*t
+ t
*s
)/2)
1825 Q_rows
.append(W
.coordinates(this_row
))
1826 Q
= matrix(field
, W
.dimension(), Q_rows
)
1832 def _embed_complex_matrix(M
):
1834 Embed the n-by-n complex matrix ``M`` into the space of real
1835 matrices of size 2n-by-2n via the map the sends each entry `z = a +
1836 bi` to the block matrix ``[[a,b],[-b,a]]``.
1840 sage: F = QuadraticField(-1,'i')
1841 sage: x1 = F(4 - 2*i)
1842 sage: x2 = F(1 + 2*i)
1845 sage: M = matrix(F,2,[[x1,x2],[x3,x4]])
1846 sage: _embed_complex_matrix(M)
1855 Embedding is a homomorphism (isomorphism, in fact)::
1857 sage: set_random_seed()
1858 sage: n = ZZ.random_element(5)
1859 sage: F = QuadraticField(-1, 'i')
1860 sage: X = random_matrix(F, n)
1861 sage: Y = random_matrix(F, n)
1862 sage: actual = _embed_complex_matrix(X) * _embed_complex_matrix(Y)
1863 sage: expected = _embed_complex_matrix(X*Y)
1864 sage: actual == expected
1870 raise ValueError("the matrix 'M' must be square")
1871 field
= M
.base_ring()
1876 blocks
.append(matrix(field
, 2, [[a
,b
],[-b
,a
]]))
1878 # We can drop the imaginaries here.
1879 return block_matrix(field
.base_ring(), n
, blocks
)
1882 def _unembed_complex_matrix(M
):
1884 The inverse of _embed_complex_matrix().
1888 sage: A = matrix(QQ,[ [ 1, 2, 3, 4],
1889 ....: [-2, 1, -4, 3],
1890 ....: [ 9, 10, 11, 12],
1891 ....: [-10, 9, -12, 11] ])
1892 sage: _unembed_complex_matrix(A)
1894 [ 10*i + 9 12*i + 11]
1898 Unembedding is the inverse of embedding::
1900 sage: set_random_seed()
1901 sage: F = QuadraticField(-1, 'i')
1902 sage: M = random_matrix(F, 3)
1903 sage: _unembed_complex_matrix(_embed_complex_matrix(M)) == M
1909 raise ValueError("the matrix 'M' must be square")
1910 if not n
.mod(2).is_zero():
1911 raise ValueError("the matrix 'M' must be a complex embedding")
1913 F
= QuadraticField(-1, 'i')
1916 # Go top-left to bottom-right (reading order), converting every
1917 # 2-by-2 block we see to a single complex element.
1919 for k
in xrange(n
/2):
1920 for j
in xrange(n
/2):
1921 submat
= M
[2*k
:2*k
+2,2*j
:2*j
+2]
1922 if submat
[0,0] != submat
[1,1]:
1923 raise ValueError('bad on-diagonal submatrix')
1924 if submat
[0,1] != -submat
[1,0]:
1925 raise ValueError('bad off-diagonal submatrix')
1926 z
= submat
[0,0] + submat
[0,1]*i
1929 return matrix(F
, n
/2, elements
)
1932 def _embed_quaternion_matrix(M
):
1934 Embed the n-by-n quaternion matrix ``M`` into the space of real
1935 matrices of size 4n-by-4n by first sending each quaternion entry
1936 `z = a + bi + cj + dk` to the block-complex matrix
1937 ``[[a + bi, c+di],[-c + di, a-bi]]`, and then embedding those into
1942 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1943 sage: i,j,k = Q.gens()
1944 sage: x = 1 + 2*i + 3*j + 4*k
1945 sage: M = matrix(Q, 1, [[x]])
1946 sage: _embed_quaternion_matrix(M)
1952 Embedding is a homomorphism (isomorphism, in fact)::
1954 sage: set_random_seed()
1955 sage: n = ZZ.random_element(5)
1956 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1957 sage: X = random_matrix(Q, n)
1958 sage: Y = random_matrix(Q, n)
1959 sage: actual = _embed_quaternion_matrix(X)*_embed_quaternion_matrix(Y)
1960 sage: expected = _embed_quaternion_matrix(X*Y)
1961 sage: actual == expected
1965 quaternions
= M
.base_ring()
1968 raise ValueError("the matrix 'M' must be square")
1970 F
= QuadraticField(-1, 'i')
1975 t
= z
.coefficient_tuple()
1980 cplx_matrix
= matrix(F
, 2, [[ a
+ b
*i
, c
+ d
*i
],
1981 [-c
+ d
*i
, a
- b
*i
]])
1982 blocks
.append(_embed_complex_matrix(cplx_matrix
))
1984 # We should have real entries by now, so use the realest field
1985 # we've got for the return value.
1986 return block_matrix(quaternions
.base_ring(), n
, blocks
)
1989 def _unembed_quaternion_matrix(M
):
1991 The inverse of _embed_quaternion_matrix().
1995 sage: M = matrix(QQ, [[ 1, 2, 3, 4],
1996 ....: [-2, 1, -4, 3],
1997 ....: [-3, 4, 1, -2],
1998 ....: [-4, -3, 2, 1]])
1999 sage: _unembed_quaternion_matrix(M)
2000 [1 + 2*i + 3*j + 4*k]
2004 Unembedding is the inverse of embedding::
2006 sage: set_random_seed()
2007 sage: Q = QuaternionAlgebra(QQ, -1, -1)
2008 sage: M = random_matrix(Q, 3)
2009 sage: _unembed_quaternion_matrix(_embed_quaternion_matrix(M)) == M
2015 raise ValueError("the matrix 'M' must be square")
2016 if not n
.mod(4).is_zero():
2017 raise ValueError("the matrix 'M' must be a complex embedding")
2019 Q
= QuaternionAlgebra(QQ
,-1,-1)
2022 # Go top-left to bottom-right (reading order), converting every
2023 # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1
2026 for l
in xrange(n
/4):
2027 for m
in xrange(n
/4):
2028 submat
= _unembed_complex_matrix(M
[4*l
:4*l
+4,4*m
:4*m
+4])
2029 if submat
[0,0] != submat
[1,1].conjugate():
2030 raise ValueError('bad on-diagonal submatrix')
2031 if submat
[0,1] != -submat
[1,0].conjugate():
2032 raise ValueError('bad off-diagonal submatrix')
2033 z
= submat
[0,0].real() + submat
[0,0].imag()*i
2034 z
+= submat
[0,1].real()*j
+ submat
[0,1].imag()*k
2037 return matrix(Q
, n
/4, elements
)
2040 # The usual inner product on R^n.
2042 return x
.vector().inner_product(y
.vector())
2044 # The inner product used for the real symmetric simple EJA.
2045 # We keep it as a separate function because e.g. the complex
2046 # algebra uses the same inner product, except divided by 2.
2047 def _matrix_ip(X
,Y
):
2048 X_mat
= X
.natural_representation()
2049 Y_mat
= Y
.natural_representation()
2050 return (X_mat
*Y_mat
).trace()
2053 class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra
):
2055 The rank-n simple EJA consisting of real symmetric n-by-n
2056 matrices, the usual symmetric Jordan product, and the trace inner
2057 product. It has dimension `(n^2 + n)/2` over the reals.
2061 sage: J = RealSymmetricEJA(2)
2062 sage: e0, e1, e2 = J.gens()
2072 The degree of this algebra is `(n^2 + n) / 2`::
2074 sage: set_random_seed()
2075 sage: n = ZZ.random_element(1,5)
2076 sage: J = RealSymmetricEJA(n)
2077 sage: J.degree() == (n^2 + n)/2
2080 The Jordan multiplication is what we think it is::
2082 sage: set_random_seed()
2083 sage: n = ZZ.random_element(1,5)
2084 sage: J = RealSymmetricEJA(n)
2085 sage: x = J.random_element()
2086 sage: y = J.random_element()
2087 sage: actual = (x*y).natural_representation()
2088 sage: X = x.natural_representation()
2089 sage: Y = y.natural_representation()
2090 sage: expected = (X*Y + Y*X)/2
2091 sage: actual == expected
2093 sage: J(expected) == x*y
2098 def __classcall_private__(cls
, n
, field
=QQ
):
2099 S
= _real_symmetric_basis(n
, field
=field
)
2100 (Qs
, T
) = _multiplication_table_from_matrix_basis(S
)
2102 fdeja
= super(RealSymmetricEJA
, cls
)
2103 return fdeja
.__classcall
_private
__(cls
,
2109 def inner_product(self
, x
, y
):
2110 return _matrix_ip(x
,y
)
2113 class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra
):
2115 The rank-n simple EJA consisting of complex Hermitian n-by-n
2116 matrices over the real numbers, the usual symmetric Jordan product,
2117 and the real-part-of-trace inner product. It has dimension `n^2` over
2122 The degree of this algebra is `n^2`::
2124 sage: set_random_seed()
2125 sage: n = ZZ.random_element(1,5)
2126 sage: J = ComplexHermitianEJA(n)
2127 sage: J.degree() == n^2
2130 The Jordan multiplication is what we think it is::
2132 sage: set_random_seed()
2133 sage: n = ZZ.random_element(1,5)
2134 sage: J = ComplexHermitianEJA(n)
2135 sage: x = J.random_element()
2136 sage: y = J.random_element()
2137 sage: actual = (x*y).natural_representation()
2138 sage: X = x.natural_representation()
2139 sage: Y = y.natural_representation()
2140 sage: expected = (X*Y + Y*X)/2
2141 sage: actual == expected
2143 sage: J(expected) == x*y
2148 def __classcall_private__(cls
, n
, field
=QQ
):
2149 S
= _complex_hermitian_basis(n
)
2150 (Qs
, T
) = _multiplication_table_from_matrix_basis(S
)
2152 fdeja
= super(ComplexHermitianEJA
, cls
)
2153 return fdeja
.__classcall
_private
__(cls
,
2159 def inner_product(self
, x
, y
):
2160 # Since a+bi on the diagonal is represented as
2165 # we'll double-count the "a" entries if we take the trace of
2167 return _matrix_ip(x
,y
)/2
2170 class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra
):
2172 The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
2173 matrices, the usual symmetric Jordan product, and the
2174 real-part-of-trace inner product. It has dimension `2n^2 - n` over
2179 The degree of this algebra is `n^2`::
2181 sage: set_random_seed()
2182 sage: n = ZZ.random_element(1,5)
2183 sage: J = QuaternionHermitianEJA(n)
2184 sage: J.degree() == 2*(n^2) - n
2187 The Jordan multiplication is what we think it is::
2189 sage: set_random_seed()
2190 sage: n = ZZ.random_element(1,5)
2191 sage: J = QuaternionHermitianEJA(n)
2192 sage: x = J.random_element()
2193 sage: y = J.random_element()
2194 sage: actual = (x*y).natural_representation()
2195 sage: X = x.natural_representation()
2196 sage: Y = y.natural_representation()
2197 sage: expected = (X*Y + Y*X)/2
2198 sage: actual == expected
2200 sage: J(expected) == x*y
2205 def __classcall_private__(cls
, n
, field
=QQ
):
2206 S
= _quaternion_hermitian_basis(n
)
2207 (Qs
, T
) = _multiplication_table_from_matrix_basis(S
)
2209 fdeja
= super(QuaternionHermitianEJA
, cls
)
2210 return fdeja
.__classcall
_private
__(cls
,
2216 def inner_product(self
, x
, y
):
2217 # Since a+bi+cj+dk on the diagonal is represented as
2219 # a + bi +cj + dk = [ a b c d]
2224 # we'll quadruple-count the "a" entries if we take the trace of
2226 return _matrix_ip(x
,y
)/4
2229 class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra
):
2231 The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
2232 with the usual inner product and jordan product ``x*y =
2233 (<x_bar,y_bar>, x0*y_bar + y0*x_bar)``. It has dimension `n` over
2238 This multiplication table can be verified by hand::
2240 sage: J = JordanSpinEJA(4)
2241 sage: e0,e1,e2,e3 = J.gens()
2259 def __classcall_private__(cls
, n
, field
=QQ
):
2261 id_matrix
= identity_matrix(field
, n
)
2263 ei
= id_matrix
.column(i
)
2264 Qi
= zero_matrix(field
, n
)
2266 Qi
.set_column(0, ei
)
2267 Qi
+= diagonal_matrix(n
, [ei
[0]]*n
)
2268 # The addition of the diagonal matrix adds an extra ei[0] in the
2269 # upper-left corner of the matrix.
2270 Qi
[0,0] = Qi
[0,0] * ~
field(2)
2273 # The rank of the spin algebra is two, unless we're in a
2274 # one-dimensional ambient space (because the rank is bounded by
2275 # the ambient dimension).
2276 fdeja
= super(JordanSpinEJA
, cls
)
2277 return fdeja
.__classcall
_private
__(cls
, field
, Qs
, rank
=min(n
,2))
2279 def inner_product(self
, x
, y
):
2280 return _usual_ip(x
,y
)