1 from sage
.matrix
.constructor
import matrix
2 from sage
.modules
.free_module
import VectorSpace
3 from sage
.modules
.with_basis
.indexed_element
import IndexedFreeModuleElement
5 # TODO: make this unnecessary somehow.
6 from sage
.misc
.lazy_import
import lazy_import
7 lazy_import('mjo.eja.eja_algebra', 'FiniteDimensionalEuclideanJordanAlgebra')
8 lazy_import('mjo.eja.eja_subalgebra',
9 'FiniteDimensionalEuclideanJordanElementSubalgebra')
10 from mjo
.eja
.eja_operator
import FiniteDimensionalEuclideanJordanAlgebraOperator
11 from mjo
.eja
.eja_utils
import _mat2vec
13 class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement
):
15 An element of a Euclidean Jordan algebra.
20 Oh man, I should not be doing this. This hides the "disabled"
21 methods ``left_matrix`` and ``matrix`` from introspection;
22 in particular it removes them from tab-completion.
24 return filter(lambda s
: s
not in ['left_matrix', 'matrix'],
32 Return ``self`` raised to the power ``n``.
34 Jordan algebras are always power-associative; see for
35 example Faraut and Koranyi, Proposition II.1.2 (ii).
37 We have to override this because our superclass uses row
38 vectors instead of column vectors! We, on the other hand,
39 assume column vectors everywhere.
43 sage: from mjo.eja.eja_algebra import random_eja
47 The definition of `x^2` is the unambiguous `x*x`::
49 sage: set_random_seed()
50 sage: x = random_eja().random_element()
54 A few examples of power-associativity::
56 sage: set_random_seed()
57 sage: x = random_eja().random_element()
58 sage: x*(x*x)*(x*x) == x^5
60 sage: (x*x)*(x*x*x) == x^5
63 We also know that powers operator-commute (Koecher, Chapter
66 sage: set_random_seed()
67 sage: x = random_eja().random_element()
68 sage: m = ZZ.random_element(0,10)
69 sage: n = ZZ.random_element(0,10)
70 sage: Lxm = (x^m).operator()
71 sage: Lxn = (x^n).operator()
72 sage: Lxm*Lxn == Lxn*Lxm
77 return self
.parent().one()
81 return (self
.operator()**(n
-1))(self
)
84 def apply_univariate_polynomial(self
, p
):
86 Apply the univariate polynomial ``p`` to this element.
88 A priori, SageMath won't allow us to apply a univariate
89 polynomial to an element of an EJA, because we don't know
90 that EJAs are rings (they are usually not associative). Of
91 course, we know that EJAs are power-associative, so the
92 operation is ultimately kosher. This function sidesteps
93 the CAS to get the answer we want and expect.
97 sage: from mjo.eja.eja_algebra import (RealCartesianProductEJA,
102 sage: R = PolynomialRing(QQ, 't')
104 sage: p = t^4 - t^3 + 5*t - 2
105 sage: J = RealCartesianProductEJA(5)
106 sage: J.one().apply_univariate_polynomial(p) == 3*J.one()
111 We should always get back an element of the algebra::
113 sage: set_random_seed()
114 sage: p = PolynomialRing(QQ, 't').random_element()
115 sage: J = random_eja()
116 sage: x = J.random_element()
117 sage: x.apply_univariate_polynomial(p) in J
121 if len(p
.variables()) > 1:
122 raise ValueError("not a univariate polynomial")
125 # Convert the coeficcients to the parent's base ring,
126 # because a priori they might live in an (unnecessarily)
127 # larger ring for which P.sum() would fail below.
128 cs
= [ R(c
) for c
in p
.coefficients(sparse
=False) ]
129 return P
.sum( cs
[k
]*(self
**k
) for k
in range(len(cs
)) )
132 def characteristic_polynomial(self
):
134 Return the characteristic polynomial of this element.
138 sage: from mjo.eja.eja_algebra import RealCartesianProductEJA
142 The rank of `R^3` is three, and the minimal polynomial of
143 the identity element is `(t-1)` from which it follows that
144 the characteristic polynomial should be `(t-1)^3`::
146 sage: J = RealCartesianProductEJA(3)
147 sage: J.one().characteristic_polynomial()
148 t^3 - 3*t^2 + 3*t - 1
150 Likewise, the characteristic of the zero element in the
151 rank-three algebra `R^{n}` should be `t^{3}`::
153 sage: J = RealCartesianProductEJA(3)
154 sage: J.zero().characteristic_polynomial()
159 The characteristic polynomial of an element should evaluate
160 to zero on that element::
162 sage: set_random_seed()
163 sage: x = RealCartesianProductEJA(3).random_element()
164 sage: p = x.characteristic_polynomial()
165 sage: x.apply_univariate_polynomial(p)
169 p
= self
.parent().characteristic_polynomial()
170 return p(*self
.to_vector())
173 def inner_product(self
, other
):
175 Return the parent algebra's inner product of myself and ``other``.
179 sage: from mjo.eja.eja_algebra import (
180 ....: ComplexHermitianEJA,
182 ....: QuaternionHermitianEJA,
183 ....: RealSymmetricEJA,
188 The inner product in the Jordan spin algebra is the usual
189 inner product on `R^n` (this example only works because the
190 basis for the Jordan algebra is the standard basis in `R^n`)::
192 sage: J = JordanSpinEJA(3)
193 sage: x = vector(QQ,[1,2,3])
194 sage: y = vector(QQ,[4,5,6])
195 sage: x.inner_product(y)
197 sage: J.from_vector(x).inner_product(J.from_vector(y))
200 The inner product on `S^n` is `<X,Y> = trace(X*Y)`, where
201 multiplication is the usual matrix multiplication in `S^n`,
202 so the inner product of the identity matrix with itself
205 sage: J = RealSymmetricEJA(3)
206 sage: J.one().inner_product(J.one())
209 Likewise, the inner product on `C^n` is `<X,Y> =
210 Re(trace(X*Y))`, where we must necessarily take the real
211 part because the product of Hermitian matrices may not be
214 sage: J = ComplexHermitianEJA(3)
215 sage: J.one().inner_product(J.one())
218 Ditto for the quaternions::
220 sage: J = QuaternionHermitianEJA(3)
221 sage: J.one().inner_product(J.one())
226 Ensure that we can always compute an inner product, and that
227 it gives us back a real number::
229 sage: set_random_seed()
230 sage: J = random_eja()
231 sage: x = J.random_element()
232 sage: y = J.random_element()
233 sage: x.inner_product(y) in RR
239 raise TypeError("'other' must live in the same algebra")
241 return P
.inner_product(self
, other
)
244 def operator_commutes_with(self
, other
):
246 Return whether or not this element operator-commutes
251 sage: from mjo.eja.eja_algebra import random_eja
255 The definition of a Jordan algebra says that any element
256 operator-commutes with its square::
258 sage: set_random_seed()
259 sage: x = random_eja().random_element()
260 sage: x.operator_commutes_with(x^2)
265 Test Lemma 1 from Chapter III of Koecher::
267 sage: set_random_seed()
268 sage: J = random_eja()
269 sage: u = J.random_element()
270 sage: v = J.random_element()
271 sage: lhs = u.operator_commutes_with(u*v)
272 sage: rhs = v.operator_commutes_with(u^2)
276 Test the first polarization identity from my notes, Koecher
277 Chapter III, or from Baes (2.3)::
279 sage: set_random_seed()
280 sage: J = random_eja()
281 sage: x = J.random_element()
282 sage: y = J.random_element()
283 sage: Lx = x.operator()
284 sage: Ly = y.operator()
285 sage: Lxx = (x*x).operator()
286 sage: Lxy = (x*y).operator()
287 sage: bool(2*Lx*Lxy + Ly*Lxx == 2*Lxy*Lx + Lxx*Ly)
290 Test the second polarization identity from my notes or from
293 sage: set_random_seed()
294 sage: J = random_eja()
295 sage: x = J.random_element()
296 sage: y = J.random_element()
297 sage: z = J.random_element()
298 sage: Lx = x.operator()
299 sage: Ly = y.operator()
300 sage: Lz = z.operator()
301 sage: Lzy = (z*y).operator()
302 sage: Lxy = (x*y).operator()
303 sage: Lxz = (x*z).operator()
304 sage: bool(Lx*Lzy + Lz*Lxy + Ly*Lxz == Lzy*Lx + Lxy*Lz + Lxz*Ly)
307 Test the third polarization identity from my notes or from
310 sage: set_random_seed()
311 sage: J = random_eja()
312 sage: u = J.random_element()
313 sage: y = J.random_element()
314 sage: z = J.random_element()
315 sage: Lu = u.operator()
316 sage: Ly = y.operator()
317 sage: Lz = z.operator()
318 sage: Lzy = (z*y).operator()
319 sage: Luy = (u*y).operator()
320 sage: Luz = (u*z).operator()
321 sage: Luyz = (u*(y*z)).operator()
322 sage: lhs = Lu*Lzy + Lz*Luy + Ly*Luz
323 sage: rhs = Luyz + Ly*Lu*Lz + Lz*Lu*Ly
324 sage: bool(lhs == rhs)
328 if not other
in self
.parent():
329 raise TypeError("'other' must live in the same algebra")
338 Return my determinant, the product of my eigenvalues.
342 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
347 sage: J = JordanSpinEJA(2)
348 sage: e0,e1 = J.gens()
349 sage: x = sum( J.gens() )
355 sage: J = JordanSpinEJA(3)
356 sage: e0,e1,e2 = J.gens()
357 sage: x = sum( J.gens() )
363 An element is invertible if and only if its determinant is
366 sage: set_random_seed()
367 sage: x = random_eja().random_element()
368 sage: x.is_invertible() == (x.det() != 0)
374 p
= P
._charpoly
_coeff
(0)
375 # The _charpoly_coeff function already adds the factor of
376 # -1 to ensure that _charpoly_coeff(0) is really what
377 # appears in front of t^{0} in the charpoly. However,
378 # we want (-1)^r times THAT for the determinant.
379 return ((-1)**r
)*p(*self
.to_vector())
384 Return the Jordan-multiplicative inverse of this element.
388 We appeal to the quadratic representation as in Koecher's
389 Theorem 12 in Chapter III, Section 5.
393 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
398 The inverse in the spin factor algebra is given in Alizadeh's
401 sage: set_random_seed()
402 sage: n = ZZ.random_element(1,10)
403 sage: J = JordanSpinEJA(n)
404 sage: x = J.random_element()
405 sage: while not x.is_invertible():
406 ....: x = J.random_element()
407 sage: x_vec = x.to_vector()
409 sage: x_bar = x_vec[1:]
410 sage: coeff = ~(x0^2 - x_bar.inner_product(x_bar))
411 sage: inv_vec = x_vec.parent()([x0] + (-x_bar).list())
412 sage: x_inverse = coeff*inv_vec
413 sage: x.inverse() == J.from_vector(x_inverse)
418 The identity element is its own inverse::
420 sage: set_random_seed()
421 sage: J = random_eja()
422 sage: J.one().inverse() == J.one()
425 If an element has an inverse, it acts like one::
427 sage: set_random_seed()
428 sage: J = random_eja()
429 sage: x = J.random_element()
430 sage: (not x.is_invertible()) or (x.inverse()*x == J.one())
433 The inverse of the inverse is what we started with::
435 sage: set_random_seed()
436 sage: J = random_eja()
437 sage: x = J.random_element()
438 sage: (not x.is_invertible()) or (x.inverse().inverse() == x)
441 The zero element is never invertible::
443 sage: set_random_seed()
444 sage: J = random_eja().zero().inverse()
445 Traceback (most recent call last):
447 ValueError: element is not invertible
450 if not self
.is_invertible():
451 raise ValueError("element is not invertible")
453 return (~self
.quadratic_representation())(self
)
456 def is_invertible(self
):
458 Return whether or not this element is invertible.
462 The usual way to do this is to check if the determinant is
463 zero, but we need the characteristic polynomial for the
464 determinant. The minimal polynomial is a lot easier to get,
465 so we use Corollary 2 in Chapter V of Koecher to check
466 whether or not the paren't algebra's zero element is a root
467 of this element's minimal polynomial.
469 Beware that we can't use the superclass method, because it
470 relies on the algebra being associative.
474 sage: from mjo.eja.eja_algebra import random_eja
478 The identity element is always invertible::
480 sage: set_random_seed()
481 sage: J = random_eja()
482 sage: J.one().is_invertible()
485 The zero element is never invertible::
487 sage: set_random_seed()
488 sage: J = random_eja()
489 sage: J.zero().is_invertible()
493 # In fact, we only need to know if the constant term is non-zero,
494 # so we can pass in the field's zero element instead.
495 zero
= self
.base_ring().zero()
496 p
= self
.minimal_polynomial()
497 return not (p(zero
) == zero
)
500 def is_nilpotent(self
):
502 Return whether or not some power of this element is zero.
506 We use Theorem 5 in Chapter III of Koecher, which says that
507 an element ``x`` is nilpotent if and only if ``x.operator()``
508 is nilpotent. And it is a basic fact of linear algebra that
509 an operator on an `n`-dimensional space is nilpotent if and
510 only if, when raised to the `n`th power, it equals the zero
511 operator (for example, see Axler Corollary 8.8).
515 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
520 sage: J = JordanSpinEJA(3)
521 sage: x = sum(J.gens())
522 sage: x.is_nilpotent()
527 The identity element is never nilpotent::
529 sage: set_random_seed()
530 sage: random_eja().one().is_nilpotent()
533 The additive identity is always nilpotent::
535 sage: set_random_seed()
536 sage: random_eja().zero().is_nilpotent()
541 zero_operator
= P
.zero().operator()
542 return self
.operator()**P
.dimension() == zero_operator
545 def is_regular(self
):
547 Return whether or not this is a regular element.
551 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
556 The identity element always has degree one, but any element
557 linearly-independent from it is regular::
559 sage: J = JordanSpinEJA(5)
560 sage: J.one().is_regular()
562 sage: e0, e1, e2, e3, e4 = J.gens() # e0 is the identity
563 sage: for x in J.gens():
564 ....: (J.one() + x).is_regular()
573 The zero element should never be regular, unless the parent
574 algebra has dimension one::
576 sage: set_random_seed()
577 sage: J = random_eja()
578 sage: J.dimension() == 1 or not J.zero().is_regular()
581 The unit element isn't regular unless the algebra happens to
582 consist of only its scalar multiples::
584 sage: set_random_seed()
585 sage: J = random_eja()
586 sage: J.dimension() == 1 or not J.one().is_regular()
590 return self
.degree() == self
.parent().rank()
595 Return the degree of this element, which is defined to be
596 the degree of its minimal polynomial.
600 For now, we skip the messy minimal polynomial computation
601 and instead return the dimension of the vector space spanned
602 by the powers of this element. The latter is a bit more
603 straightforward to compute.
607 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
612 sage: J = JordanSpinEJA(4)
613 sage: J.one().degree()
615 sage: e0,e1,e2,e3 = J.gens()
616 sage: (e0 - e1).degree()
619 In the spin factor algebra (of rank two), all elements that
620 aren't multiples of the identity are regular::
622 sage: set_random_seed()
623 sage: n = ZZ.random_element(1,10)
624 sage: J = JordanSpinEJA(n)
625 sage: x = J.random_element()
626 sage: x == x.coefficient(0)*J.one() or x.degree() == 2
631 The zero and unit elements are both of degree one::
633 sage: set_random_seed()
634 sage: J = random_eja()
635 sage: J.zero().degree()
637 sage: J.one().degree()
640 Our implementation agrees with the definition::
642 sage: set_random_seed()
643 sage: x = random_eja().random_element()
644 sage: x.degree() == x.minimal_polynomial().degree()
648 return self
.subalgebra_generated_by().dimension()
651 def left_matrix(self
):
653 Our parent class defines ``left_matrix`` and ``matrix``
654 methods whose names are misleading. We don't want them.
656 raise NotImplementedError("use operator().matrix() instead")
661 def minimal_polynomial(self
):
663 Return the minimal polynomial of this element,
664 as a function of the variable `t`.
668 We restrict ourselves to the associative subalgebra
669 generated by this element, and then return the minimal
670 polynomial of this element's operator matrix (in that
671 subalgebra). This works by Baes Proposition 2.3.16.
675 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
680 The minimal polynomial of the identity and zero elements are
683 sage: set_random_seed()
684 sage: J = random_eja()
685 sage: J.one().minimal_polynomial()
687 sage: J.zero().minimal_polynomial()
690 The degree of an element is (by one definition) the degree
691 of its minimal polynomial::
693 sage: set_random_seed()
694 sage: x = random_eja().random_element()
695 sage: x.degree() == x.minimal_polynomial().degree()
698 The minimal polynomial and the characteristic polynomial coincide
699 and are known (see Alizadeh, Example 11.11) for all elements of
700 the spin factor algebra that aren't scalar multiples of the
703 sage: set_random_seed()
704 sage: n = ZZ.random_element(2,10)
705 sage: J = JordanSpinEJA(n)
706 sage: y = J.random_element()
707 sage: while y == y.coefficient(0)*J.one():
708 ....: y = J.random_element()
709 sage: y0 = y.to_vector()[0]
710 sage: y_bar = y.to_vector()[1:]
711 sage: actual = y.minimal_polynomial()
712 sage: t = PolynomialRing(J.base_ring(),'t').gen(0)
713 sage: expected = t^2 - 2*y0*t + (y0^2 - norm(y_bar)^2)
714 sage: bool(actual == expected)
717 The minimal polynomial should always kill its element::
719 sage: set_random_seed()
720 sage: x = random_eja().random_element()
721 sage: p = x.minimal_polynomial()
722 sage: x.apply_univariate_polynomial(p)
726 A
= self
.subalgebra_generated_by()
727 return A(self
).operator().minimal_polynomial()
731 def natural_representation(self
):
733 Return a more-natural representation of this element.
735 Every finite-dimensional Euclidean Jordan Algebra is a
736 direct sum of five simple algebras, four of which comprise
737 Hermitian matrices. This method returns the original
738 "natural" representation of this element as a Hermitian
739 matrix, if it has one. If not, you get the usual representation.
743 sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
744 ....: QuaternionHermitianEJA)
748 sage: J = ComplexHermitianEJA(3)
751 sage: J.one().natural_representation()
761 sage: J = QuaternionHermitianEJA(3)
764 sage: J.one().natural_representation()
765 [1 0 0 0 0 0 0 0 0 0 0 0]
766 [0 1 0 0 0 0 0 0 0 0 0 0]
767 [0 0 1 0 0 0 0 0 0 0 0 0]
768 [0 0 0 1 0 0 0 0 0 0 0 0]
769 [0 0 0 0 1 0 0 0 0 0 0 0]
770 [0 0 0 0 0 1 0 0 0 0 0 0]
771 [0 0 0 0 0 0 1 0 0 0 0 0]
772 [0 0 0 0 0 0 0 1 0 0 0 0]
773 [0 0 0 0 0 0 0 0 1 0 0 0]
774 [0 0 0 0 0 0 0 0 0 1 0 0]
775 [0 0 0 0 0 0 0 0 0 0 1 0]
776 [0 0 0 0 0 0 0 0 0 0 0 1]
779 B
= self
.parent().natural_basis()
780 W
= B
[0].matrix_space()
781 return W
.linear_combination(zip(B
,self
.to_vector()))
786 Return the left-multiplication-by-this-element
787 operator on the ambient algebra.
791 sage: from mjo.eja.eja_algebra import random_eja
795 sage: set_random_seed()
796 sage: J = random_eja()
797 sage: x = J.random_element()
798 sage: y = J.random_element()
799 sage: x.operator()(y) == x*y
801 sage: y.operator()(x) == x*y
806 left_mult_by_self
= lambda y
: self
*y
807 L
= P
.module_morphism(function
=left_mult_by_self
, codomain
=P
)
808 return FiniteDimensionalEuclideanJordanAlgebraOperator(
814 def quadratic_representation(self
, other
=None):
816 Return the quadratic representation of this element.
820 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
825 The explicit form in the spin factor algebra is given by
826 Alizadeh's Example 11.12::
828 sage: set_random_seed()
829 sage: n = ZZ.random_element(1,10)
830 sage: J = JordanSpinEJA(n)
831 sage: x = J.random_element()
832 sage: x_vec = x.to_vector()
834 sage: x_bar = x_vec[1:]
835 sage: A = matrix(QQ, 1, [x_vec.inner_product(x_vec)])
836 sage: B = 2*x0*x_bar.row()
837 sage: C = 2*x0*x_bar.column()
838 sage: D = matrix.identity(QQ, n-1)
839 sage: D = (x0^2 - x_bar.inner_product(x_bar))*D
840 sage: D = D + 2*x_bar.tensor_product(x_bar)
841 sage: Q = matrix.block(2,2,[A,B,C,D])
842 sage: Q == x.quadratic_representation().matrix()
845 Test all of the properties from Theorem 11.2 in Alizadeh::
847 sage: set_random_seed()
848 sage: J = random_eja()
849 sage: x = J.random_element()
850 sage: y = J.random_element()
851 sage: Lx = x.operator()
852 sage: Lxx = (x*x).operator()
853 sage: Qx = x.quadratic_representation()
854 sage: Qy = y.quadratic_representation()
855 sage: Qxy = x.quadratic_representation(y)
856 sage: Qex = J.one().quadratic_representation(x)
857 sage: n = ZZ.random_element(10)
858 sage: Qxn = (x^n).quadratic_representation()
862 sage: 2*Qxy == (x+y).quadratic_representation() - Qx - Qy
865 Property 2 (multiply on the right for :trac:`28272`):
867 sage: alpha = QQ.random_element()
868 sage: (alpha*x).quadratic_representation() == Qx*(alpha^2)
873 sage: not x.is_invertible() or ( Qx(x.inverse()) == x )
876 sage: not x.is_invertible() or (
879 ....: x.inverse().quadratic_representation() )
882 sage: Qxy(J.one()) == x*y
887 sage: not x.is_invertible() or (
888 ....: x.quadratic_representation(x.inverse())*Qx
889 ....: == Qx*x.quadratic_representation(x.inverse()) )
892 sage: not x.is_invertible() or (
893 ....: x.quadratic_representation(x.inverse())*Qx
895 ....: 2*x.operator()*Qex - Qx )
898 sage: 2*x.operator()*Qex - Qx == Lxx
903 sage: Qy(x).quadratic_representation() == Qy*Qx*Qy
913 sage: not x.is_invertible() or (
914 ....: Qx*x.inverse().operator() == Lx )
919 sage: not x.operator_commutes_with(y) or (
920 ....: Qx(y)^n == Qxn(y^n) )
926 elif not other
in self
.parent():
927 raise TypeError("'other' must live in the same algebra")
931 return ( L
*M
+ M
*L
- (self
*other
).operator() )
936 def subalgebra_generated_by(self
):
938 Return the associative subalgebra of the parent EJA generated
943 sage: from mjo.eja.eja_algebra import random_eja
947 This subalgebra, being composed of only powers, is associative::
949 sage: set_random_seed()
950 sage: x0 = random_eja().random_element()
951 sage: A = x0.subalgebra_generated_by()
952 sage: x = A.random_element()
953 sage: y = A.random_element()
954 sage: z = A.random_element()
955 sage: (x*y)*z == x*(y*z)
958 Squaring in the subalgebra should work the same as in
961 sage: set_random_seed()
962 sage: x = random_eja().random_element()
963 sage: A = x.subalgebra_generated_by()
964 sage: A(x^2) == A(x)*A(x)
968 return FiniteDimensionalEuclideanJordanElementSubalgebra(self
)
971 def subalgebra_idempotent(self
):
973 Find an idempotent in the associative subalgebra I generate
974 using Proposition 2.3.5 in Baes.
978 sage: from mjo.eja.eja_algebra import random_eja
982 sage: set_random_seed()
983 sage: J = random_eja()
984 sage: x = J.random_element()
985 sage: while x.is_nilpotent():
986 ....: x = J.random_element()
987 sage: c = x.subalgebra_idempotent()
992 if self
.is_nilpotent():
993 raise ValueError("this only works with non-nilpotent elements!")
995 J
= self
.subalgebra_generated_by()
998 # The image of the matrix of left-u^m-multiplication
999 # will be minimal for some natural number s...
1001 minimal_dim
= J
.dimension()
1002 for i
in xrange(1, minimal_dim
):
1003 this_dim
= (u
**i
).operator().matrix().image().dimension()
1004 if this_dim
< minimal_dim
:
1005 minimal_dim
= this_dim
1008 # Now minimal_matrix should correspond to the smallest
1009 # non-zero subspace in Baes's (or really, Koecher's)
1012 # However, we need to restrict the matrix to work on the
1013 # subspace... or do we? Can't we just solve, knowing that
1014 # A(c) = u^(s+1) should have a solution in the big space,
1017 # Beware, solve_right() means that we're using COLUMN vectors.
1018 # Our FiniteDimensionalAlgebraElement superclass uses rows.
1020 A
= u_next
.operator().matrix()
1021 c
= J
.from_vector(A
.solve_right(u_next
.to_vector()))
1023 # Now c is the idempotent we want, but it still lives in the subalgebra.
1024 return c
.superalgebra_element()
1029 Return my trace, the sum of my eigenvalues.
1033 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
1034 ....: RealCartesianProductEJA,
1039 sage: J = JordanSpinEJA(3)
1040 sage: x = sum(J.gens())
1046 sage: J = RealCartesianProductEJA(5)
1047 sage: J.one().trace()
1052 The trace of an element is a real number::
1054 sage: set_random_seed()
1055 sage: J = random_eja()
1056 sage: J.random_element().trace() in J.base_ring()
1062 p
= P
._charpoly
_coeff
(r
-1)
1063 # The _charpoly_coeff function already adds the factor of
1064 # -1 to ensure that _charpoly_coeff(r-1) is really what
1065 # appears in front of t^{r-1} in the charpoly. However,
1066 # we want the negative of THAT for the trace.
1067 return -p(*self
.to_vector())
1070 def trace_inner_product(self
, other
):
1072 Return the trace inner product of myself and ``other``.
1076 sage: from mjo.eja.eja_algebra import random_eja
1080 The trace inner product is commutative::
1082 sage: set_random_seed()
1083 sage: J = random_eja()
1084 sage: x = J.random_element(); y = J.random_element()
1085 sage: x.trace_inner_product(y) == y.trace_inner_product(x)
1088 The trace inner product is bilinear::
1090 sage: set_random_seed()
1091 sage: J = random_eja()
1092 sage: x = J.random_element()
1093 sage: y = J.random_element()
1094 sage: z = J.random_element()
1095 sage: a = QQ.random_element();
1096 sage: actual = (a*(x+z)).trace_inner_product(y)
1097 sage: expected = ( a*x.trace_inner_product(y) +
1098 ....: a*z.trace_inner_product(y) )
1099 sage: actual == expected
1101 sage: actual = x.trace_inner_product(a*(y+z))
1102 sage: expected = ( a*x.trace_inner_product(y) +
1103 ....: a*x.trace_inner_product(z) )
1104 sage: actual == expected
1107 The trace inner product satisfies the compatibility
1108 condition in the definition of a Euclidean Jordan algebra::
1110 sage: set_random_seed()
1111 sage: J = random_eja()
1112 sage: x = J.random_element()
1113 sage: y = J.random_element()
1114 sage: z = J.random_element()
1115 sage: (x*y).trace_inner_product(z) == y.trace_inner_product(x*z)
1119 if not other
in self
.parent():
1120 raise TypeError("'other' must live in the same algebra")
1122 return (self
*other
).trace()