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
.map import Map
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
17 class FiniteDimensionalEuclideanJordanAlgebraOperator(Map
):
18 def __init__(self
, domain_eja
, codomain_eja
, mat
):
20 isinstance(domain_eja
, FiniteDimensionalEuclideanJordanAlgebra
) and
21 isinstance(codomain_eja
, FiniteDimensionalEuclideanJordanAlgebra
) ):
22 raise ValueError('(co)domains must be finite-dimensional Euclidean '
25 F
= domain_eja
.base_ring()
26 if not (F
== codomain_eja
.base_ring()):
27 raise ValueError("domain and codomain must have the same base ring")
29 # We need to supply something here to avoid getting the
30 # default Homset of the parent FiniteDimensionalAlgebra class,
31 # which messes up e.g. equality testing. We use FreeModules(F)
32 # instead of VectorSpaces(F) because our characteristic polynomial
33 # algorithm will need to F to be a polynomial ring at some point.
34 # When F is a field, FreeModules(F) returns VectorSpaces(F) anyway.
35 parent
= Hom(domain_eja
, codomain_eja
, FreeModules(F
))
37 # The Map initializer will set our parent to a homset, which
38 # is explicitly NOT what we want, because these ain't algebra
40 super(FiniteDimensionalEuclideanJordanAlgebraOperator
,self
).__init
__(parent
)
42 # Keep a matrix around to do all of the real work. It would
43 # be nice if we could use a VectorSpaceMorphism instead, but
44 # those use row vectors that we don't want to accidentally
45 # expose to our users.
51 Allow this operator to be called only on elements of an EJA.
55 sage: J = JordanSpinEJA(3)
56 sage: x = J.linear_combination(zip(range(len(J.gens())), J.gens()))
57 sage: id = identity_matrix(J.base_ring(), J.dimension())
58 sage: f = FiniteDimensionalEuclideanJordanAlgebraOperator(J,J,id)
63 return self
.codomain()(self
.matrix()*x
.vector())
66 def _add_(self
, other
):
68 Add the ``other`` EJA operator to this one.
72 When we add two EJA operators, we get another one back::
74 sage: J = RealSymmetricEJA(2)
75 sage: id = identity_matrix(J.base_ring(), J.dimension())
76 sage: f = FiniteDimensionalEuclideanJordanAlgebraOperator(J,J,id)
77 sage: g = FiniteDimensionalEuclideanJordanAlgebraOperator(J,J,id)
79 Linear operator between finite-dimensional Euclidean Jordan
80 algebras represented by the matrix:
84 Domain: Euclidean Jordan algebra of degree 3 over Rational Field
85 Codomain: Euclidean Jordan algebra of degree 3 over Rational Field
87 If you try to add two identical vector space operators but on
88 different EJAs, that should blow up::
90 sage: J1 = RealSymmetricEJA(2)
91 sage: J2 = JordanSpinEJA(3)
92 sage: id = identity_matrix(QQ, 3)
93 sage: f = FiniteDimensionalEuclideanJordanAlgebraOperator(J1,J1,id)
94 sage: g = FiniteDimensionalEuclideanJordanAlgebraOperator(J2,J2,id)
96 Traceback (most recent call last):
98 TypeError: unsupported operand parent(s) for +: ...
101 return FiniteDimensionalEuclideanJordanAlgebraOperator(
104 self
.matrix() + other
.matrix())
107 def _composition_(self
, other
, homset
):
109 Compose two EJA operators to get another one (and NOT a formal
110 composite object) back.
114 sage: J1 = JordanSpinEJA(3)
115 sage: J2 = RealCartesianProductEJA(2)
116 sage: J3 = RealSymmetricEJA(1)
117 sage: mat1 = matrix(QQ, [[1,2,3],
119 sage: mat2 = matrix(QQ, [[7,8]])
120 sage: g = FiniteDimensionalEuclideanJordanAlgebraOperator(J1,
123 sage: f = FiniteDimensionalEuclideanJordanAlgebraOperator(J2,
127 Linear operator between finite-dimensional Euclidean Jordan
128 algebras represented by the matrix:
130 Domain: Euclidean Jordan algebra of degree 3 over Rational Field
131 Codomain: Euclidean Jordan algebra of degree 1 over Rational Field
134 return FiniteDimensionalEuclideanJordanAlgebraOperator(
137 self
.matrix()*other
.matrix())
140 def __eq__(self
, other
):
141 if self
.domain() != other
.domain():
143 if self
.codomain() != other
.codomain():
145 if self
.matrix() != other
.matrix():
149 def __invert__(self
):
151 Invert this EJA operator.
155 sage: J = RealSymmetricEJA(2)
156 sage: id = identity_matrix(J.base_ring(), J.dimension())
157 sage: f = FiniteDimensionalEuclideanJordanAlgebraOperator(J,J,id)
159 Linear operator between finite-dimensional Euclidean Jordan
160 algebras represented by the matrix:
164 Domain: Euclidean Jordan algebra of degree 3 over Rational Field
165 Codomain: Euclidean Jordan algebra of degree 3 over Rational Field
168 return FiniteDimensionalEuclideanJordanAlgebraOperator(
174 def __mul__(self
, other
):
176 Compose two EJA operators, or scale myself by an element of the
177 ambient vector space.
179 We need to override the real ``__mul__`` function to prevent the
180 coercion framework from throwing an error when it fails to convert
181 a base ring element into a morphism.
185 We can scale an operator on a rational algebra by a rational number::
187 sage: J = RealSymmetricEJA(2)
188 sage: e0,e1,e2 = J.gens()
189 sage: x = 2*e0 + 4*e1 + 16*e2
191 Linear operator between finite-dimensional Euclidean Jordan algebras
192 represented by the matrix:
196 Domain: Euclidean Jordan algebra of degree 3 over Rational Field
197 Codomain: Euclidean Jordan algebra of degree 3 over Rational Field
198 sage: x.operator()*(1/2)
199 Linear operator between finite-dimensional Euclidean Jordan algebras
200 represented by the matrix:
204 Domain: Euclidean Jordan algebra of degree 3 over Rational Field
205 Codomain: Euclidean Jordan algebra of degree 3 over Rational Field
208 if other
in self
.codomain().base_ring():
209 return FiniteDimensionalEuclideanJordanAlgebraOperator(
214 # This should eventually delegate to _composition_ after performing
215 # some sanity checks for us.
216 mor
= super(FiniteDimensionalEuclideanJordanAlgebraOperator
,self
)
217 return mor
.__mul
__(other
)
222 Negate this EJA operator.
226 sage: J = RealSymmetricEJA(2)
227 sage: id = identity_matrix(J.base_ring(), J.dimension())
228 sage: f = FiniteDimensionalEuclideanJordanAlgebraOperator(J,J,id)
230 Linear operator between finite-dimensional Euclidean Jordan
231 algebras represented by the matrix:
235 Domain: Euclidean Jordan algebra of degree 3 over Rational Field
236 Codomain: Euclidean Jordan algebra of degree 3 over Rational Field
239 return FiniteDimensionalEuclideanJordanAlgebraOperator(
245 def __pow__(self
, n
):
247 Raise this EJA operator to the power ``n``.
251 Ensure that we get back another EJA operator that can be added,
252 subtracted, et cetera::
254 sage: J = RealSymmetricEJA(2)
255 sage: id = identity_matrix(J.base_ring(), J.dimension())
256 sage: f = FiniteDimensionalEuclideanJordanAlgebraOperator(J,J,id)
257 sage: f^0 + f^1 + f^2
258 Linear operator between finite-dimensional Euclidean Jordan
259 algebras represented by the matrix:
263 Domain: Euclidean Jordan algebra of degree 3 over Rational Field
264 Codomain: Euclidean Jordan algebra of degree 3 over Rational Field
270 # Raising a vector space morphism to the zero power gives
271 # you back a special IdentityMorphism that is useless to us.
272 rows
= self
.codomain().dimension()
273 cols
= self
.domain().dimension()
274 mat
= matrix
.identity(self
.base_ring(), rows
, cols
)
276 mat
= self
.matrix()**n
278 return FiniteDimensionalEuclideanJordanAlgebraOperator(
287 A text representation of this linear operator on a Euclidean
292 sage: J = JordanSpinEJA(2)
293 sage: id = identity_matrix(J.base_ring(), J.dimension())
294 sage: FiniteDimensionalEuclideanJordanAlgebraOperator(J,J,id)
295 Linear operator between finite-dimensional Euclidean Jordan
296 algebras represented by the matrix:
299 Domain: Euclidean Jordan algebra of degree 2 over Rational Field
300 Codomain: Euclidean Jordan algebra of degree 2 over Rational Field
303 msg
= ("Linear operator between finite-dimensional Euclidean Jordan "
304 "algebras represented by the matrix:\n",
308 return ''.join(msg
).format(self
.matrix(),
313 def _sub_(self
, other
):
315 Subtract ``other`` from this EJA operator.
319 sage: J = RealSymmetricEJA(2)
320 sage: id = identity_matrix(J.base_ring(),J.dimension())
321 sage: f = FiniteDimensionalEuclideanJordanAlgebraOperator(J,J,id)
323 Linear operator between finite-dimensional Euclidean Jordan
324 algebras represented by the matrix:
328 Domain: Euclidean Jordan algebra of degree 3 over Rational Field
329 Codomain: Euclidean Jordan algebra of degree 3 over Rational Field
332 return (self
+ (-other
))
337 Return the matrix representation of this operator with respect
338 to the default bases of its (co)domain.
342 sage: J = RealSymmetricEJA(2)
343 sage: mat = matrix(J.base_ring(), J.dimension(), range(9))
344 sage: f = FiniteDimensionalEuclideanJordanAlgebraOperator(J,J,mat)
354 class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra
):
356 def __classcall_private__(cls
,
360 assume_associative
=False,
365 mult_table
= [b
.base_extend(field
) for b
in mult_table
]
368 if not (is_Matrix(b
) and b
.dimensions() == (n
, n
)):
369 raise ValueError("input is not a multiplication table")
370 mult_table
= tuple(mult_table
)
372 cat
= FiniteDimensionalAlgebrasWithBasis(field
)
373 cat
.or_subcategory(category
)
374 if assume_associative
:
375 cat
= cat
.Associative()
377 names
= normalize_names(n
, names
)
379 fda
= super(FiniteDimensionalEuclideanJordanAlgebra
, cls
)
380 return fda
.__classcall
__(cls
,
383 assume_associative
=assume_associative
,
387 natural_basis
=natural_basis
)
394 assume_associative
=False,
401 By definition, Jordan multiplication commutes::
403 sage: set_random_seed()
404 sage: J = random_eja()
405 sage: x = J.random_element()
406 sage: y = J.random_element()
412 self
._natural
_basis
= natural_basis
413 self
._multiplication
_table
= mult_table
414 fda
= super(FiniteDimensionalEuclideanJordanAlgebra
, self
)
423 Return a string representation of ``self``.
425 fmt
= "Euclidean Jordan algebra of degree {} over {}"
426 return fmt
.format(self
.degree(), self
.base_ring())
429 def _a_regular_element(self
):
431 Guess a regular element. Needed to compute the basis for our
432 characteristic polynomial coefficients.
435 z
= self
.sum( (i
+1)*gs
[i
] for i
in range(len(gs
)) )
436 if not z
.is_regular():
437 raise ValueError("don't know a regular element")
442 def _charpoly_basis_space(self
):
444 Return the vector space spanned by the basis used in our
445 characteristic polynomial coefficients. This is used not only to
446 compute those coefficients, but also any time we need to
447 evaluate the coefficients (like when we compute the trace or
450 z
= self
._a
_regular
_element
()
451 V
= self
.vector_space()
452 V1
= V
.span_of_basis( (z
**k
).vector() for k
in range(self
.rank()) )
453 b
= (V1
.basis() + V1
.complement().basis())
454 return V
.span_of_basis(b
)
458 def _charpoly_coeff(self
, i
):
460 Return the coefficient polynomial "a_{i}" of this algebra's
461 general characteristic polynomial.
463 Having this be a separate cached method lets us compute and
464 store the trace/determinant (a_{r-1} and a_{0} respectively)
465 separate from the entire characteristic polynomial.
467 (A_of_x
, x
, xr
, detA
) = self
._charpoly
_matrix
_system
()
468 R
= A_of_x
.base_ring()
470 # Guaranteed by theory
473 # Danger: the in-place modification is done for performance
474 # reasons (reconstructing a matrix with huge polynomial
475 # entries is slow), but I don't know how cached_method works,
476 # so it's highly possible that we're modifying some global
477 # list variable by reference, here. In other words, you
478 # probably shouldn't call this method twice on the same
479 # algebra, at the same time, in two threads
480 Ai_orig
= A_of_x
.column(i
)
481 A_of_x
.set_column(i
,xr
)
482 numerator
= A_of_x
.det()
483 A_of_x
.set_column(i
,Ai_orig
)
485 # We're relying on the theory here to ensure that each a_i is
486 # indeed back in R, and the added negative signs are to make
487 # the whole charpoly expression sum to zero.
488 return R(-numerator
/detA
)
492 def _charpoly_matrix_system(self
):
494 Compute the matrix whose entries A_ij are polynomials in
495 X1,...,XN, the vector ``x`` of variables X1,...,XN, the vector
496 corresponding to `x^r` and the determinent of the matrix A =
497 [A_ij]. In other words, all of the fixed (cachable) data needed
498 to compute the coefficients of the characteristic polynomial.
503 # Construct a new algebra over a multivariate polynomial ring...
504 names
= ['X' + str(i
) for i
in range(1,n
+1)]
505 R
= PolynomialRing(self
.base_ring(), names
)
506 J
= FiniteDimensionalEuclideanJordanAlgebra(R
,
507 self
._multiplication
_table
,
510 idmat
= identity_matrix(J
.base_ring(), n
)
512 W
= self
._charpoly
_basis
_space
()
513 W
= W
.change_ring(R
.fraction_field())
515 # Starting with the standard coordinates x = (X1,X2,...,Xn)
516 # and then converting the entries to W-coordinates allows us
517 # to pass in the standard coordinates to the charpoly and get
518 # back the right answer. Specifically, with x = (X1,X2,...,Xn),
521 # W.coordinates(x^2) eval'd at (standard z-coords)
525 # W-coords of (standard coords of x^2 eval'd at std-coords of z)
527 # We want the middle equivalent thing in our matrix, but use
528 # the first equivalent thing instead so that we can pass in
529 # standard coordinates.
530 x
= J(vector(R
, R
.gens()))
531 l1
= [column_matrix(W
.coordinates((x
**k
).vector())) for k
in range(r
)]
532 l2
= [idmat
.column(k
-1).column() for k
in range(r
+1, n
+1)]
533 A_of_x
= block_matrix(R
, 1, n
, (l1
+ l2
))
534 xr
= W
.coordinates((x
**r
).vector())
535 return (A_of_x
, x
, xr
, A_of_x
.det())
539 def characteristic_polynomial(self
):
544 This implementation doesn't guarantee that the polynomial
545 denominator in the coefficients is not identically zero, so
546 theoretically it could crash. The way that this is handled
547 in e.g. Faraut and Koranyi is to use a basis that guarantees
548 the denominator is non-zero. But, doing so requires knowledge
549 of at least one regular element, and we don't even know how
550 to do that. The trade-off is that, if we use the standard basis,
551 the resulting polynomial will accept the "usual" coordinates. In
552 other words, we don't have to do a change of basis before e.g.
553 computing the trace or determinant.
557 The characteristic polynomial in the spin algebra is given in
558 Alizadeh, Example 11.11::
560 sage: J = JordanSpinEJA(3)
561 sage: p = J.characteristic_polynomial(); p
562 X1^2 - X2^2 - X3^2 + (-2*t)*X1 + t^2
563 sage: xvec = J.one().vector()
571 # The list of coefficient polynomials a_1, a_2, ..., a_n.
572 a
= [ self
._charpoly
_coeff
(i
) for i
in range(n
) ]
574 # We go to a bit of trouble here to reorder the
575 # indeterminates, so that it's easier to evaluate the
576 # characteristic polynomial at x's coordinates and get back
577 # something in terms of t, which is what we want.
579 S
= PolynomialRing(self
.base_ring(),'t')
581 S
= PolynomialRing(S
, R
.variable_names())
584 # Note: all entries past the rth should be zero. The
585 # coefficient of the highest power (x^r) is 1, but it doesn't
586 # appear in the solution vector which contains coefficients
587 # for the other powers (to make them sum to x^r).
589 a
[r
] = 1 # corresponds to x^r
591 # When the rank is equal to the dimension, trying to
592 # assign a[r] goes out-of-bounds.
593 a
.append(1) # corresponds to x^r
595 return sum( a
[k
]*(t
**k
) for k
in range(len(a
)) )
598 def inner_product(self
, x
, y
):
600 The inner product associated with this Euclidean Jordan algebra.
602 Defaults to the trace inner product, but can be overridden by
603 subclasses if they are sure that the necessary properties are
608 The inner product must satisfy its axiom for this algebra to truly
609 be a Euclidean Jordan Algebra::
611 sage: set_random_seed()
612 sage: J = random_eja()
613 sage: x = J.random_element()
614 sage: y = J.random_element()
615 sage: z = J.random_element()
616 sage: (x*y).inner_product(z) == y.inner_product(x*z)
620 if (not x
in self
) or (not y
in self
):
621 raise TypeError("arguments must live in this algebra")
622 return x
.trace_inner_product(y
)
625 def natural_basis(self
):
627 Return a more-natural representation of this algebra's basis.
629 Every finite-dimensional Euclidean Jordan Algebra is a direct
630 sum of five simple algebras, four of which comprise Hermitian
631 matrices. This method returns the original "natural" basis
632 for our underlying vector space. (Typically, the natural basis
633 is used to construct the multiplication table in the first place.)
635 Note that this will always return a matrix. The standard basis
636 in `R^n` will be returned as `n`-by-`1` column matrices.
640 sage: J = RealSymmetricEJA(2)
643 sage: J.natural_basis()
651 sage: J = JordanSpinEJA(2)
654 sage: J.natural_basis()
661 if self
._natural
_basis
is None:
662 return tuple( b
.vector().column() for b
in self
.basis() )
664 return self
._natural
_basis
669 Return the rank of this EJA.
671 if self
._rank
is None:
672 raise ValueError("no rank specified at genesis")
676 def vector_space(self
):
678 Return the vector space that underlies this algebra.
682 sage: J = RealSymmetricEJA(2)
683 sage: J.vector_space()
684 Vector space of dimension 3 over Rational Field
687 return self
.zero().vector().parent().ambient_vector_space()
690 class Element(FiniteDimensionalAlgebraElement
):
692 An element of a Euclidean Jordan algebra.
697 Oh man, I should not be doing this. This hides the "disabled"
698 methods ``left_matrix`` and ``matrix`` from introspection;
699 in particular it removes them from tab-completion.
701 return filter(lambda s
: s
not in ['left_matrix', 'matrix'],
702 dir(self
.__class
__) )
705 def __init__(self
, A
, elt
=None):
709 The identity in `S^n` is converted to the identity in the EJA::
711 sage: J = RealSymmetricEJA(3)
712 sage: I = identity_matrix(QQ,3)
713 sage: J(I) == J.one()
716 This skew-symmetric matrix can't be represented in the EJA::
718 sage: J = RealSymmetricEJA(3)
719 sage: A = matrix(QQ,3, lambda i,j: i-j)
721 Traceback (most recent call last):
723 ArithmeticError: vector is not in free module
726 # Goal: if we're given a matrix, and if it lives in our
727 # parent algebra's "natural ambient space," convert it
728 # into an algebra element.
730 # The catch is, we make a recursive call after converting
731 # the given matrix into a vector that lives in the algebra.
732 # This we need to try the parent class initializer first,
733 # to avoid recursing forever if we're given something that
734 # already fits into the algebra, but also happens to live
735 # in the parent's "natural ambient space" (this happens with
738 FiniteDimensionalAlgebraElement
.__init
__(self
, A
, elt
)
740 natural_basis
= A
.natural_basis()
741 if elt
in natural_basis
[0].matrix_space():
742 # Thanks for nothing! Matrix spaces aren't vector
743 # spaces in Sage, so we have to figure out its
744 # natural-basis coordinates ourselves.
745 V
= VectorSpace(elt
.base_ring(), elt
.nrows()**2)
746 W
= V
.span( _mat2vec(s
) for s
in natural_basis
)
747 coords
= W
.coordinates(_mat2vec(elt
))
748 FiniteDimensionalAlgebraElement
.__init
__(self
, A
, coords
)
750 def __pow__(self
, n
):
752 Return ``self`` raised to the power ``n``.
754 Jordan algebras are always power-associative; see for
755 example Faraut and Koranyi, Proposition II.1.2 (ii).
759 We have to override this because our superclass uses row vectors
760 instead of column vectors! We, on the other hand, assume column
765 sage: set_random_seed()
766 sage: x = random_eja().random_element()
767 sage: x.operator()(x) == (x^2)
770 A few examples of power-associativity::
772 sage: set_random_seed()
773 sage: x = random_eja().random_element()
774 sage: x*(x*x)*(x*x) == x^5
776 sage: (x*x)*(x*x*x) == x^5
779 We also know that powers operator-commute (Koecher, Chapter
782 sage: set_random_seed()
783 sage: x = random_eja().random_element()
784 sage: m = ZZ.random_element(0,10)
785 sage: n = ZZ.random_element(0,10)
786 sage: Lxm = (x^m).operator()
787 sage: Lxn = (x^n).operator()
788 sage: Lxm*Lxn == Lxn*Lxm
793 return self
.parent().one()
797 return (self
.operator()**(n
-1))(self
)
800 def apply_univariate_polynomial(self
, p
):
802 Apply the univariate polynomial ``p`` to this element.
804 A priori, SageMath won't allow us to apply a univariate
805 polynomial to an element of an EJA, because we don't know
806 that EJAs are rings (they are usually not associative). Of
807 course, we know that EJAs are power-associative, so the
808 operation is ultimately kosher. This function sidesteps
809 the CAS to get the answer we want and expect.
813 sage: R = PolynomialRing(QQ, 't')
815 sage: p = t^4 - t^3 + 5*t - 2
816 sage: J = RealCartesianProductEJA(5)
817 sage: J.one().apply_univariate_polynomial(p) == 3*J.one()
822 We should always get back an element of the algebra::
824 sage: set_random_seed()
825 sage: p = PolynomialRing(QQ, 't').random_element()
826 sage: J = random_eja()
827 sage: x = J.random_element()
828 sage: x.apply_univariate_polynomial(p) in J
832 if len(p
.variables()) > 1:
833 raise ValueError("not a univariate polynomial")
836 # Convert the coeficcients to the parent's base ring,
837 # because a priori they might live in an (unnecessarily)
838 # larger ring for which P.sum() would fail below.
839 cs
= [ R(c
) for c
in p
.coefficients(sparse
=False) ]
840 return P
.sum( cs
[k
]*(self
**k
) for k
in range(len(cs
)) )
843 def characteristic_polynomial(self
):
845 Return the characteristic polynomial of this element.
849 The rank of `R^3` is three, and the minimal polynomial of
850 the identity element is `(t-1)` from which it follows that
851 the characteristic polynomial should be `(t-1)^3`::
853 sage: J = RealCartesianProductEJA(3)
854 sage: J.one().characteristic_polynomial()
855 t^3 - 3*t^2 + 3*t - 1
857 Likewise, the characteristic of the zero element in the
858 rank-three algebra `R^{n}` should be `t^{3}`::
860 sage: J = RealCartesianProductEJA(3)
861 sage: J.zero().characteristic_polynomial()
864 The characteristic polynomial of an element should evaluate
865 to zero on that element::
867 sage: set_random_seed()
868 sage: x = RealCartesianProductEJA(3).random_element()
869 sage: p = x.characteristic_polynomial()
870 sage: x.apply_univariate_polynomial(p)
874 p
= self
.parent().characteristic_polynomial()
875 return p(*self
.vector())
878 def inner_product(self
, other
):
880 Return the parent algebra's inner product of myself and ``other``.
884 The inner product in the Jordan spin algebra is the usual
885 inner product on `R^n` (this example only works because the
886 basis for the Jordan algebra is the standard basis in `R^n`)::
888 sage: J = JordanSpinEJA(3)
889 sage: x = vector(QQ,[1,2,3])
890 sage: y = vector(QQ,[4,5,6])
891 sage: x.inner_product(y)
893 sage: J(x).inner_product(J(y))
896 The inner product on `S^n` is `<X,Y> = trace(X*Y)`, where
897 multiplication is the usual matrix multiplication in `S^n`,
898 so the inner product of the identity matrix with itself
901 sage: J = RealSymmetricEJA(3)
902 sage: J.one().inner_product(J.one())
905 Likewise, the inner product on `C^n` is `<X,Y> =
906 Re(trace(X*Y))`, where we must necessarily take the real
907 part because the product of Hermitian matrices may not be
910 sage: J = ComplexHermitianEJA(3)
911 sage: J.one().inner_product(J.one())
914 Ditto for the quaternions::
916 sage: J = QuaternionHermitianEJA(3)
917 sage: J.one().inner_product(J.one())
922 Ensure that we can always compute an inner product, and that
923 it gives us back a real number::
925 sage: set_random_seed()
926 sage: J = random_eja()
927 sage: x = J.random_element()
928 sage: y = J.random_element()
929 sage: x.inner_product(y) in RR
935 raise TypeError("'other' must live in the same algebra")
937 return P
.inner_product(self
, other
)
940 def operator_commutes_with(self
, other
):
942 Return whether or not this element operator-commutes
947 The definition of a Jordan algebra says that any element
948 operator-commutes with its square::
950 sage: set_random_seed()
951 sage: x = random_eja().random_element()
952 sage: x.operator_commutes_with(x^2)
957 Test Lemma 1 from Chapter III of Koecher::
959 sage: set_random_seed()
960 sage: J = random_eja()
961 sage: u = J.random_element()
962 sage: v = J.random_element()
963 sage: lhs = u.operator_commutes_with(u*v)
964 sage: rhs = v.operator_commutes_with(u^2)
968 Test the first polarization identity from my notes, Koecher Chapter
969 III, or from Baes (2.3)::
971 sage: set_random_seed()
972 sage: J = random_eja()
973 sage: x = J.random_element()
974 sage: y = J.random_element()
975 sage: Lx = x.operator()
976 sage: Ly = y.operator()
977 sage: Lxx = (x*x).operator()
978 sage: Lxy = (x*y).operator()
979 sage: bool(2*Lx*Lxy + Ly*Lxx == 2*Lxy*Lx + Lxx*Ly)
982 Test the second polarization identity from my notes or from
985 sage: set_random_seed()
986 sage: J = random_eja()
987 sage: x = J.random_element()
988 sage: y = J.random_element()
989 sage: z = J.random_element()
990 sage: Lx = x.operator()
991 sage: Ly = y.operator()
992 sage: Lz = z.operator()
993 sage: Lzy = (z*y).operator()
994 sage: Lxy = (x*y).operator()
995 sage: Lxz = (x*z).operator()
996 sage: bool(Lx*Lzy + Lz*Lxy + Ly*Lxz == Lzy*Lx + Lxy*Lz + Lxz*Ly)
999 Test the third polarization identity from my notes or from
1002 sage: set_random_seed()
1003 sage: J = random_eja()
1004 sage: u = J.random_element()
1005 sage: y = J.random_element()
1006 sage: z = J.random_element()
1007 sage: Lu = u.operator()
1008 sage: Ly = y.operator()
1009 sage: Lz = z.operator()
1010 sage: Lzy = (z*y).operator()
1011 sage: Luy = (u*y).operator()
1012 sage: Luz = (u*z).operator()
1013 sage: Luyz = (u*(y*z)).operator()
1014 sage: lhs = Lu*Lzy + Lz*Luy + Ly*Luz
1015 sage: rhs = Luyz + Ly*Lu*Lz + Lz*Lu*Ly
1016 sage: bool(lhs == rhs)
1020 if not other
in self
.parent():
1021 raise TypeError("'other' must live in the same algebra")
1024 B
= other
.operator()
1030 Return my determinant, the product of my eigenvalues.
1034 sage: J = JordanSpinEJA(2)
1035 sage: e0,e1 = J.gens()
1036 sage: x = sum( J.gens() )
1042 sage: J = JordanSpinEJA(3)
1043 sage: e0,e1,e2 = J.gens()
1044 sage: x = sum( J.gens() )
1050 An element is invertible if and only if its determinant is
1053 sage: set_random_seed()
1054 sage: x = random_eja().random_element()
1055 sage: x.is_invertible() == (x.det() != 0)
1061 p
= P
._charpoly
_coeff
(0)
1062 # The _charpoly_coeff function already adds the factor of
1063 # -1 to ensure that _charpoly_coeff(0) is really what
1064 # appears in front of t^{0} in the charpoly. However,
1065 # we want (-1)^r times THAT for the determinant.
1066 return ((-1)**r
)*p(*self
.vector())
1071 Return the Jordan-multiplicative inverse of this element.
1075 We appeal to the quadratic representation as in Koecher's
1076 Theorem 12 in Chapter III, Section 5.
1080 The inverse in the spin factor algebra is given in Alizadeh's
1083 sage: set_random_seed()
1084 sage: n = ZZ.random_element(1,10)
1085 sage: J = JordanSpinEJA(n)
1086 sage: x = J.random_element()
1087 sage: while not x.is_invertible():
1088 ....: x = J.random_element()
1089 sage: x_vec = x.vector()
1091 sage: x_bar = x_vec[1:]
1092 sage: coeff = ~(x0^2 - x_bar.inner_product(x_bar))
1093 sage: inv_vec = x_vec.parent()([x0] + (-x_bar).list())
1094 sage: x_inverse = coeff*inv_vec
1095 sage: x.inverse() == J(x_inverse)
1100 The identity element is its own inverse::
1102 sage: set_random_seed()
1103 sage: J = random_eja()
1104 sage: J.one().inverse() == J.one()
1107 If an element has an inverse, it acts like one::
1109 sage: set_random_seed()
1110 sage: J = random_eja()
1111 sage: x = J.random_element()
1112 sage: (not x.is_invertible()) or (x.inverse()*x == J.one())
1115 The inverse of the inverse is what we started with::
1117 sage: set_random_seed()
1118 sage: J = random_eja()
1119 sage: x = J.random_element()
1120 sage: (not x.is_invertible()) or (x.inverse().inverse() == x)
1123 The zero element is never invertible::
1125 sage: set_random_seed()
1126 sage: J = random_eja().zero().inverse()
1127 Traceback (most recent call last):
1129 ValueError: element is not invertible
1132 if not self
.is_invertible():
1133 raise ValueError("element is not invertible")
1135 return (~self
.quadratic_representation())(self
)
1138 def is_invertible(self
):
1140 Return whether or not this element is invertible.
1142 We can't use the superclass method because it relies on
1143 the algebra being associative.
1147 The usual way to do this is to check if the determinant is
1148 zero, but we need the characteristic polynomial for the
1149 determinant. The minimal polynomial is a lot easier to get,
1150 so we use Corollary 2 in Chapter V of Koecher to check
1151 whether or not the paren't algebra's zero element is a root
1152 of this element's minimal polynomial.
1156 The identity element is always invertible::
1158 sage: set_random_seed()
1159 sage: J = random_eja()
1160 sage: J.one().is_invertible()
1163 The zero element is never invertible::
1165 sage: set_random_seed()
1166 sage: J = random_eja()
1167 sage: J.zero().is_invertible()
1171 zero
= self
.parent().zero()
1172 p
= self
.minimal_polynomial()
1173 return not (p(zero
) == zero
)
1176 def is_nilpotent(self
):
1178 Return whether or not some power of this element is zero.
1180 The superclass method won't work unless we're in an
1181 associative algebra, and we aren't. However, we generate
1182 an assocoative subalgebra and we're nilpotent there if and
1183 only if we're nilpotent here (probably).
1187 The identity element is never nilpotent::
1189 sage: set_random_seed()
1190 sage: random_eja().one().is_nilpotent()
1193 The additive identity is always nilpotent::
1195 sage: set_random_seed()
1196 sage: random_eja().zero().is_nilpotent()
1200 # The element we're going to call "is_nilpotent()" on.
1201 # Either myself, interpreted as an element of a finite-
1202 # dimensional algebra, or an element of an associative
1206 if self
.parent().is_associative():
1207 elt
= FiniteDimensionalAlgebraElement(self
.parent(), self
)
1209 V
= self
.span_of_powers()
1210 assoc_subalg
= self
.subalgebra_generated_by()
1211 # Mis-design warning: the basis used for span_of_powers()
1212 # and subalgebra_generated_by() must be the same, and in
1214 elt
= assoc_subalg(V
.coordinates(self
.vector()))
1216 # Recursive call, but should work since elt lives in an
1217 # associative algebra.
1218 return elt
.is_nilpotent()
1221 def is_regular(self
):
1223 Return whether or not this is a regular element.
1227 The identity element always has degree one, but any element
1228 linearly-independent from it is regular::
1230 sage: J = JordanSpinEJA(5)
1231 sage: J.one().is_regular()
1233 sage: e0, e1, e2, e3, e4 = J.gens() # e0 is the identity
1234 sage: for x in J.gens():
1235 ....: (J.one() + x).is_regular()
1243 return self
.degree() == self
.parent().rank()
1248 Compute the degree of this element the straightforward way
1249 according to the definition; by appending powers to a list
1250 and figuring out its dimension (that is, whether or not
1251 they're linearly dependent).
1255 sage: J = JordanSpinEJA(4)
1256 sage: J.one().degree()
1258 sage: e0,e1,e2,e3 = J.gens()
1259 sage: (e0 - e1).degree()
1262 In the spin factor algebra (of rank two), all elements that
1263 aren't multiples of the identity are regular::
1265 sage: set_random_seed()
1266 sage: n = ZZ.random_element(1,10)
1267 sage: J = JordanSpinEJA(n)
1268 sage: x = J.random_element()
1269 sage: x == x.coefficient(0)*J.one() or x.degree() == 2
1273 return self
.span_of_powers().dimension()
1276 def left_matrix(self
):
1278 Our parent class defines ``left_matrix`` and ``matrix``
1279 methods whose names are misleading. We don't want them.
1281 raise NotImplementedError("use operator().matrix() instead")
1283 matrix
= left_matrix
1286 def minimal_polynomial(self
):
1288 Return the minimal polynomial of this element,
1289 as a function of the variable `t`.
1293 We restrict ourselves to the associative subalgebra
1294 generated by this element, and then return the minimal
1295 polynomial of this element's operator matrix (in that
1296 subalgebra). This works by Baes Proposition 2.3.16.
1300 The minimal polynomial of the identity and zero elements are
1303 sage: set_random_seed()
1304 sage: J = random_eja()
1305 sage: J.one().minimal_polynomial()
1307 sage: J.zero().minimal_polynomial()
1310 The degree of an element is (by one definition) the degree
1311 of its minimal polynomial::
1313 sage: set_random_seed()
1314 sage: x = random_eja().random_element()
1315 sage: x.degree() == x.minimal_polynomial().degree()
1318 The minimal polynomial and the characteristic polynomial coincide
1319 and are known (see Alizadeh, Example 11.11) for all elements of
1320 the spin factor algebra that aren't scalar multiples of the
1323 sage: set_random_seed()
1324 sage: n = ZZ.random_element(2,10)
1325 sage: J = JordanSpinEJA(n)
1326 sage: y = J.random_element()
1327 sage: while y == y.coefficient(0)*J.one():
1328 ....: y = J.random_element()
1329 sage: y0 = y.vector()[0]
1330 sage: y_bar = y.vector()[1:]
1331 sage: actual = y.minimal_polynomial()
1332 sage: t = PolynomialRing(J.base_ring(),'t').gen(0)
1333 sage: expected = t^2 - 2*y0*t + (y0^2 - norm(y_bar)^2)
1334 sage: bool(actual == expected)
1337 The minimal polynomial should always kill its element::
1339 sage: set_random_seed()
1340 sage: x = random_eja().random_element()
1341 sage: p = x.minimal_polynomial()
1342 sage: x.apply_univariate_polynomial(p)
1346 V
= self
.span_of_powers()
1347 assoc_subalg
= self
.subalgebra_generated_by()
1348 # Mis-design warning: the basis used for span_of_powers()
1349 # and subalgebra_generated_by() must be the same, and in
1351 elt
= assoc_subalg(V
.coordinates(self
.vector()))
1353 # We get back a symbolic polynomial in 'x' but want a real
1354 # polynomial in 't'.
1355 p_of_x
= elt
.operator_matrix().minimal_polynomial()
1356 return p_of_x
.change_variable_name('t')
1359 def natural_representation(self
):
1361 Return a more-natural representation of this element.
1363 Every finite-dimensional Euclidean Jordan Algebra is a
1364 direct sum of five simple algebras, four of which comprise
1365 Hermitian matrices. This method returns the original
1366 "natural" representation of this element as a Hermitian
1367 matrix, if it has one. If not, you get the usual representation.
1371 sage: J = ComplexHermitianEJA(3)
1374 sage: J.one().natural_representation()
1384 sage: J = QuaternionHermitianEJA(3)
1387 sage: J.one().natural_representation()
1388 [1 0 0 0 0 0 0 0 0 0 0 0]
1389 [0 1 0 0 0 0 0 0 0 0 0 0]
1390 [0 0 1 0 0 0 0 0 0 0 0 0]
1391 [0 0 0 1 0 0 0 0 0 0 0 0]
1392 [0 0 0 0 1 0 0 0 0 0 0 0]
1393 [0 0 0 0 0 1 0 0 0 0 0 0]
1394 [0 0 0 0 0 0 1 0 0 0 0 0]
1395 [0 0 0 0 0 0 0 1 0 0 0 0]
1396 [0 0 0 0 0 0 0 0 1 0 0 0]
1397 [0 0 0 0 0 0 0 0 0 1 0 0]
1398 [0 0 0 0 0 0 0 0 0 0 1 0]
1399 [0 0 0 0 0 0 0 0 0 0 0 1]
1402 B
= self
.parent().natural_basis()
1403 W
= B
[0].matrix_space()
1404 return W
.linear_combination(zip(self
.vector(), B
))
1409 Return the left-multiplication-by-this-element
1410 operator on the ambient algebra.
1414 sage: set_random_seed()
1415 sage: J = random_eja()
1416 sage: x = J.random_element()
1417 sage: y = J.random_element()
1418 sage: x.operator()(y) == x*y
1420 sage: y.operator()(x) == x*y
1425 return FiniteDimensionalEuclideanJordanAlgebraOperator(
1427 self
.operator_matrix() )
1431 def operator_matrix(self
):
1433 Return the matrix that represents left- (or right-)
1434 multiplication by this element in the parent algebra.
1436 We implement this ourselves to work around the fact that
1437 our parent class represents everything with row vectors.
1441 Ensure that our operator's ``matrix`` method agrees with
1442 this implementation::
1444 sage: set_random_seed()
1445 sage: J = random_eja()
1446 sage: x = J.random_element()
1447 sage: x.operator().matrix() == x.operator_matrix()
1451 fda_elt
= FiniteDimensionalAlgebraElement(self
.parent(), self
)
1452 return fda_elt
.matrix().transpose()
1455 def quadratic_representation(self
, other
=None):
1457 Return the quadratic representation of this element.
1461 The explicit form in the spin factor algebra is given by
1462 Alizadeh's Example 11.12::
1464 sage: set_random_seed()
1465 sage: n = ZZ.random_element(1,10)
1466 sage: J = JordanSpinEJA(n)
1467 sage: x = J.random_element()
1468 sage: x_vec = x.vector()
1470 sage: x_bar = x_vec[1:]
1471 sage: A = matrix(QQ, 1, [x_vec.inner_product(x_vec)])
1472 sage: B = 2*x0*x_bar.row()
1473 sage: C = 2*x0*x_bar.column()
1474 sage: D = identity_matrix(QQ, n-1)
1475 sage: D = (x0^2 - x_bar.inner_product(x_bar))*D
1476 sage: D = D + 2*x_bar.tensor_product(x_bar)
1477 sage: Q = block_matrix(2,2,[A,B,C,D])
1478 sage: Q == x.quadratic_representation().matrix()
1481 Test all of the properties from Theorem 11.2 in Alizadeh::
1483 sage: set_random_seed()
1484 sage: J = random_eja()
1485 sage: x = J.random_element()
1486 sage: y = J.random_element()
1487 sage: Lx = x.operator()
1488 sage: Lxx = (x*x).operator()
1489 sage: Qx = x.quadratic_representation()
1490 sage: Qy = y.quadratic_representation()
1491 sage: Qxy = x.quadratic_representation(y)
1492 sage: Qex = J.one().quadratic_representation(x)
1493 sage: n = ZZ.random_element(10)
1494 sage: Qxn = (x^n).quadratic_representation()
1498 sage: 2*Qxy == (x+y).quadratic_representation() - Qx - Qy
1501 Property 2 (multiply on the right for :trac:`28272`):
1503 sage: alpha = QQ.random_element()
1504 sage: (alpha*x).quadratic_representation() == Qx*(alpha^2)
1509 sage: not x.is_invertible() or ( Qx(x.inverse()) == x )
1512 sage: not x.is_invertible() or (
1515 ....: x.inverse().quadratic_representation() )
1518 sage: Qxy(J.one()) == x*y
1523 sage: not x.is_invertible() or (
1524 ....: x.quadratic_representation(x.inverse())*Qx
1525 ....: == Qx*x.quadratic_representation(x.inverse()) )
1528 sage: not x.is_invertible() or (
1529 ....: x.quadratic_representation(x.inverse())*Qx
1531 ....: 2*x.operator()*Qex - Qx )
1534 sage: 2*x.operator()*Qex - Qx == Lxx
1539 sage: Qy(x).quadratic_representation() == Qy*Qx*Qy
1549 sage: not x.is_invertible() or (
1550 ....: Qx*x.inverse().operator() == Lx )
1555 sage: not x.operator_commutes_with(y) or (
1556 ....: Qx(y)^n == Qxn(y^n) )
1562 elif not other
in self
.parent():
1563 raise TypeError("'other' must live in the same algebra")
1566 M
= other
.operator()
1567 return ( L
*M
+ M
*L
- (self
*other
).operator() )
1570 def span_of_powers(self
):
1572 Return the vector space spanned by successive powers of
1575 # The dimension of the subalgebra can't be greater than
1576 # the big algebra, so just put everything into a list
1577 # and let span() get rid of the excess.
1579 # We do the extra ambient_vector_space() in case we're messing
1580 # with polynomials and the direct parent is a module.
1581 V
= self
.parent().vector_space()
1582 return V
.span( (self
**d
).vector() for d
in xrange(V
.dimension()) )
1585 def subalgebra_generated_by(self
):
1587 Return the associative subalgebra of the parent EJA generated
1592 sage: set_random_seed()
1593 sage: x = random_eja().random_element()
1594 sage: x.subalgebra_generated_by().is_associative()
1597 Squaring in the subalgebra should work the same as in
1600 sage: set_random_seed()
1601 sage: x = random_eja().random_element()
1602 sage: u = x.subalgebra_generated_by().random_element()
1603 sage: u.operator()(u) == u^2
1607 # First get the subspace spanned by the powers of myself...
1608 V
= self
.span_of_powers()
1609 F
= self
.base_ring()
1611 # Now figure out the entries of the right-multiplication
1612 # matrix for the successive basis elements b0, b1,... of
1615 for b_right
in V
.basis():
1616 eja_b_right
= self
.parent()(b_right
)
1618 # The first row of the right-multiplication matrix by
1619 # b1 is what we get if we apply that matrix to b1. The
1620 # second row of the right multiplication matrix by b1
1621 # is what we get when we apply that matrix to b2...
1623 # IMPORTANT: this assumes that all vectors are COLUMN
1624 # vectors, unlike our superclass (which uses row vectors).
1625 for b_left
in V
.basis():
1626 eja_b_left
= self
.parent()(b_left
)
1627 # Multiply in the original EJA, but then get the
1628 # coordinates from the subalgebra in terms of its
1630 this_row
= V
.coordinates((eja_b_left
*eja_b_right
).vector())
1631 b_right_rows
.append(this_row
)
1632 b_right_matrix
= matrix(F
, b_right_rows
)
1633 mats
.append(b_right_matrix
)
1635 # It's an algebra of polynomials in one element, and EJAs
1636 # are power-associative.
1638 # TODO: choose generator names intelligently.
1639 return FiniteDimensionalEuclideanJordanAlgebra(F
, mats
, assume_associative
=True, names
='f')
1642 def subalgebra_idempotent(self
):
1644 Find an idempotent in the associative subalgebra I generate
1645 using Proposition 2.3.5 in Baes.
1649 sage: set_random_seed()
1650 sage: J = random_eja()
1651 sage: x = J.random_element()
1652 sage: while x.is_nilpotent():
1653 ....: x = J.random_element()
1654 sage: c = x.subalgebra_idempotent()
1659 if self
.is_nilpotent():
1660 raise ValueError("this only works with non-nilpotent elements!")
1662 V
= self
.span_of_powers()
1663 J
= self
.subalgebra_generated_by()
1664 # Mis-design warning: the basis used for span_of_powers()
1665 # and subalgebra_generated_by() must be the same, and in
1667 u
= J(V
.coordinates(self
.vector()))
1669 # The image of the matrix of left-u^m-multiplication
1670 # will be minimal for some natural number s...
1672 minimal_dim
= V
.dimension()
1673 for i
in xrange(1, V
.dimension()):
1674 this_dim
= (u
**i
).operator_matrix().image().dimension()
1675 if this_dim
< minimal_dim
:
1676 minimal_dim
= this_dim
1679 # Now minimal_matrix should correspond to the smallest
1680 # non-zero subspace in Baes's (or really, Koecher's)
1683 # However, we need to restrict the matrix to work on the
1684 # subspace... or do we? Can't we just solve, knowing that
1685 # A(c) = u^(s+1) should have a solution in the big space,
1688 # Beware, solve_right() means that we're using COLUMN vectors.
1689 # Our FiniteDimensionalAlgebraElement superclass uses rows.
1691 A
= u_next
.operator_matrix()
1692 c_coordinates
= A
.solve_right(u_next
.vector())
1694 # Now c_coordinates is the idempotent we want, but it's in
1695 # the coordinate system of the subalgebra.
1697 # We need the basis for J, but as elements of the parent algebra.
1699 basis
= [self
.parent(v
) for v
in V
.basis()]
1700 return self
.parent().linear_combination(zip(c_coordinates
, basis
))
1705 Return my trace, the sum of my eigenvalues.
1709 sage: J = JordanSpinEJA(3)
1710 sage: x = sum(J.gens())
1716 sage: J = RealCartesianProductEJA(5)
1717 sage: J.one().trace()
1722 The trace of an element is a real number::
1724 sage: set_random_seed()
1725 sage: J = random_eja()
1726 sage: J.random_element().trace() in J.base_ring()
1732 p
= P
._charpoly
_coeff
(r
-1)
1733 # The _charpoly_coeff function already adds the factor of
1734 # -1 to ensure that _charpoly_coeff(r-1) is really what
1735 # appears in front of t^{r-1} in the charpoly. However,
1736 # we want the negative of THAT for the trace.
1737 return -p(*self
.vector())
1740 def trace_inner_product(self
, other
):
1742 Return the trace inner product of myself and ``other``.
1746 The trace inner product is commutative::
1748 sage: set_random_seed()
1749 sage: J = random_eja()
1750 sage: x = J.random_element(); y = J.random_element()
1751 sage: x.trace_inner_product(y) == y.trace_inner_product(x)
1754 The trace inner product is bilinear::
1756 sage: set_random_seed()
1757 sage: J = random_eja()
1758 sage: x = J.random_element()
1759 sage: y = J.random_element()
1760 sage: z = J.random_element()
1761 sage: a = QQ.random_element();
1762 sage: actual = (a*(x+z)).trace_inner_product(y)
1763 sage: expected = ( a*x.trace_inner_product(y) +
1764 ....: a*z.trace_inner_product(y) )
1765 sage: actual == expected
1767 sage: actual = x.trace_inner_product(a*(y+z))
1768 sage: expected = ( a*x.trace_inner_product(y) +
1769 ....: a*x.trace_inner_product(z) )
1770 sage: actual == expected
1773 The trace inner product satisfies the compatibility
1774 condition in the definition of a Euclidean Jordan algebra::
1776 sage: set_random_seed()
1777 sage: J = random_eja()
1778 sage: x = J.random_element()
1779 sage: y = J.random_element()
1780 sage: z = J.random_element()
1781 sage: (x*y).trace_inner_product(z) == y.trace_inner_product(x*z)
1785 if not other
in self
.parent():
1786 raise TypeError("'other' must live in the same algebra")
1788 return (self
*other
).trace()
1791 class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1793 Return the Euclidean Jordan Algebra corresponding to the set
1794 `R^n` under the Hadamard product.
1796 Note: this is nothing more than the Cartesian product of ``n``
1797 copies of the spin algebra. Once Cartesian product algebras
1798 are implemented, this can go.
1802 This multiplication table can be verified by hand::
1804 sage: J = RealCartesianProductEJA(3)
1805 sage: e0,e1,e2 = J.gens()
1821 def __classcall_private__(cls
, n
, field
=QQ
):
1822 # The FiniteDimensionalAlgebra constructor takes a list of
1823 # matrices, the ith representing right multiplication by the ith
1824 # basis element in the vector space. So if e_1 = (1,0,0), then
1825 # right (Hadamard) multiplication of x by e_1 picks out the first
1826 # component of x; and likewise for the ith basis element e_i.
1827 Qs
= [ matrix(field
, n
, n
, lambda k
,j
: 1*(k
== j
== i
))
1828 for i
in xrange(n
) ]
1830 fdeja
= super(RealCartesianProductEJA
, cls
)
1831 return fdeja
.__classcall
_private
__(cls
, field
, Qs
, rank
=n
)
1833 def inner_product(self
, x
, y
):
1834 return _usual_ip(x
,y
)
1839 Return a "random" finite-dimensional Euclidean Jordan Algebra.
1843 For now, we choose a random natural number ``n`` (greater than zero)
1844 and then give you back one of the following:
1846 * The cartesian product of the rational numbers ``n`` times; this is
1847 ``QQ^n`` with the Hadamard product.
1849 * The Jordan spin algebra on ``QQ^n``.
1851 * The ``n``-by-``n`` rational symmetric matrices with the symmetric
1854 * The ``n``-by-``n`` complex-rational Hermitian matrices embedded
1855 in the space of ``2n``-by-``2n`` real symmetric matrices.
1857 * The ``n``-by-``n`` quaternion-rational Hermitian matrices embedded
1858 in the space of ``4n``-by-``4n`` real symmetric matrices.
1860 Later this might be extended to return Cartesian products of the
1866 Euclidean Jordan algebra of degree...
1870 # The max_n component lets us choose different upper bounds on the
1871 # value "n" that gets passed to the constructor. This is needed
1872 # because e.g. R^{10} is reasonable to test, while the Hermitian
1873 # 10-by-10 quaternion matrices are not.
1874 (constructor
, max_n
) = choice([(RealCartesianProductEJA
, 6),
1876 (RealSymmetricEJA
, 5),
1877 (ComplexHermitianEJA
, 4),
1878 (QuaternionHermitianEJA
, 3)])
1879 n
= ZZ
.random_element(1, max_n
)
1880 return constructor(n
, field
=QQ
)
1884 def _real_symmetric_basis(n
, field
=QQ
):
1886 Return a basis for the space of real symmetric n-by-n matrices.
1888 # The basis of symmetric matrices, as matrices, in their R^(n-by-n)
1892 for j
in xrange(i
+1):
1893 Eij
= matrix(field
, n
, lambda k
,l
: k
==i
and l
==j
)
1897 # Beware, orthogonal but not normalized!
1898 Sij
= Eij
+ Eij
.transpose()
1903 def _complex_hermitian_basis(n
, field
=QQ
):
1905 Returns a basis for the space of complex Hermitian n-by-n matrices.
1909 sage: set_random_seed()
1910 sage: n = ZZ.random_element(1,5)
1911 sage: all( M.is_symmetric() for M in _complex_hermitian_basis(n) )
1915 F
= QuadraticField(-1, 'I')
1918 # This is like the symmetric case, but we need to be careful:
1920 # * We want conjugate-symmetry, not just symmetry.
1921 # * The diagonal will (as a result) be real.
1925 for j
in xrange(i
+1):
1926 Eij
= matrix(field
, n
, lambda k
,l
: k
==i
and l
==j
)
1928 Sij
= _embed_complex_matrix(Eij
)
1931 # Beware, orthogonal but not normalized! The second one
1932 # has a minus because it's conjugated.
1933 Sij_real
= _embed_complex_matrix(Eij
+ Eij
.transpose())
1935 Sij_imag
= _embed_complex_matrix(I
*Eij
- I
*Eij
.transpose())
1940 def _quaternion_hermitian_basis(n
, field
=QQ
):
1942 Returns a basis for the space of quaternion Hermitian n-by-n matrices.
1946 sage: set_random_seed()
1947 sage: n = ZZ.random_element(1,5)
1948 sage: all( M.is_symmetric() for M in _quaternion_hermitian_basis(n) )
1952 Q
= QuaternionAlgebra(QQ
,-1,-1)
1955 # This is like the symmetric case, but we need to be careful:
1957 # * We want conjugate-symmetry, not just symmetry.
1958 # * The diagonal will (as a result) be real.
1962 for j
in xrange(i
+1):
1963 Eij
= matrix(Q
, n
, lambda k
,l
: k
==i
and l
==j
)
1965 Sij
= _embed_quaternion_matrix(Eij
)
1968 # Beware, orthogonal but not normalized! The second,
1969 # third, and fourth ones have a minus because they're
1971 Sij_real
= _embed_quaternion_matrix(Eij
+ Eij
.transpose())
1973 Sij_I
= _embed_quaternion_matrix(I
*Eij
- I
*Eij
.transpose())
1975 Sij_J
= _embed_quaternion_matrix(J
*Eij
- J
*Eij
.transpose())
1977 Sij_K
= _embed_quaternion_matrix(K
*Eij
- K
*Eij
.transpose())
1983 return vector(m
.base_ring(), m
.list())
1986 return matrix(v
.base_ring(), sqrt(v
.degree()), v
.list())
1988 def _multiplication_table_from_matrix_basis(basis
):
1990 At least three of the five simple Euclidean Jordan algebras have the
1991 symmetric multiplication (A,B) |-> (AB + BA)/2, where the
1992 multiplication on the right is matrix multiplication. Given a basis
1993 for the underlying matrix space, this function returns a
1994 multiplication table (obtained by looping through the basis
1995 elements) for an algebra of those matrices. A reordered copy
1996 of the basis is also returned to work around the fact that
1997 the ``span()`` in this function will change the order of the basis
1998 from what we think it is, to... something else.
2000 # In S^2, for example, we nominally have four coordinates even
2001 # though the space is of dimension three only. The vector space V
2002 # is supposed to hold the entire long vector, and the subspace W
2003 # of V will be spanned by the vectors that arise from symmetric
2004 # matrices. Thus for S^2, dim(V) == 4 and dim(W) == 3.
2005 field
= basis
[0].base_ring()
2006 dimension
= basis
[0].nrows()
2008 V
= VectorSpace(field
, dimension
**2)
2009 W
= V
.span( _mat2vec(s
) for s
in basis
)
2011 # Taking the span above reorders our basis (thanks, jerk!) so we
2012 # need to put our "matrix basis" in the same order as the
2013 # (reordered) vector basis.
2014 S
= tuple( _vec2mat(b
) for b
in W
.basis() )
2018 # Brute force the multiplication-by-s matrix by looping
2019 # through all elements of the basis and doing the computation
2020 # to find out what the corresponding row should be. BEWARE:
2021 # these multiplication tables won't be symmetric! It therefore
2022 # becomes REALLY IMPORTANT that the underlying algebra
2023 # constructor uses ROW vectors and not COLUMN vectors. That's
2024 # why we're computing rows here and not columns.
2027 this_row
= _mat2vec((s
*t
+ t
*s
)/2)
2028 Q_rows
.append(W
.coordinates(this_row
))
2029 Q
= matrix(field
, W
.dimension(), Q_rows
)
2035 def _embed_complex_matrix(M
):
2037 Embed the n-by-n complex matrix ``M`` into the space of real
2038 matrices of size 2n-by-2n via the map the sends each entry `z = a +
2039 bi` to the block matrix ``[[a,b],[-b,a]]``.
2043 sage: F = QuadraticField(-1,'i')
2044 sage: x1 = F(4 - 2*i)
2045 sage: x2 = F(1 + 2*i)
2048 sage: M = matrix(F,2,[[x1,x2],[x3,x4]])
2049 sage: _embed_complex_matrix(M)
2058 Embedding is a homomorphism (isomorphism, in fact)::
2060 sage: set_random_seed()
2061 sage: n = ZZ.random_element(5)
2062 sage: F = QuadraticField(-1, 'i')
2063 sage: X = random_matrix(F, n)
2064 sage: Y = random_matrix(F, n)
2065 sage: actual = _embed_complex_matrix(X) * _embed_complex_matrix(Y)
2066 sage: expected = _embed_complex_matrix(X*Y)
2067 sage: actual == expected
2073 raise ValueError("the matrix 'M' must be square")
2074 field
= M
.base_ring()
2079 blocks
.append(matrix(field
, 2, [[a
,b
],[-b
,a
]]))
2081 # We can drop the imaginaries here.
2082 return block_matrix(field
.base_ring(), n
, blocks
)
2085 def _unembed_complex_matrix(M
):
2087 The inverse of _embed_complex_matrix().
2091 sage: A = matrix(QQ,[ [ 1, 2, 3, 4],
2092 ....: [-2, 1, -4, 3],
2093 ....: [ 9, 10, 11, 12],
2094 ....: [-10, 9, -12, 11] ])
2095 sage: _unembed_complex_matrix(A)
2097 [ 10*i + 9 12*i + 11]
2101 Unembedding is the inverse of embedding::
2103 sage: set_random_seed()
2104 sage: F = QuadraticField(-1, 'i')
2105 sage: M = random_matrix(F, 3)
2106 sage: _unembed_complex_matrix(_embed_complex_matrix(M)) == M
2112 raise ValueError("the matrix 'M' must be square")
2113 if not n
.mod(2).is_zero():
2114 raise ValueError("the matrix 'M' must be a complex embedding")
2116 F
= QuadraticField(-1, 'i')
2119 # Go top-left to bottom-right (reading order), converting every
2120 # 2-by-2 block we see to a single complex element.
2122 for k
in xrange(n
/2):
2123 for j
in xrange(n
/2):
2124 submat
= M
[2*k
:2*k
+2,2*j
:2*j
+2]
2125 if submat
[0,0] != submat
[1,1]:
2126 raise ValueError('bad on-diagonal submatrix')
2127 if submat
[0,1] != -submat
[1,0]:
2128 raise ValueError('bad off-diagonal submatrix')
2129 z
= submat
[0,0] + submat
[0,1]*i
2132 return matrix(F
, n
/2, elements
)
2135 def _embed_quaternion_matrix(M
):
2137 Embed the n-by-n quaternion matrix ``M`` into the space of real
2138 matrices of size 4n-by-4n by first sending each quaternion entry
2139 `z = a + bi + cj + dk` to the block-complex matrix
2140 ``[[a + bi, c+di],[-c + di, a-bi]]`, and then embedding those into
2145 sage: Q = QuaternionAlgebra(QQ,-1,-1)
2146 sage: i,j,k = Q.gens()
2147 sage: x = 1 + 2*i + 3*j + 4*k
2148 sage: M = matrix(Q, 1, [[x]])
2149 sage: _embed_quaternion_matrix(M)
2155 Embedding is a homomorphism (isomorphism, in fact)::
2157 sage: set_random_seed()
2158 sage: n = ZZ.random_element(5)
2159 sage: Q = QuaternionAlgebra(QQ,-1,-1)
2160 sage: X = random_matrix(Q, n)
2161 sage: Y = random_matrix(Q, n)
2162 sage: actual = _embed_quaternion_matrix(X)*_embed_quaternion_matrix(Y)
2163 sage: expected = _embed_quaternion_matrix(X*Y)
2164 sage: actual == expected
2168 quaternions
= M
.base_ring()
2171 raise ValueError("the matrix 'M' must be square")
2173 F
= QuadraticField(-1, 'i')
2178 t
= z
.coefficient_tuple()
2183 cplx_matrix
= matrix(F
, 2, [[ a
+ b
*i
, c
+ d
*i
],
2184 [-c
+ d
*i
, a
- b
*i
]])
2185 blocks
.append(_embed_complex_matrix(cplx_matrix
))
2187 # We should have real entries by now, so use the realest field
2188 # we've got for the return value.
2189 return block_matrix(quaternions
.base_ring(), n
, blocks
)
2192 def _unembed_quaternion_matrix(M
):
2194 The inverse of _embed_quaternion_matrix().
2198 sage: M = matrix(QQ, [[ 1, 2, 3, 4],
2199 ....: [-2, 1, -4, 3],
2200 ....: [-3, 4, 1, -2],
2201 ....: [-4, -3, 2, 1]])
2202 sage: _unembed_quaternion_matrix(M)
2203 [1 + 2*i + 3*j + 4*k]
2207 Unembedding is the inverse of embedding::
2209 sage: set_random_seed()
2210 sage: Q = QuaternionAlgebra(QQ, -1, -1)
2211 sage: M = random_matrix(Q, 3)
2212 sage: _unembed_quaternion_matrix(_embed_quaternion_matrix(M)) == M
2218 raise ValueError("the matrix 'M' must be square")
2219 if not n
.mod(4).is_zero():
2220 raise ValueError("the matrix 'M' must be a complex embedding")
2222 Q
= QuaternionAlgebra(QQ
,-1,-1)
2225 # Go top-left to bottom-right (reading order), converting every
2226 # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1
2229 for l
in xrange(n
/4):
2230 for m
in xrange(n
/4):
2231 submat
= _unembed_complex_matrix(M
[4*l
:4*l
+4,4*m
:4*m
+4])
2232 if submat
[0,0] != submat
[1,1].conjugate():
2233 raise ValueError('bad on-diagonal submatrix')
2234 if submat
[0,1] != -submat
[1,0].conjugate():
2235 raise ValueError('bad off-diagonal submatrix')
2236 z
= submat
[0,0].real() + submat
[0,0].imag()*i
2237 z
+= submat
[0,1].real()*j
+ submat
[0,1].imag()*k
2240 return matrix(Q
, n
/4, elements
)
2243 # The usual inner product on R^n.
2245 return x
.vector().inner_product(y
.vector())
2247 # The inner product used for the real symmetric simple EJA.
2248 # We keep it as a separate function because e.g. the complex
2249 # algebra uses the same inner product, except divided by 2.
2250 def _matrix_ip(X
,Y
):
2251 X_mat
= X
.natural_representation()
2252 Y_mat
= Y
.natural_representation()
2253 return (X_mat
*Y_mat
).trace()
2256 class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra
):
2258 The rank-n simple EJA consisting of real symmetric n-by-n
2259 matrices, the usual symmetric Jordan product, and the trace inner
2260 product. It has dimension `(n^2 + n)/2` over the reals.
2264 sage: J = RealSymmetricEJA(2)
2265 sage: e0, e1, e2 = J.gens()
2275 The degree of this algebra is `(n^2 + n) / 2`::
2277 sage: set_random_seed()
2278 sage: n = ZZ.random_element(1,5)
2279 sage: J = RealSymmetricEJA(n)
2280 sage: J.degree() == (n^2 + n)/2
2283 The Jordan multiplication is what we think it is::
2285 sage: set_random_seed()
2286 sage: n = ZZ.random_element(1,5)
2287 sage: J = RealSymmetricEJA(n)
2288 sage: x = J.random_element()
2289 sage: y = J.random_element()
2290 sage: actual = (x*y).natural_representation()
2291 sage: X = x.natural_representation()
2292 sage: Y = y.natural_representation()
2293 sage: expected = (X*Y + Y*X)/2
2294 sage: actual == expected
2296 sage: J(expected) == x*y
2301 def __classcall_private__(cls
, n
, field
=QQ
):
2302 S
= _real_symmetric_basis(n
, field
=field
)
2303 (Qs
, T
) = _multiplication_table_from_matrix_basis(S
)
2305 fdeja
= super(RealSymmetricEJA
, cls
)
2306 return fdeja
.__classcall
_private
__(cls
,
2312 def inner_product(self
, x
, y
):
2313 return _matrix_ip(x
,y
)
2316 class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra
):
2318 The rank-n simple EJA consisting of complex Hermitian n-by-n
2319 matrices over the real numbers, the usual symmetric Jordan product,
2320 and the real-part-of-trace inner product. It has dimension `n^2` over
2325 The degree of this algebra is `n^2`::
2327 sage: set_random_seed()
2328 sage: n = ZZ.random_element(1,5)
2329 sage: J = ComplexHermitianEJA(n)
2330 sage: J.degree() == n^2
2333 The Jordan multiplication is what we think it is::
2335 sage: set_random_seed()
2336 sage: n = ZZ.random_element(1,5)
2337 sage: J = ComplexHermitianEJA(n)
2338 sage: x = J.random_element()
2339 sage: y = J.random_element()
2340 sage: actual = (x*y).natural_representation()
2341 sage: X = x.natural_representation()
2342 sage: Y = y.natural_representation()
2343 sage: expected = (X*Y + Y*X)/2
2344 sage: actual == expected
2346 sage: J(expected) == x*y
2351 def __classcall_private__(cls
, n
, field
=QQ
):
2352 S
= _complex_hermitian_basis(n
)
2353 (Qs
, T
) = _multiplication_table_from_matrix_basis(S
)
2355 fdeja
= super(ComplexHermitianEJA
, cls
)
2356 return fdeja
.__classcall
_private
__(cls
,
2362 def inner_product(self
, x
, y
):
2363 # Since a+bi on the diagonal is represented as
2368 # we'll double-count the "a" entries if we take the trace of
2370 return _matrix_ip(x
,y
)/2
2373 class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra
):
2375 The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
2376 matrices, the usual symmetric Jordan product, and the
2377 real-part-of-trace inner product. It has dimension `2n^2 - n` over
2382 The degree of this algebra is `n^2`::
2384 sage: set_random_seed()
2385 sage: n = ZZ.random_element(1,5)
2386 sage: J = QuaternionHermitianEJA(n)
2387 sage: J.degree() == 2*(n^2) - n
2390 The Jordan multiplication is what we think it is::
2392 sage: set_random_seed()
2393 sage: n = ZZ.random_element(1,5)
2394 sage: J = QuaternionHermitianEJA(n)
2395 sage: x = J.random_element()
2396 sage: y = J.random_element()
2397 sage: actual = (x*y).natural_representation()
2398 sage: X = x.natural_representation()
2399 sage: Y = y.natural_representation()
2400 sage: expected = (X*Y + Y*X)/2
2401 sage: actual == expected
2403 sage: J(expected) == x*y
2408 def __classcall_private__(cls
, n
, field
=QQ
):
2409 S
= _quaternion_hermitian_basis(n
)
2410 (Qs
, T
) = _multiplication_table_from_matrix_basis(S
)
2412 fdeja
= super(QuaternionHermitianEJA
, cls
)
2413 return fdeja
.__classcall
_private
__(cls
,
2419 def inner_product(self
, x
, y
):
2420 # Since a+bi+cj+dk on the diagonal is represented as
2422 # a + bi +cj + dk = [ a b c d]
2427 # we'll quadruple-count the "a" entries if we take the trace of
2429 return _matrix_ip(x
,y
)/4
2432 class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra
):
2434 The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
2435 with the usual inner product and jordan product ``x*y =
2436 (<x_bar,y_bar>, x0*y_bar + y0*x_bar)``. It has dimension `n` over
2441 This multiplication table can be verified by hand::
2443 sage: J = JordanSpinEJA(4)
2444 sage: e0,e1,e2,e3 = J.gens()
2462 def __classcall_private__(cls
, n
, field
=QQ
):
2464 id_matrix
= identity_matrix(field
, n
)
2466 ei
= id_matrix
.column(i
)
2467 Qi
= zero_matrix(field
, n
)
2469 Qi
.set_column(0, ei
)
2470 Qi
+= diagonal_matrix(n
, [ei
[0]]*n
)
2471 # The addition of the diagonal matrix adds an extra ei[0] in the
2472 # upper-left corner of the matrix.
2473 Qi
[0,0] = Qi
[0,0] * ~
field(2)
2476 # The rank of the spin algebra is two, unless we're in a
2477 # one-dimensional ambient space (because the rank is bounded by
2478 # the ambient dimension).
2479 fdeja
= super(JordanSpinEJA
, cls
)
2480 return fdeja
.__classcall
_private
__(cls
, field
, Qs
, rank
=min(n
,2))
2482 def inner_product(self
, x
, y
):
2483 return _usual_ip(x
,y
)