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
15 class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra
):
17 def __classcall_private__(cls
,
21 assume_associative
=False,
26 mult_table
= [b
.base_extend(field
) for b
in mult_table
]
29 if not (is_Matrix(b
) and b
.dimensions() == (n
, n
)):
30 raise ValueError("input is not a multiplication table")
31 mult_table
= tuple(mult_table
)
33 cat
= MagmaticAlgebras(field
).FiniteDimensional().WithBasis()
34 cat
.or_subcategory(category
)
35 if assume_associative
:
36 cat
= cat
.Associative()
38 names
= normalize_names(n
, names
)
40 fda
= super(FiniteDimensionalEuclideanJordanAlgebra
, cls
)
41 return fda
.__classcall
__(cls
,
44 assume_associative
=assume_associative
,
48 natural_basis
=natural_basis
)
55 assume_associative
=False,
62 By definition, Jordan multiplication commutes::
64 sage: set_random_seed()
65 sage: J = random_eja()
66 sage: x = J.random_element()
67 sage: y = J.random_element()
73 self
._natural
_basis
= natural_basis
74 fda
= super(FiniteDimensionalEuclideanJordanAlgebra
, self
)
83 Return a string representation of ``self``.
85 fmt
= "Euclidean Jordan algebra of degree {} over {}"
86 return fmt
.format(self
.degree(), self
.base_ring())
89 def inner_product(self
, x
, y
):
91 The inner product associated with this Euclidean Jordan algebra.
93 Defaults to the trace inner product, but can be overridden by
94 subclasses if they are sure that the necessary properties are
99 The inner product must satisfy its axiom for this algebra to truly
100 be a Euclidean Jordan Algebra::
102 sage: set_random_seed()
103 sage: J = random_eja()
104 sage: x = J.random_element()
105 sage: y = J.random_element()
106 sage: z = J.random_element()
107 sage: (x*y).inner_product(z) == y.inner_product(x*z)
111 if (not x
in self
) or (not y
in self
):
112 raise TypeError("arguments must live in this algebra")
113 return x
.trace_inner_product(y
)
116 def natural_basis(self
):
118 Return a more-natural representation of this algebra's basis.
120 Every finite-dimensional Euclidean Jordan Algebra is a direct
121 sum of five simple algebras, four of which comprise Hermitian
122 matrices. This method returns the original "natural" basis
123 for our underlying vector space. (Typically, the natural basis
124 is used to construct the multiplication table in the first place.)
126 Note that this will always return a matrix. The standard basis
127 in `R^n` will be returned as `n`-by-`1` column matrices.
131 sage: J = RealSymmetricEJA(2)
134 sage: J.natural_basis()
142 sage: J = JordanSpinEJA(2)
145 sage: J.natural_basis()
152 if self
._natural
_basis
is None:
153 return tuple( b
.vector().column() for b
in self
.basis() )
155 return self
._natural
_basis
160 Return the rank of this EJA.
162 if self
._rank
is None:
163 raise ValueError("no rank specified at genesis")
168 class Element(FiniteDimensionalAlgebraElement
):
170 An element of a Euclidean Jordan algebra.
173 def __init__(self
, A
, elt
=None):
177 The identity in `S^n` is converted to the identity in the EJA::
179 sage: J = RealSymmetricEJA(3)
180 sage: I = identity_matrix(QQ,3)
181 sage: J(I) == J.one()
184 This skew-symmetric matrix can't be represented in the EJA::
186 sage: J = RealSymmetricEJA(3)
187 sage: A = matrix(QQ,3, lambda i,j: i-j)
189 Traceback (most recent call last):
191 ArithmeticError: vector is not in free module
194 # Goal: if we're given a matrix, and if it lives in our
195 # parent algebra's "natural ambient space," convert it
196 # into an algebra element.
198 # The catch is, we make a recursive call after converting
199 # the given matrix into a vector that lives in the algebra.
200 # This we need to try the parent class initializer first,
201 # to avoid recursing forever if we're given something that
202 # already fits into the algebra, but also happens to live
203 # in the parent's "natural ambient space" (this happens with
206 FiniteDimensionalAlgebraElement
.__init
__(self
, A
, elt
)
208 natural_basis
= A
.natural_basis()
209 if elt
in natural_basis
[0].matrix_space():
210 # Thanks for nothing! Matrix spaces aren't vector
211 # spaces in Sage, so we have to figure out its
212 # natural-basis coordinates ourselves.
213 V
= VectorSpace(elt
.base_ring(), elt
.nrows()**2)
214 W
= V
.span( _mat2vec(s
) for s
in natural_basis
)
215 coords
= W
.coordinates(_mat2vec(elt
))
216 FiniteDimensionalAlgebraElement
.__init
__(self
, A
, coords
)
218 def __pow__(self
, n
):
220 Return ``self`` raised to the power ``n``.
222 Jordan algebras are always power-associative; see for
223 example Faraut and Koranyi, Proposition II.1.2 (ii).
227 We have to override this because our superclass uses row vectors
228 instead of column vectors! We, on the other hand, assume column
233 sage: set_random_seed()
234 sage: x = random_eja().random_element()
235 sage: x.operator_matrix()*x.vector() == (x^2).vector()
238 A few examples of power-associativity::
240 sage: set_random_seed()
241 sage: x = random_eja().random_element()
242 sage: x*(x*x)*(x*x) == x^5
244 sage: (x*x)*(x*x*x) == x^5
247 We also know that powers operator-commute (Koecher, Chapter
250 sage: set_random_seed()
251 sage: x = random_eja().random_element()
252 sage: m = ZZ.random_element(0,10)
253 sage: n = ZZ.random_element(0,10)
254 sage: Lxm = (x^m).operator_matrix()
255 sage: Lxn = (x^n).operator_matrix()
256 sage: Lxm*Lxn == Lxn*Lxm
266 return A( (self
.operator_matrix()**(n
-1))*self
.vector() )
269 def characteristic_polynomial(self
):
271 Return my characteristic polynomial (if I'm a regular
274 Eventually this should be implemented in terms of the parent
275 algebra's characteristic polynomial that works for ALL
278 if self
.is_regular():
279 return self
.minimal_polynomial()
281 raise NotImplementedError('irregular element')
284 def inner_product(self
, other
):
286 Return the parent algebra's inner product of myself and ``other``.
290 The inner product in the Jordan spin algebra is the usual
291 inner product on `R^n` (this example only works because the
292 basis for the Jordan algebra is the standard basis in `R^n`)::
294 sage: J = JordanSpinEJA(3)
295 sage: x = vector(QQ,[1,2,3])
296 sage: y = vector(QQ,[4,5,6])
297 sage: x.inner_product(y)
299 sage: J(x).inner_product(J(y))
302 The inner product on `S^n` is `<X,Y> = trace(X*Y)`, where
303 multiplication is the usual matrix multiplication in `S^n`,
304 so the inner product of the identity matrix with itself
307 sage: J = RealSymmetricEJA(3)
308 sage: J.one().inner_product(J.one())
311 Likewise, the inner product on `C^n` is `<X,Y> =
312 Re(trace(X*Y))`, where we must necessarily take the real
313 part because the product of Hermitian matrices may not be
316 sage: J = ComplexHermitianEJA(3)
317 sage: J.one().inner_product(J.one())
320 Ditto for the quaternions::
322 sage: J = QuaternionHermitianEJA(3)
323 sage: J.one().inner_product(J.one())
328 Ensure that we can always compute an inner product, and that
329 it gives us back a real number::
331 sage: set_random_seed()
332 sage: J = random_eja()
333 sage: x = J.random_element()
334 sage: y = J.random_element()
335 sage: x.inner_product(y) in RR
341 raise TypeError("'other' must live in the same algebra")
343 return P
.inner_product(self
, other
)
346 def operator_commutes_with(self
, other
):
348 Return whether or not this element operator-commutes
353 The definition of a Jordan algebra says that any element
354 operator-commutes with its square::
356 sage: set_random_seed()
357 sage: x = random_eja().random_element()
358 sage: x.operator_commutes_with(x^2)
363 Test Lemma 1 from Chapter III of Koecher::
365 sage: set_random_seed()
366 sage: J = random_eja()
367 sage: u = J.random_element()
368 sage: v = J.random_element()
369 sage: lhs = u.operator_commutes_with(u*v)
370 sage: rhs = v.operator_commutes_with(u^2)
375 if not other
in self
.parent():
376 raise TypeError("'other' must live in the same algebra")
378 A
= self
.operator_matrix()
379 B
= other
.operator_matrix()
385 Return my determinant, the product of my eigenvalues.
389 sage: J = JordanSpinEJA(2)
390 sage: e0,e1 = J.gens()
394 sage: J = JordanSpinEJA(3)
395 sage: e0,e1,e2 = J.gens()
396 sage: x = e0 + e1 + e2
401 cs
= self
.characteristic_polynomial().coefficients(sparse
=False)
404 return cs
[0] * (-1)**r
406 raise ValueError('charpoly had no coefficients')
411 Return the Jordan-multiplicative inverse of this element.
413 We can't use the superclass method because it relies on the
414 algebra being associative.
418 The inverse in the spin factor algebra is given in Alizadeh's
421 sage: set_random_seed()
422 sage: n = ZZ.random_element(1,10)
423 sage: J = JordanSpinEJA(n)
424 sage: x = J.random_element()
425 sage: while x.is_zero():
426 ....: x = J.random_element()
427 sage: x_vec = x.vector()
429 sage: x_bar = x_vec[1:]
430 sage: coeff = 1/(x0^2 - x_bar.inner_product(x_bar))
431 sage: inv_vec = x_vec.parent()([x0] + (-x_bar).list())
432 sage: x_inverse = coeff*inv_vec
433 sage: x.inverse() == J(x_inverse)
438 The identity element is its own inverse::
440 sage: set_random_seed()
441 sage: J = random_eja()
442 sage: J.one().inverse() == J.one()
445 If an element has an inverse, it acts like one. TODO: this
446 can be a lot less ugly once ``is_invertible`` doesn't crash
447 on irregular elements::
449 sage: set_random_seed()
450 sage: J = random_eja()
451 sage: x = J.random_element()
453 ....: x.inverse()*x == J.one()
459 if self
.parent().is_associative():
460 elt
= FiniteDimensionalAlgebraElement(self
.parent(), self
)
463 # TODO: we can do better once the call to is_invertible()
464 # doesn't crash on irregular elements.
465 #if not self.is_invertible():
466 # raise ValueError('element is not invertible')
468 # We do this a little different than the usual recursive
469 # call to a finite-dimensional algebra element, because we
470 # wind up with an inverse that lives in the subalgebra and
471 # we need information about the parent to convert it back.
472 V
= self
.span_of_powers()
473 assoc_subalg
= self
.subalgebra_generated_by()
474 # Mis-design warning: the basis used for span_of_powers()
475 # and subalgebra_generated_by() must be the same, and in
477 elt
= assoc_subalg(V
.coordinates(self
.vector()))
479 # This will be in the subalgebra's coordinates...
480 fda_elt
= FiniteDimensionalAlgebraElement(assoc_subalg
, elt
)
481 subalg_inverse
= fda_elt
.inverse()
483 # So we have to convert back...
484 basis
= [ self
.parent(v
) for v
in V
.basis() ]
485 pairs
= zip(subalg_inverse
.vector(), basis
)
486 return self
.parent().linear_combination(pairs
)
489 def is_invertible(self
):
491 Return whether or not this element is invertible.
493 We can't use the superclass method because it relies on
494 the algebra being associative.
496 return not self
.det().is_zero()
499 def is_nilpotent(self
):
501 Return whether or not some power of this element is zero.
503 The superclass method won't work unless we're in an
504 associative algebra, and we aren't. However, we generate
505 an assocoative subalgebra and we're nilpotent there if and
506 only if we're nilpotent here (probably).
510 The identity element is never nilpotent::
512 sage: set_random_seed()
513 sage: random_eja().one().is_nilpotent()
516 The additive identity is always nilpotent::
518 sage: set_random_seed()
519 sage: random_eja().zero().is_nilpotent()
523 # The element we're going to call "is_nilpotent()" on.
524 # Either myself, interpreted as an element of a finite-
525 # dimensional algebra, or an element of an associative
529 if self
.parent().is_associative():
530 elt
= FiniteDimensionalAlgebraElement(self
.parent(), self
)
532 V
= self
.span_of_powers()
533 assoc_subalg
= self
.subalgebra_generated_by()
534 # Mis-design warning: the basis used for span_of_powers()
535 # and subalgebra_generated_by() must be the same, and in
537 elt
= assoc_subalg(V
.coordinates(self
.vector()))
539 # Recursive call, but should work since elt lives in an
540 # associative algebra.
541 return elt
.is_nilpotent()
544 def is_regular(self
):
546 Return whether or not this is a regular element.
550 The identity element always has degree one, but any element
551 linearly-independent from it is regular::
553 sage: J = JordanSpinEJA(5)
554 sage: J.one().is_regular()
556 sage: e0, e1, e2, e3, e4 = J.gens() # e0 is the identity
557 sage: for x in J.gens():
558 ....: (J.one() + x).is_regular()
566 return self
.degree() == self
.parent().rank()
571 Compute the degree of this element the straightforward way
572 according to the definition; by appending powers to a list
573 and figuring out its dimension (that is, whether or not
574 they're linearly dependent).
578 sage: J = JordanSpinEJA(4)
579 sage: J.one().degree()
581 sage: e0,e1,e2,e3 = J.gens()
582 sage: (e0 - e1).degree()
585 In the spin factor algebra (of rank two), all elements that
586 aren't multiples of the identity are regular::
588 sage: set_random_seed()
589 sage: n = ZZ.random_element(1,10)
590 sage: J = JordanSpinEJA(n)
591 sage: x = J.random_element()
592 sage: x == x.coefficient(0)*J.one() or x.degree() == 2
596 return self
.span_of_powers().dimension()
599 def minimal_polynomial(self
):
603 sage: set_random_seed()
604 sage: x = random_eja().random_element()
605 sage: x.degree() == x.minimal_polynomial().degree()
610 sage: set_random_seed()
611 sage: x = random_eja().random_element()
612 sage: x.degree() == x.minimal_polynomial().degree()
615 The minimal polynomial and the characteristic polynomial coincide
616 and are known (see Alizadeh, Example 11.11) for all elements of
617 the spin factor algebra that aren't scalar multiples of the
620 sage: set_random_seed()
621 sage: n = ZZ.random_element(2,10)
622 sage: J = JordanSpinEJA(n)
623 sage: y = J.random_element()
624 sage: while y == y.coefficient(0)*J.one():
625 ....: y = J.random_element()
626 sage: y0 = y.vector()[0]
627 sage: y_bar = y.vector()[1:]
628 sage: actual = y.minimal_polynomial()
629 sage: x = SR.symbol('x', domain='real')
630 sage: expected = x^2 - 2*y0*x + (y0^2 - norm(y_bar)^2)
631 sage: bool(actual == expected)
635 # The element we're going to call "minimal_polynomial()" on.
636 # Either myself, interpreted as an element of a finite-
637 # dimensional algebra, or an element of an associative
641 if self
.parent().is_associative():
642 elt
= FiniteDimensionalAlgebraElement(self
.parent(), self
)
644 V
= self
.span_of_powers()
645 assoc_subalg
= self
.subalgebra_generated_by()
646 # Mis-design warning: the basis used for span_of_powers()
647 # and subalgebra_generated_by() must be the same, and in
649 elt
= assoc_subalg(V
.coordinates(self
.vector()))
651 # Recursive call, but should work since elt lives in an
652 # associative algebra.
653 return elt
.minimal_polynomial()
656 def natural_representation(self
):
658 Return a more-natural representation of this element.
660 Every finite-dimensional Euclidean Jordan Algebra is a
661 direct sum of five simple algebras, four of which comprise
662 Hermitian matrices. This method returns the original
663 "natural" representation of this element as a Hermitian
664 matrix, if it has one. If not, you get the usual representation.
668 sage: J = ComplexHermitianEJA(3)
671 sage: J.one().natural_representation()
681 sage: J = QuaternionHermitianEJA(3)
684 sage: J.one().natural_representation()
685 [1 0 0 0 0 0 0 0 0 0 0 0]
686 [0 1 0 0 0 0 0 0 0 0 0 0]
687 [0 0 1 0 0 0 0 0 0 0 0 0]
688 [0 0 0 1 0 0 0 0 0 0 0 0]
689 [0 0 0 0 1 0 0 0 0 0 0 0]
690 [0 0 0 0 0 1 0 0 0 0 0 0]
691 [0 0 0 0 0 0 1 0 0 0 0 0]
692 [0 0 0 0 0 0 0 1 0 0 0 0]
693 [0 0 0 0 0 0 0 0 1 0 0 0]
694 [0 0 0 0 0 0 0 0 0 1 0 0]
695 [0 0 0 0 0 0 0 0 0 0 1 0]
696 [0 0 0 0 0 0 0 0 0 0 0 1]
699 B
= self
.parent().natural_basis()
700 W
= B
[0].matrix_space()
701 return W
.linear_combination(zip(self
.vector(), B
))
704 def operator_matrix(self
):
706 Return the matrix that represents left- (or right-)
707 multiplication by this element in the parent algebra.
709 We have to override this because the superclass method
710 returns a matrix that acts on row vectors (that is, on
715 Test the first polarization identity from my notes, Koecher Chapter
716 III, or from Baes (2.3)::
718 sage: set_random_seed()
719 sage: J = random_eja()
720 sage: x = J.random_element()
721 sage: y = J.random_element()
722 sage: Lx = x.operator_matrix()
723 sage: Ly = y.operator_matrix()
724 sage: Lxx = (x*x).operator_matrix()
725 sage: Lxy = (x*y).operator_matrix()
726 sage: bool(2*Lx*Lxy + Ly*Lxx == 2*Lxy*Lx + Lxx*Ly)
729 Test the second polarization identity from my notes or from
732 sage: set_random_seed()
733 sage: J = random_eja()
734 sage: x = J.random_element()
735 sage: y = J.random_element()
736 sage: z = J.random_element()
737 sage: Lx = x.operator_matrix()
738 sage: Ly = y.operator_matrix()
739 sage: Lz = z.operator_matrix()
740 sage: Lzy = (z*y).operator_matrix()
741 sage: Lxy = (x*y).operator_matrix()
742 sage: Lxz = (x*z).operator_matrix()
743 sage: bool(Lx*Lzy + Lz*Lxy + Ly*Lxz == Lzy*Lx + Lxy*Lz + Lxz*Ly)
746 Test the third polarization identity from my notes or from
749 sage: set_random_seed()
750 sage: J = random_eja()
751 sage: u = J.random_element()
752 sage: y = J.random_element()
753 sage: z = J.random_element()
754 sage: Lu = u.operator_matrix()
755 sage: Ly = y.operator_matrix()
756 sage: Lz = z.operator_matrix()
757 sage: Lzy = (z*y).operator_matrix()
758 sage: Luy = (u*y).operator_matrix()
759 sage: Luz = (u*z).operator_matrix()
760 sage: Luyz = (u*(y*z)).operator_matrix()
761 sage: lhs = Lu*Lzy + Lz*Luy + Ly*Luz
762 sage: rhs = Luyz + Ly*Lu*Lz + Lz*Lu*Ly
763 sage: bool(lhs == rhs)
767 fda_elt
= FiniteDimensionalAlgebraElement(self
.parent(), self
)
768 return fda_elt
.matrix().transpose()
771 def quadratic_representation(self
, other
=None):
773 Return the quadratic representation of this element.
777 The explicit form in the spin factor algebra is given by
778 Alizadeh's Example 11.12::
780 sage: set_random_seed()
781 sage: n = ZZ.random_element(1,10)
782 sage: J = JordanSpinEJA(n)
783 sage: x = J.random_element()
784 sage: x_vec = x.vector()
786 sage: x_bar = x_vec[1:]
787 sage: A = matrix(QQ, 1, [x_vec.inner_product(x_vec)])
788 sage: B = 2*x0*x_bar.row()
789 sage: C = 2*x0*x_bar.column()
790 sage: D = identity_matrix(QQ, n-1)
791 sage: D = (x0^2 - x_bar.inner_product(x_bar))*D
792 sage: D = D + 2*x_bar.tensor_product(x_bar)
793 sage: Q = block_matrix(2,2,[A,B,C,D])
794 sage: Q == x.quadratic_representation()
797 Test all of the properties from Theorem 11.2 in Alizadeh::
799 sage: set_random_seed()
800 sage: J = random_eja()
801 sage: x = J.random_element()
802 sage: y = J.random_element()
806 sage: actual = x.quadratic_representation(y)
807 sage: expected = ( (x+y).quadratic_representation()
808 ....: -x.quadratic_representation()
809 ....: -y.quadratic_representation() ) / 2
810 sage: actual == expected
815 sage: alpha = QQ.random_element()
816 sage: actual = (alpha*x).quadratic_representation()
817 sage: expected = (alpha^2)*x.quadratic_representation()
818 sage: actual == expected
823 sage: Qy = y.quadratic_representation()
824 sage: actual = J(Qy*x.vector()).quadratic_representation()
825 sage: expected = Qy*x.quadratic_representation()*Qy
826 sage: actual == expected
831 sage: k = ZZ.random_element(1,10)
832 sage: actual = (x^k).quadratic_representation()
833 sage: expected = (x.quadratic_representation())^k
834 sage: actual == expected
840 elif not other
in self
.parent():
841 raise TypeError("'other' must live in the same algebra")
843 L
= self
.operator_matrix()
844 M
= other
.operator_matrix()
845 return ( L
*M
+ M
*L
- (self
*other
).operator_matrix() )
848 def span_of_powers(self
):
850 Return the vector space spanned by successive powers of
853 # The dimension of the subalgebra can't be greater than
854 # the big algebra, so just put everything into a list
855 # and let span() get rid of the excess.
856 V
= self
.vector().parent()
857 return V
.span( (self
**d
).vector() for d
in xrange(V
.dimension()) )
860 def subalgebra_generated_by(self
):
862 Return the associative subalgebra of the parent EJA generated
867 sage: set_random_seed()
868 sage: x = random_eja().random_element()
869 sage: x.subalgebra_generated_by().is_associative()
872 Squaring in the subalgebra should be the same thing as
873 squaring in the superalgebra::
875 sage: set_random_seed()
876 sage: x = random_eja().random_element()
877 sage: u = x.subalgebra_generated_by().random_element()
878 sage: u.operator_matrix()*u.vector() == (u**2).vector()
882 # First get the subspace spanned by the powers of myself...
883 V
= self
.span_of_powers()
886 # Now figure out the entries of the right-multiplication
887 # matrix for the successive basis elements b0, b1,... of
890 for b_right
in V
.basis():
891 eja_b_right
= self
.parent()(b_right
)
893 # The first row of the right-multiplication matrix by
894 # b1 is what we get if we apply that matrix to b1. The
895 # second row of the right multiplication matrix by b1
896 # is what we get when we apply that matrix to b2...
898 # IMPORTANT: this assumes that all vectors are COLUMN
899 # vectors, unlike our superclass (which uses row vectors).
900 for b_left
in V
.basis():
901 eja_b_left
= self
.parent()(b_left
)
902 # Multiply in the original EJA, but then get the
903 # coordinates from the subalgebra in terms of its
905 this_row
= V
.coordinates((eja_b_left
*eja_b_right
).vector())
906 b_right_rows
.append(this_row
)
907 b_right_matrix
= matrix(F
, b_right_rows
)
908 mats
.append(b_right_matrix
)
910 # It's an algebra of polynomials in one element, and EJAs
911 # are power-associative.
913 # TODO: choose generator names intelligently.
914 return FiniteDimensionalEuclideanJordanAlgebra(F
, mats
, assume_associative
=True, names
='f')
917 def subalgebra_idempotent(self
):
919 Find an idempotent in the associative subalgebra I generate
920 using Proposition 2.3.5 in Baes.
924 sage: set_random_seed()
926 sage: c = J.random_element().subalgebra_idempotent()
929 sage: J = JordanSpinEJA(5)
930 sage: c = J.random_element().subalgebra_idempotent()
935 if self
.is_nilpotent():
936 raise ValueError("this only works with non-nilpotent elements!")
938 V
= self
.span_of_powers()
939 J
= self
.subalgebra_generated_by()
940 # Mis-design warning: the basis used for span_of_powers()
941 # and subalgebra_generated_by() must be the same, and in
943 u
= J(V
.coordinates(self
.vector()))
945 # The image of the matrix of left-u^m-multiplication
946 # will be minimal for some natural number s...
948 minimal_dim
= V
.dimension()
949 for i
in xrange(1, V
.dimension()):
950 this_dim
= (u
**i
).operator_matrix().image().dimension()
951 if this_dim
< minimal_dim
:
952 minimal_dim
= this_dim
955 # Now minimal_matrix should correspond to the smallest
956 # non-zero subspace in Baes's (or really, Koecher's)
959 # However, we need to restrict the matrix to work on the
960 # subspace... or do we? Can't we just solve, knowing that
961 # A(c) = u^(s+1) should have a solution in the big space,
964 # Beware, solve_right() means that we're using COLUMN vectors.
965 # Our FiniteDimensionalAlgebraElement superclass uses rows.
967 A
= u_next
.operator_matrix()
968 c_coordinates
= A
.solve_right(u_next
.vector())
970 # Now c_coordinates is the idempotent we want, but it's in
971 # the coordinate system of the subalgebra.
973 # We need the basis for J, but as elements of the parent algebra.
975 basis
= [self
.parent(v
) for v
in V
.basis()]
976 return self
.parent().linear_combination(zip(c_coordinates
, basis
))
981 Return my trace, the sum of my eigenvalues.
985 sage: J = JordanSpinEJA(3)
986 sage: e0,e1,e2 = J.gens()
987 sage: x = e0 + e1 + e2
992 cs
= self
.characteristic_polynomial().coefficients(sparse
=False)
996 raise ValueError('charpoly had fewer than 2 coefficients')
999 def trace_inner_product(self
, other
):
1001 Return the trace inner product of myself and ``other``.
1003 if not other
in self
.parent():
1004 raise TypeError("'other' must live in the same algebra")
1006 return (self
*other
).trace()
1009 def eja_rn(dimension
, field
=QQ
):
1011 Return the Euclidean Jordan Algebra corresponding to the set
1012 `R^n` under the Hadamard product.
1016 This multiplication table can be verified by hand::
1019 sage: e0,e1,e2 = J.gens()
1034 # The FiniteDimensionalAlgebra constructor takes a list of
1035 # matrices, the ith representing right multiplication by the ith
1036 # basis element in the vector space. So if e_1 = (1,0,0), then
1037 # right (Hadamard) multiplication of x by e_1 picks out the first
1038 # component of x; and likewise for the ith basis element e_i.
1039 Qs
= [ matrix(field
, dimension
, dimension
, lambda k
,j
: 1*(k
== j
== i
))
1040 for i
in xrange(dimension
) ]
1042 return FiniteDimensionalEuclideanJordanAlgebra(field
,
1050 Return a "random" finite-dimensional Euclidean Jordan Algebra.
1054 For now, we choose a random natural number ``n`` (greater than zero)
1055 and then give you back one of the following:
1057 * The cartesian product of the rational numbers ``n`` times; this is
1058 ``QQ^n`` with the Hadamard product.
1060 * The Jordan spin algebra on ``QQ^n``.
1062 * The ``n``-by-``n`` rational symmetric matrices with the symmetric
1065 * The ``n``-by-``n`` complex-rational Hermitian matrices embedded
1066 in the space of ``2n``-by-``2n`` real symmetric matrices.
1068 * The ``n``-by-``n`` quaternion-rational Hermitian matrices embedded
1069 in the space of ``4n``-by-``4n`` real symmetric matrices.
1071 Later this might be extended to return Cartesian products of the
1077 Euclidean Jordan algebra of degree...
1080 n
= ZZ
.random_element(1,5)
1081 constructor
= choice([eja_rn
,
1084 ComplexHermitianEJA
,
1085 QuaternionHermitianEJA
])
1086 return constructor(n
, field
=QQ
)
1090 def _real_symmetric_basis(n
, field
=QQ
):
1092 Return a basis for the space of real symmetric n-by-n matrices.
1094 # The basis of symmetric matrices, as matrices, in their R^(n-by-n)
1098 for j
in xrange(i
+1):
1099 Eij
= matrix(field
, n
, lambda k
,l
: k
==i
and l
==j
)
1103 # Beware, orthogonal but not normalized!
1104 Sij
= Eij
+ Eij
.transpose()
1109 def _complex_hermitian_basis(n
, field
=QQ
):
1111 Returns a basis for the space of complex Hermitian n-by-n matrices.
1115 sage: set_random_seed()
1116 sage: n = ZZ.random_element(1,5)
1117 sage: all( M.is_symmetric() for M in _complex_hermitian_basis(n) )
1121 F
= QuadraticField(-1, 'I')
1124 # This is like the symmetric case, but we need to be careful:
1126 # * We want conjugate-symmetry, not just symmetry.
1127 # * The diagonal will (as a result) be real.
1131 for j
in xrange(i
+1):
1132 Eij
= matrix(field
, n
, lambda k
,l
: k
==i
and l
==j
)
1134 Sij
= _embed_complex_matrix(Eij
)
1137 # Beware, orthogonal but not normalized! The second one
1138 # has a minus because it's conjugated.
1139 Sij_real
= _embed_complex_matrix(Eij
+ Eij
.transpose())
1141 Sij_imag
= _embed_complex_matrix(I
*Eij
- I
*Eij
.transpose())
1146 def _quaternion_hermitian_basis(n
, field
=QQ
):
1148 Returns a basis for the space of quaternion Hermitian n-by-n matrices.
1152 sage: set_random_seed()
1153 sage: n = ZZ.random_element(1,5)
1154 sage: all( M.is_symmetric() for M in _quaternion_hermitian_basis(n) )
1158 Q
= QuaternionAlgebra(QQ
,-1,-1)
1161 # This is like the symmetric case, but we need to be careful:
1163 # * We want conjugate-symmetry, not just symmetry.
1164 # * The diagonal will (as a result) be real.
1168 for j
in xrange(i
+1):
1169 Eij
= matrix(Q
, n
, lambda k
,l
: k
==i
and l
==j
)
1171 Sij
= _embed_quaternion_matrix(Eij
)
1174 # Beware, orthogonal but not normalized! The second,
1175 # third, and fourth ones have a minus because they're
1177 Sij_real
= _embed_quaternion_matrix(Eij
+ Eij
.transpose())
1179 Sij_I
= _embed_quaternion_matrix(I
*Eij
- I
*Eij
.transpose())
1181 Sij_J
= _embed_quaternion_matrix(J
*Eij
- J
*Eij
.transpose())
1183 Sij_K
= _embed_quaternion_matrix(K
*Eij
- K
*Eij
.transpose())
1189 return vector(m
.base_ring(), m
.list())
1192 return matrix(v
.base_ring(), sqrt(v
.degree()), v
.list())
1194 def _multiplication_table_from_matrix_basis(basis
):
1196 At least three of the five simple Euclidean Jordan algebras have the
1197 symmetric multiplication (A,B) |-> (AB + BA)/2, where the
1198 multiplication on the right is matrix multiplication. Given a basis
1199 for the underlying matrix space, this function returns a
1200 multiplication table (obtained by looping through the basis
1201 elements) for an algebra of those matrices. A reordered copy
1202 of the basis is also returned to work around the fact that
1203 the ``span()`` in this function will change the order of the basis
1204 from what we think it is, to... something else.
1206 # In S^2, for example, we nominally have four coordinates even
1207 # though the space is of dimension three only. The vector space V
1208 # is supposed to hold the entire long vector, and the subspace W
1209 # of V will be spanned by the vectors that arise from symmetric
1210 # matrices. Thus for S^2, dim(V) == 4 and dim(W) == 3.
1211 field
= basis
[0].base_ring()
1212 dimension
= basis
[0].nrows()
1214 V
= VectorSpace(field
, dimension
**2)
1215 W
= V
.span( _mat2vec(s
) for s
in basis
)
1217 # Taking the span above reorders our basis (thanks, jerk!) so we
1218 # need to put our "matrix basis" in the same order as the
1219 # (reordered) vector basis.
1220 S
= tuple( _vec2mat(b
) for b
in W
.basis() )
1224 # Brute force the multiplication-by-s matrix by looping
1225 # through all elements of the basis and doing the computation
1226 # to find out what the corresponding row should be. BEWARE:
1227 # these multiplication tables won't be symmetric! It therefore
1228 # becomes REALLY IMPORTANT that the underlying algebra
1229 # constructor uses ROW vectors and not COLUMN vectors. That's
1230 # why we're computing rows here and not columns.
1233 this_row
= _mat2vec((s
*t
+ t
*s
)/2)
1234 Q_rows
.append(W
.coordinates(this_row
))
1235 Q
= matrix(field
, W
.dimension(), Q_rows
)
1241 def _embed_complex_matrix(M
):
1243 Embed the n-by-n complex matrix ``M`` into the space of real
1244 matrices of size 2n-by-2n via the map the sends each entry `z = a +
1245 bi` to the block matrix ``[[a,b],[-b,a]]``.
1249 sage: F = QuadraticField(-1,'i')
1250 sage: x1 = F(4 - 2*i)
1251 sage: x2 = F(1 + 2*i)
1254 sage: M = matrix(F,2,[[x1,x2],[x3,x4]])
1255 sage: _embed_complex_matrix(M)
1264 Embedding is a homomorphism (isomorphism, in fact)::
1266 sage: set_random_seed()
1267 sage: n = ZZ.random_element(5)
1268 sage: F = QuadraticField(-1, 'i')
1269 sage: X = random_matrix(F, n)
1270 sage: Y = random_matrix(F, n)
1271 sage: actual = _embed_complex_matrix(X) * _embed_complex_matrix(Y)
1272 sage: expected = _embed_complex_matrix(X*Y)
1273 sage: actual == expected
1279 raise ValueError("the matrix 'M' must be square")
1280 field
= M
.base_ring()
1285 blocks
.append(matrix(field
, 2, [[a
,b
],[-b
,a
]]))
1287 # We can drop the imaginaries here.
1288 return block_matrix(field
.base_ring(), n
, blocks
)
1291 def _unembed_complex_matrix(M
):
1293 The inverse of _embed_complex_matrix().
1297 sage: A = matrix(QQ,[ [ 1, 2, 3, 4],
1298 ....: [-2, 1, -4, 3],
1299 ....: [ 9, 10, 11, 12],
1300 ....: [-10, 9, -12, 11] ])
1301 sage: _unembed_complex_matrix(A)
1303 [ 10*i + 9 12*i + 11]
1307 Unembedding is the inverse of embedding::
1309 sage: set_random_seed()
1310 sage: F = QuadraticField(-1, 'i')
1311 sage: M = random_matrix(F, 3)
1312 sage: _unembed_complex_matrix(_embed_complex_matrix(M)) == M
1318 raise ValueError("the matrix 'M' must be square")
1319 if not n
.mod(2).is_zero():
1320 raise ValueError("the matrix 'M' must be a complex embedding")
1322 F
= QuadraticField(-1, 'i')
1325 # Go top-left to bottom-right (reading order), converting every
1326 # 2-by-2 block we see to a single complex element.
1328 for k
in xrange(n
/2):
1329 for j
in xrange(n
/2):
1330 submat
= M
[2*k
:2*k
+2,2*j
:2*j
+2]
1331 if submat
[0,0] != submat
[1,1]:
1332 raise ValueError('bad on-diagonal submatrix')
1333 if submat
[0,1] != -submat
[1,0]:
1334 raise ValueError('bad off-diagonal submatrix')
1335 z
= submat
[0,0] + submat
[0,1]*i
1338 return matrix(F
, n
/2, elements
)
1341 def _embed_quaternion_matrix(M
):
1343 Embed the n-by-n quaternion matrix ``M`` into the space of real
1344 matrices of size 4n-by-4n by first sending each quaternion entry
1345 `z = a + bi + cj + dk` to the block-complex matrix
1346 ``[[a + bi, c+di],[-c + di, a-bi]]`, and then embedding those into
1351 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1352 sage: i,j,k = Q.gens()
1353 sage: x = 1 + 2*i + 3*j + 4*k
1354 sage: M = matrix(Q, 1, [[x]])
1355 sage: _embed_quaternion_matrix(M)
1361 Embedding is a homomorphism (isomorphism, in fact)::
1363 sage: set_random_seed()
1364 sage: n = ZZ.random_element(5)
1365 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1366 sage: X = random_matrix(Q, n)
1367 sage: Y = random_matrix(Q, n)
1368 sage: actual = _embed_quaternion_matrix(X)*_embed_quaternion_matrix(Y)
1369 sage: expected = _embed_quaternion_matrix(X*Y)
1370 sage: actual == expected
1374 quaternions
= M
.base_ring()
1377 raise ValueError("the matrix 'M' must be square")
1379 F
= QuadraticField(-1, 'i')
1384 t
= z
.coefficient_tuple()
1389 cplx_matrix
= matrix(F
, 2, [[ a
+ b
*i
, c
+ d
*i
],
1390 [-c
+ d
*i
, a
- b
*i
]])
1391 blocks
.append(_embed_complex_matrix(cplx_matrix
))
1393 # We should have real entries by now, so use the realest field
1394 # we've got for the return value.
1395 return block_matrix(quaternions
.base_ring(), n
, blocks
)
1398 def _unembed_quaternion_matrix(M
):
1400 The inverse of _embed_quaternion_matrix().
1404 sage: M = matrix(QQ, [[ 1, 2, 3, 4],
1405 ....: [-2, 1, -4, 3],
1406 ....: [-3, 4, 1, -2],
1407 ....: [-4, -3, 2, 1]])
1408 sage: _unembed_quaternion_matrix(M)
1409 [1 + 2*i + 3*j + 4*k]
1413 Unembedding is the inverse of embedding::
1415 sage: set_random_seed()
1416 sage: Q = QuaternionAlgebra(QQ, -1, -1)
1417 sage: M = random_matrix(Q, 3)
1418 sage: _unembed_quaternion_matrix(_embed_quaternion_matrix(M)) == M
1424 raise ValueError("the matrix 'M' must be square")
1425 if not n
.mod(4).is_zero():
1426 raise ValueError("the matrix 'M' must be a complex embedding")
1428 Q
= QuaternionAlgebra(QQ
,-1,-1)
1431 # Go top-left to bottom-right (reading order), converting every
1432 # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1
1435 for l
in xrange(n
/4):
1436 for m
in xrange(n
/4):
1437 submat
= _unembed_complex_matrix(M
[4*l
:4*l
+4,4*m
:4*m
+4])
1438 if submat
[0,0] != submat
[1,1].conjugate():
1439 raise ValueError('bad on-diagonal submatrix')
1440 if submat
[0,1] != -submat
[1,0].conjugate():
1441 raise ValueError('bad off-diagonal submatrix')
1442 z
= submat
[0,0].real() + submat
[0,0].imag()*i
1443 z
+= submat
[0,1].real()*j
+ submat
[0,1].imag()*k
1446 return matrix(Q
, n
/4, elements
)
1449 # The usual inner product on R^n.
1451 return x
.vector().inner_product(y
.vector())
1453 # The inner product used for the real symmetric simple EJA.
1454 # We keep it as a separate function because e.g. the complex
1455 # algebra uses the same inner product, except divided by 2.
1456 def _matrix_ip(X
,Y
):
1457 X_mat
= X
.natural_representation()
1458 Y_mat
= Y
.natural_representation()
1459 return (X_mat
*Y_mat
).trace()
1462 class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1464 The rank-n simple EJA consisting of real symmetric n-by-n
1465 matrices, the usual symmetric Jordan product, and the trace inner
1466 product. It has dimension `(n^2 + n)/2` over the reals.
1470 sage: J = RealSymmetricEJA(2)
1471 sage: e0, e1, e2 = J.gens()
1481 The degree of this algebra is `(n^2 + n) / 2`::
1483 sage: set_random_seed()
1484 sage: n = ZZ.random_element(1,5)
1485 sage: J = RealSymmetricEJA(n)
1486 sage: J.degree() == (n^2 + n)/2
1489 The Jordan multiplication is what we think it is::
1491 sage: set_random_seed()
1492 sage: n = ZZ.random_element(1,5)
1493 sage: J = RealSymmetricEJA(n)
1494 sage: x = J.random_element()
1495 sage: y = J.random_element()
1496 sage: actual = (x*y).natural_representation()
1497 sage: X = x.natural_representation()
1498 sage: Y = y.natural_representation()
1499 sage: expected = (X*Y + Y*X)/2
1500 sage: actual == expected
1502 sage: J(expected) == x*y
1507 def __classcall_private__(cls
, n
, field
=QQ
):
1508 S
= _real_symmetric_basis(n
, field
=field
)
1509 (Qs
, T
) = _multiplication_table_from_matrix_basis(S
)
1511 fdeja
= super(RealSymmetricEJA
, cls
)
1512 return fdeja
.__classcall
_private
__(cls
,
1518 def inner_product(self
, x
, y
):
1519 return _matrix_ip(x
,y
)
1522 class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1524 The rank-n simple EJA consisting of complex Hermitian n-by-n
1525 matrices over the real numbers, the usual symmetric Jordan product,
1526 and the real-part-of-trace inner product. It has dimension `n^2` over
1531 The degree of this algebra is `n^2`::
1533 sage: set_random_seed()
1534 sage: n = ZZ.random_element(1,5)
1535 sage: J = ComplexHermitianEJA(n)
1536 sage: J.degree() == n^2
1539 The Jordan multiplication is what we think it is::
1541 sage: set_random_seed()
1542 sage: n = ZZ.random_element(1,5)
1543 sage: J = ComplexHermitianEJA(n)
1544 sage: x = J.random_element()
1545 sage: y = J.random_element()
1546 sage: actual = (x*y).natural_representation()
1547 sage: X = x.natural_representation()
1548 sage: Y = y.natural_representation()
1549 sage: expected = (X*Y + Y*X)/2
1550 sage: actual == expected
1552 sage: J(expected) == x*y
1557 def __classcall_private__(cls
, n
, field
=QQ
):
1558 S
= _complex_hermitian_basis(n
)
1559 (Qs
, T
) = _multiplication_table_from_matrix_basis(S
)
1561 fdeja
= super(ComplexHermitianEJA
, cls
)
1562 return fdeja
.__classcall
_private
__(cls
,
1568 def inner_product(self
, x
, y
):
1569 # Since a+bi on the diagonal is represented as
1574 # we'll double-count the "a" entries if we take the trace of
1576 return _matrix_ip(x
,y
)/2
1579 class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1581 The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
1582 matrices, the usual symmetric Jordan product, and the
1583 real-part-of-trace inner product. It has dimension `2n^2 - n` over
1588 The degree of this algebra is `n^2`::
1590 sage: set_random_seed()
1591 sage: n = ZZ.random_element(1,5)
1592 sage: J = QuaternionHermitianEJA(n)
1593 sage: J.degree() == 2*(n^2) - n
1596 The Jordan multiplication is what we think it is::
1598 sage: set_random_seed()
1599 sage: n = ZZ.random_element(1,5)
1600 sage: J = QuaternionHermitianEJA(n)
1601 sage: x = J.random_element()
1602 sage: y = J.random_element()
1603 sage: actual = (x*y).natural_representation()
1604 sage: X = x.natural_representation()
1605 sage: Y = y.natural_representation()
1606 sage: expected = (X*Y + Y*X)/2
1607 sage: actual == expected
1609 sage: J(expected) == x*y
1614 def __classcall_private__(cls
, n
, field
=QQ
):
1615 S
= _quaternion_hermitian_basis(n
)
1616 (Qs
, T
) = _multiplication_table_from_matrix_basis(S
)
1618 fdeja
= super(QuaternionHermitianEJA
, cls
)
1619 return fdeja
.__classcall
_private
__(cls
,
1625 def inner_product(self
, x
, y
):
1626 # Since a+bi+cj+dk on the diagonal is represented as
1628 # a + bi +cj + dk = [ a b c d]
1633 # we'll quadruple-count the "a" entries if we take the trace of
1635 return _matrix_ip(x
,y
)/4
1638 class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1640 The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
1641 with the usual inner product and jordan product ``x*y =
1642 (<x_bar,y_bar>, x0*y_bar + y0*x_bar)``. It has dimension `n` over
1647 This multiplication table can be verified by hand::
1649 sage: J = JordanSpinEJA(4)
1650 sage: e0,e1,e2,e3 = J.gens()
1668 def __classcall_private__(cls
, n
, field
=QQ
):
1670 id_matrix
= identity_matrix(field
, n
)
1672 ei
= id_matrix
.column(i
)
1673 Qi
= zero_matrix(field
, n
)
1675 Qi
.set_column(0, ei
)
1676 Qi
+= diagonal_matrix(n
, [ei
[0]]*n
)
1677 # The addition of the diagonal matrix adds an extra ei[0] in the
1678 # upper-left corner of the matrix.
1679 Qi
[0,0] = Qi
[0,0] * ~
field(2)
1682 fdeja
= super(JordanSpinEJA
, cls
)
1683 return fdeja
.__classcall
_private
__(cls
, field
, Qs
)
1687 Return the rank of this Jordan Spin Algebra.
1689 The rank of the spin algebra is two, unless we're in a
1690 one-dimensional ambient space (because the rank is bounded by
1691 the ambient dimension).
1693 return min(self
.dimension(),2)
1695 def inner_product(self
, x
, y
):
1696 return _usual_ip(x
,y
)