1 # -*- coding: utf-8 -*-
3 from itertools
import izip
5 from sage
.matrix
.constructor
import matrix
6 from sage
.modules
.free_module
import VectorSpace
7 from sage
.modules
.with_basis
.indexed_element
import IndexedFreeModuleElement
9 # TODO: make this unnecessary somehow.
10 from sage
.misc
.lazy_import
import lazy_import
11 lazy_import('mjo.eja.eja_algebra', 'FiniteDimensionalEuclideanJordanAlgebra')
12 lazy_import('mjo.eja.eja_subalgebra',
13 'FiniteDimensionalEuclideanJordanElementSubalgebra')
14 from mjo
.eja
.eja_operator
import FiniteDimensionalEuclideanJordanAlgebraOperator
15 from mjo
.eja
.eja_utils
import _mat2vec
17 class FiniteDimensionalEuclideanJordanAlgebraElement(IndexedFreeModuleElement
):
19 An element of a Euclidean Jordan algebra.
24 Oh man, I should not be doing this. This hides the "disabled"
25 methods ``left_matrix`` and ``matrix`` from introspection;
26 in particular it removes them from tab-completion.
28 return filter(lambda s
: s
not in ['left_matrix', 'matrix'],
36 Return ``self`` raised to the power ``n``.
38 Jordan algebras are always power-associative; see for
39 example Faraut and Korányi, Proposition II.1.2 (ii).
41 We have to override this because our superclass uses row
42 vectors instead of column vectors! We, on the other hand,
43 assume column vectors everywhere.
47 sage: from mjo.eja.eja_algebra import random_eja
51 The definition of `x^2` is the unambiguous `x*x`::
53 sage: set_random_seed()
54 sage: x = random_eja().random_element()
58 A few examples of power-associativity::
60 sage: set_random_seed()
61 sage: x = random_eja().random_element()
62 sage: x*(x*x)*(x*x) == x^5
64 sage: (x*x)*(x*x*x) == x^5
67 We also know that powers operator-commute (Koecher, Chapter
70 sage: set_random_seed()
71 sage: x = random_eja().random_element()
72 sage: m = ZZ.random_element(0,10)
73 sage: n = ZZ.random_element(0,10)
74 sage: Lxm = (x^m).operator()
75 sage: Lxn = (x^n).operator()
76 sage: Lxm*Lxn == Lxn*Lxm
81 return self
.parent().one()
85 return (self
**(n
-1))*self
88 def apply_univariate_polynomial(self
, p
):
90 Apply the univariate polynomial ``p`` to this element.
92 A priori, SageMath won't allow us to apply a univariate
93 polynomial to an element of an EJA, because we don't know
94 that EJAs are rings (they are usually not associative). Of
95 course, we know that EJAs are power-associative, so the
96 operation is ultimately kosher. This function sidesteps
97 the CAS to get the answer we want and expect.
101 sage: from mjo.eja.eja_algebra import (RealCartesianProductEJA,
106 sage: R = PolynomialRing(QQ, 't')
108 sage: p = t^4 - t^3 + 5*t - 2
109 sage: J = RealCartesianProductEJA(5)
110 sage: J.one().apply_univariate_polynomial(p) == 3*J.one()
115 We should always get back an element of the algebra::
117 sage: set_random_seed()
118 sage: p = PolynomialRing(QQ, 't').random_element()
119 sage: J = random_eja()
120 sage: x = J.random_element()
121 sage: x.apply_univariate_polynomial(p) in J
125 if len(p
.variables()) > 1:
126 raise ValueError("not a univariate polynomial")
129 # Convert the coeficcients to the parent's base ring,
130 # because a priori they might live in an (unnecessarily)
131 # larger ring for which P.sum() would fail below.
132 cs
= [ R(c
) for c
in p
.coefficients(sparse
=False) ]
133 return P
.sum( cs
[k
]*(self
**k
) for k
in range(len(cs
)) )
136 def characteristic_polynomial(self
):
138 Return the characteristic polynomial of this element.
142 sage: from mjo.eja.eja_algebra import RealCartesianProductEJA
146 The rank of `R^3` is three, and the minimal polynomial of
147 the identity element is `(t-1)` from which it follows that
148 the characteristic polynomial should be `(t-1)^3`::
150 sage: J = RealCartesianProductEJA(3)
151 sage: J.one().characteristic_polynomial()
152 t^3 - 3*t^2 + 3*t - 1
154 Likewise, the characteristic of the zero element in the
155 rank-three algebra `R^{n}` should be `t^{3}`::
157 sage: J = RealCartesianProductEJA(3)
158 sage: J.zero().characteristic_polynomial()
163 The characteristic polynomial of an element should evaluate
164 to zero on that element::
166 sage: set_random_seed()
167 sage: x = RealCartesianProductEJA(3).random_element()
168 sage: p = x.characteristic_polynomial()
169 sage: x.apply_univariate_polynomial(p)
172 The characteristic polynomials of the zero and unit elements
173 should be what we think they are in a subalgebra, too::
175 sage: J = RealCartesianProductEJA(3)
176 sage: p1 = J.one().characteristic_polynomial()
177 sage: q1 = J.zero().characteristic_polynomial()
178 sage: e0,e1,e2 = J.gens()
179 sage: A = (e0 + 2*e1 + 3*e2).subalgebra_generated_by() # dim 3
180 sage: p2 = A.one().characteristic_polynomial()
181 sage: q2 = A.zero().characteristic_polynomial()
188 p
= self
.parent().characteristic_polynomial()
189 return p(*self
.to_vector())
192 def inner_product(self
, other
):
194 Return the parent algebra's inner product of myself and ``other``.
198 sage: from mjo.eja.eja_algebra import (
199 ....: ComplexHermitianEJA,
201 ....: QuaternionHermitianEJA,
202 ....: RealSymmetricEJA,
207 The inner product in the Jordan spin algebra is the usual
208 inner product on `R^n` (this example only works because the
209 basis for the Jordan algebra is the standard basis in `R^n`)::
211 sage: J = JordanSpinEJA(3)
212 sage: x = vector(QQ,[1,2,3])
213 sage: y = vector(QQ,[4,5,6])
214 sage: x.inner_product(y)
216 sage: J.from_vector(x).inner_product(J.from_vector(y))
219 The inner product on `S^n` is `<X,Y> = trace(X*Y)`, where
220 multiplication is the usual matrix multiplication in `S^n`,
221 so the inner product of the identity matrix with itself
224 sage: J = RealSymmetricEJA(3)
225 sage: J.one().inner_product(J.one())
228 Likewise, the inner product on `C^n` is `<X,Y> =
229 Re(trace(X*Y))`, where we must necessarily take the real
230 part because the product of Hermitian matrices may not be
233 sage: J = ComplexHermitianEJA(3)
234 sage: J.one().inner_product(J.one())
237 Ditto for the quaternions::
239 sage: J = QuaternionHermitianEJA(3)
240 sage: J.one().inner_product(J.one())
245 Ensure that we can always compute an inner product, and that
246 it gives us back a real number::
248 sage: set_random_seed()
249 sage: J = random_eja()
250 sage: x,y = J.random_elements(2)
251 sage: x.inner_product(y) in RLF
257 raise TypeError("'other' must live in the same algebra")
259 return P
.inner_product(self
, other
)
262 def operator_commutes_with(self
, other
):
264 Return whether or not this element operator-commutes
269 sage: from mjo.eja.eja_algebra import random_eja
273 The definition of a Jordan algebra says that any element
274 operator-commutes with its square::
276 sage: set_random_seed()
277 sage: x = random_eja().random_element()
278 sage: x.operator_commutes_with(x^2)
283 Test Lemma 1 from Chapter III of Koecher::
285 sage: set_random_seed()
286 sage: u,v = random_eja().random_elements(2)
287 sage: lhs = u.operator_commutes_with(u*v)
288 sage: rhs = v.operator_commutes_with(u^2)
292 Test the first polarization identity from my notes, Koecher
293 Chapter III, or from Baes (2.3)::
295 sage: set_random_seed()
296 sage: x,y = random_eja().random_elements(2)
297 sage: Lx = x.operator()
298 sage: Ly = y.operator()
299 sage: Lxx = (x*x).operator()
300 sage: Lxy = (x*y).operator()
301 sage: bool(2*Lx*Lxy + Ly*Lxx == 2*Lxy*Lx + Lxx*Ly)
304 Test the second polarization identity from my notes or from
307 sage: set_random_seed()
308 sage: x,y,z = random_eja().random_elements(3)
309 sage: Lx = x.operator()
310 sage: Ly = y.operator()
311 sage: Lz = z.operator()
312 sage: Lzy = (z*y).operator()
313 sage: Lxy = (x*y).operator()
314 sage: Lxz = (x*z).operator()
315 sage: bool(Lx*Lzy + Lz*Lxy + Ly*Lxz == Lzy*Lx + Lxy*Lz + Lxz*Ly)
318 Test the third polarization identity from my notes or from
321 sage: set_random_seed()
322 sage: u,y,z = random_eja().random_elements(3)
323 sage: Lu = u.operator()
324 sage: Ly = y.operator()
325 sage: Lz = z.operator()
326 sage: Lzy = (z*y).operator()
327 sage: Luy = (u*y).operator()
328 sage: Luz = (u*z).operator()
329 sage: Luyz = (u*(y*z)).operator()
330 sage: lhs = Lu*Lzy + Lz*Luy + Ly*Luz
331 sage: rhs = Luyz + Ly*Lu*Lz + Lz*Lu*Ly
332 sage: bool(lhs == rhs)
336 if not other
in self
.parent():
337 raise TypeError("'other' must live in the same algebra")
346 Return my determinant, the product of my eigenvalues.
350 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
355 sage: J = JordanSpinEJA(2)
356 sage: e0,e1 = J.gens()
357 sage: x = sum( J.gens() )
363 sage: J = JordanSpinEJA(3)
364 sage: e0,e1,e2 = J.gens()
365 sage: x = sum( J.gens() )
371 An element is invertible if and only if its determinant is
374 sage: set_random_seed()
375 sage: x = random_eja().random_element()
376 sage: x.is_invertible() == (x.det() != 0)
379 Ensure that the determinant is multiplicative on an associative
380 subalgebra as in Faraut and Korányi's Proposition II.2.2::
382 sage: set_random_seed()
383 sage: J = random_eja().random_element().subalgebra_generated_by()
384 sage: x,y = J.random_elements(2)
385 sage: (x*y).det() == x.det()*y.det()
391 p
= P
._charpoly
_coeff
(0)
392 # The _charpoly_coeff function already adds the factor of
393 # -1 to ensure that _charpoly_coeff(0) is really what
394 # appears in front of t^{0} in the charpoly. However,
395 # we want (-1)^r times THAT for the determinant.
396 return ((-1)**r
)*p(*self
.to_vector())
401 Return the Jordan-multiplicative inverse of this element.
405 We appeal to the quadratic representation as in Koecher's
406 Theorem 12 in Chapter III, Section 5.
410 sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
416 The inverse in the spin factor algebra is given in Alizadeh's
419 sage: set_random_seed()
420 sage: J = JordanSpinEJA.random_instance()
421 sage: x = J.random_element()
422 sage: while not x.is_invertible():
423 ....: x = J.random_element()
424 sage: x_vec = x.to_vector()
426 sage: x_bar = x_vec[1:]
427 sage: coeff = ~(x0^2 - x_bar.inner_product(x_bar))
428 sage: inv_vec = x_vec.parent()([x0] + (-x_bar).list())
429 sage: x_inverse = coeff*inv_vec
430 sage: x.inverse() == J.from_vector(x_inverse)
433 Trying to invert a non-invertible element throws an error:
435 sage: JordanSpinEJA(3).zero().inverse()
436 Traceback (most recent call last):
438 ValueError: element is not invertible
442 The identity element is its own inverse::
444 sage: set_random_seed()
445 sage: J = random_eja()
446 sage: J.one().inverse() == J.one()
449 If an element has an inverse, it acts like one::
451 sage: set_random_seed()
452 sage: J = random_eja()
453 sage: x = J.random_element()
454 sage: (not x.is_invertible()) or (x.inverse()*x == J.one())
457 The inverse of the inverse is what we started with::
459 sage: set_random_seed()
460 sage: J = random_eja()
461 sage: x = J.random_element()
462 sage: (not x.is_invertible()) or (x.inverse().inverse() == x)
465 Proposition II.2.3 in Faraut and Korányi says that the inverse
466 of an element is the inverse of its left-multiplication operator
467 applied to the algebra's identity, when that inverse exists::
469 sage: set_random_seed()
470 sage: J = random_eja()
471 sage: x = J.random_element()
472 sage: (not x.operator().is_invertible()) or (
473 ....: x.operator().inverse()(J.one()) == x.inverse() )
476 Proposition II.2.4 in Faraut and Korányi gives a formula for
477 the inverse based on the characteristic polynomial and the
478 Cayley-Hamilton theorem for Euclidean Jordan algebras::
480 sage: set_random_seed()
481 sage: J = ComplexHermitianEJA(3)
482 sage: x = J.random_element()
483 sage: while not x.is_invertible():
484 ....: x = J.random_element()
486 sage: a = x.characteristic_polynomial().coefficients(sparse=False)
487 sage: expected = (-1)^(r+1)/x.det()
488 sage: expected *= sum( a[i+1]*x^i for i in range(r) )
489 sage: x.inverse() == expected
493 if not self
.is_invertible():
494 raise ValueError("element is not invertible")
496 return (~self
.quadratic_representation())(self
)
499 def is_invertible(self
):
501 Return whether or not this element is invertible.
505 The usual way to do this is to check if the determinant is
506 zero, but we need the characteristic polynomial for the
507 determinant. The minimal polynomial is a lot easier to get,
508 so we use Corollary 2 in Chapter V of Koecher to check
509 whether or not the paren't algebra's zero element is a root
510 of this element's minimal polynomial.
512 Beware that we can't use the superclass method, because it
513 relies on the algebra being associative.
517 sage: from mjo.eja.eja_algebra import random_eja
521 The identity element is always invertible::
523 sage: set_random_seed()
524 sage: J = random_eja()
525 sage: J.one().is_invertible()
528 The zero element is never invertible in a non-trivial algebra::
530 sage: set_random_seed()
531 sage: J = random_eja()
532 sage: (not J.is_trivial()) and J.zero().is_invertible()
537 if self
.parent().is_trivial():
542 # In fact, we only need to know if the constant term is non-zero,
543 # so we can pass in the field's zero element instead.
544 zero
= self
.base_ring().zero()
545 p
= self
.minimal_polynomial()
546 return not (p(zero
) == zero
)
549 def is_nilpotent(self
):
551 Return whether or not some power of this element is zero.
555 We use Theorem 5 in Chapter III of Koecher, which says that
556 an element ``x`` is nilpotent if and only if ``x.operator()``
557 is nilpotent. And it is a basic fact of linear algebra that
558 an operator on an `n`-dimensional space is nilpotent if and
559 only if, when raised to the `n`th power, it equals the zero
560 operator (for example, see Axler Corollary 8.8).
564 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
569 sage: J = JordanSpinEJA(3)
570 sage: x = sum(J.gens())
571 sage: x.is_nilpotent()
576 The identity element is never nilpotent, except in a trivial EJA::
578 sage: set_random_seed()
579 sage: J = random_eja()
580 sage: J.one().is_nilpotent() and not J.is_trivial()
583 The additive identity is always nilpotent::
585 sage: set_random_seed()
586 sage: random_eja().zero().is_nilpotent()
591 zero_operator
= P
.zero().operator()
592 return self
.operator()**P
.dimension() == zero_operator
595 def is_regular(self
):
597 Return whether or not this is a regular element.
601 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
606 The identity element always has degree one, but any element
607 linearly-independent from it is regular::
609 sage: J = JordanSpinEJA(5)
610 sage: J.one().is_regular()
612 sage: e0, e1, e2, e3, e4 = J.gens() # e0 is the identity
613 sage: for x in J.gens():
614 ....: (J.one() + x).is_regular()
623 The zero element should never be regular, unless the parent
624 algebra has dimension less than or equal to one::
626 sage: set_random_seed()
627 sage: J = random_eja()
628 sage: J.dimension() <= 1 or not J.zero().is_regular()
631 The unit element isn't regular unless the algebra happens to
632 consist of only its scalar multiples::
634 sage: set_random_seed()
635 sage: J = random_eja()
636 sage: J.dimension() <= 1 or not J.one().is_regular()
640 return self
.degree() == self
.parent().rank()
645 Return the degree of this element, which is defined to be
646 the degree of its minimal polynomial.
650 For now, we skip the messy minimal polynomial computation
651 and instead return the dimension of the vector space spanned
652 by the powers of this element. The latter is a bit more
653 straightforward to compute.
657 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
662 sage: J = JordanSpinEJA(4)
663 sage: J.one().degree()
665 sage: e0,e1,e2,e3 = J.gens()
666 sage: (e0 - e1).degree()
669 In the spin factor algebra (of rank two), all elements that
670 aren't multiples of the identity are regular::
672 sage: set_random_seed()
673 sage: J = JordanSpinEJA.random_instance()
674 sage: x = J.random_element()
675 sage: x == x.coefficient(0)*J.one() or x.degree() == 2
680 The zero and unit elements are both of degree one in nontrivial
683 sage: set_random_seed()
684 sage: J = random_eja()
685 sage: d = J.zero().degree()
686 sage: (J.is_trivial() and d == 0) or d == 1
688 sage: d = J.one().degree()
689 sage: (J.is_trivial() and d == 0) or d == 1
692 Our implementation agrees with the definition::
694 sage: set_random_seed()
695 sage: x = random_eja().random_element()
696 sage: x.degree() == x.minimal_polynomial().degree()
700 if self
.is_zero() and not self
.parent().is_trivial():
701 # The minimal polynomial of zero in a nontrivial algebra
702 # is "t"; in a trivial algebra it's "1" by convention
703 # (it's an empty product).
705 return self
.subalgebra_generated_by().dimension()
708 def left_matrix(self
):
710 Our parent class defines ``left_matrix`` and ``matrix``
711 methods whose names are misleading. We don't want them.
713 raise NotImplementedError("use operator().matrix() instead")
718 def minimal_polynomial(self
):
720 Return the minimal polynomial of this element,
721 as a function of the variable `t`.
725 We restrict ourselves to the associative subalgebra
726 generated by this element, and then return the minimal
727 polynomial of this element's operator matrix (in that
728 subalgebra). This works by Baes Proposition 2.3.16.
732 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
733 ....: RealSymmetricEJA,
739 Keeping in mind that the polynomial ``1`` evaluates the identity
740 element (also the zero element) of the trivial algebra, it is clear
741 that the polynomial ``1`` is the minimal polynomial of the only
742 element in a trivial algebra::
744 sage: J = TrivialEJA()
745 sage: J.one().minimal_polynomial()
747 sage: J.zero().minimal_polynomial()
752 The minimal polynomial of the identity and zero elements are
755 sage: set_random_seed()
756 sage: J = random_eja(nontrivial=True)
757 sage: J.one().minimal_polynomial()
759 sage: J.zero().minimal_polynomial()
762 The degree of an element is (by one definition) the degree
763 of its minimal polynomial::
765 sage: set_random_seed()
766 sage: x = random_eja().random_element()
767 sage: x.degree() == x.minimal_polynomial().degree()
770 The minimal polynomial and the characteristic polynomial coincide
771 and are known (see Alizadeh, Example 11.11) for all elements of
772 the spin factor algebra that aren't scalar multiples of the
773 identity. We require the dimension of the algebra to be at least
774 two here so that said elements actually exist::
776 sage: set_random_seed()
777 sage: n_max = max(2, JordanSpinEJA._max_test_case_size())
778 sage: n = ZZ.random_element(2, n_max)
779 sage: J = JordanSpinEJA(n)
780 sage: y = J.random_element()
781 sage: while y == y.coefficient(0)*J.one():
782 ....: y = J.random_element()
783 sage: y0 = y.to_vector()[0]
784 sage: y_bar = y.to_vector()[1:]
785 sage: actual = y.minimal_polynomial()
786 sage: t = PolynomialRing(J.base_ring(),'t').gen(0)
787 sage: expected = t^2 - 2*y0*t + (y0^2 - norm(y_bar)^2)
788 sage: bool(actual == expected)
791 The minimal polynomial should always kill its element::
793 sage: set_random_seed()
794 sage: x = random_eja().random_element()
795 sage: p = x.minimal_polynomial()
796 sage: x.apply_univariate_polynomial(p)
799 The minimal polynomial is invariant under a change of basis,
800 and in particular, a re-scaling of the basis::
802 sage: set_random_seed()
803 sage: n_max = RealSymmetricEJA._max_test_case_size()
804 sage: n = ZZ.random_element(1, n_max)
805 sage: J1 = RealSymmetricEJA(n,QQ)
806 sage: J2 = RealSymmetricEJA(n,QQ,normalize_basis=False)
807 sage: X = random_matrix(QQ,n)
808 sage: X = X*X.transpose()
811 sage: x1.minimal_polynomial() == x2.minimal_polynomial()
816 # We would generate a zero-dimensional subalgebra
817 # where the minimal polynomial would be constant.
818 # That might be correct, but only if *this* algebra
820 if not self
.parent().is_trivial():
821 # Pretty sure we know what the minimal polynomial of
822 # the zero operator is going to be. This ensures
823 # consistency of e.g. the polynomial variable returned
824 # in the "normal" case without us having to think about it.
825 return self
.operator().minimal_polynomial()
827 A
= self
.subalgebra_generated_by()
828 return A(self
).operator().minimal_polynomial()
832 def natural_representation(self
):
834 Return a more-natural representation of this element.
836 Every finite-dimensional Euclidean Jordan Algebra is a
837 direct sum of five simple algebras, four of which comprise
838 Hermitian matrices. This method returns the original
839 "natural" representation of this element as a Hermitian
840 matrix, if it has one. If not, you get the usual representation.
844 sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
845 ....: QuaternionHermitianEJA)
849 sage: J = ComplexHermitianEJA(3)
852 sage: J.one().natural_representation()
862 sage: J = QuaternionHermitianEJA(3)
865 sage: J.one().natural_representation()
866 [1 0 0 0 0 0 0 0 0 0 0 0]
867 [0 1 0 0 0 0 0 0 0 0 0 0]
868 [0 0 1 0 0 0 0 0 0 0 0 0]
869 [0 0 0 1 0 0 0 0 0 0 0 0]
870 [0 0 0 0 1 0 0 0 0 0 0 0]
871 [0 0 0 0 0 1 0 0 0 0 0 0]
872 [0 0 0 0 0 0 1 0 0 0 0 0]
873 [0 0 0 0 0 0 0 1 0 0 0 0]
874 [0 0 0 0 0 0 0 0 1 0 0 0]
875 [0 0 0 0 0 0 0 0 0 1 0 0]
876 [0 0 0 0 0 0 0 0 0 0 1 0]
877 [0 0 0 0 0 0 0 0 0 0 0 1]
880 B
= self
.parent().natural_basis()
881 W
= self
.parent().natural_basis_space()
882 return W
.linear_combination(izip(B
,self
.to_vector()))
887 The norm of this element with respect to :meth:`inner_product`.
891 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
892 ....: RealCartesianProductEJA)
896 sage: J = RealCartesianProductEJA(2)
897 sage: x = sum(J.gens())
903 sage: J = JordanSpinEJA(4)
904 sage: x = sum(J.gens())
909 return self
.inner_product(self
).sqrt()
914 Return the left-multiplication-by-this-element
915 operator on the ambient algebra.
919 sage: from mjo.eja.eja_algebra import random_eja
923 sage: set_random_seed()
924 sage: J = random_eja()
925 sage: x,y = J.random_elements(2)
926 sage: x.operator()(y) == x*y
928 sage: y.operator()(x) == x*y
933 left_mult_by_self
= lambda y
: self
*y
934 L
= P
.module_morphism(function
=left_mult_by_self
, codomain
=P
)
935 return FiniteDimensionalEuclideanJordanAlgebraOperator(
941 def quadratic_representation(self
, other
=None):
943 Return the quadratic representation of this element.
947 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
952 The explicit form in the spin factor algebra is given by
953 Alizadeh's Example 11.12::
955 sage: set_random_seed()
956 sage: x = JordanSpinEJA.random_instance().random_element()
957 sage: x_vec = x.to_vector()
958 sage: n = x_vec.degree()
960 sage: x_bar = x_vec[1:]
961 sage: A = matrix(QQ, 1, [x_vec.inner_product(x_vec)])
962 sage: B = 2*x0*x_bar.row()
963 sage: C = 2*x0*x_bar.column()
964 sage: D = matrix.identity(QQ, n-1)
965 sage: D = (x0^2 - x_bar.inner_product(x_bar))*D
966 sage: D = D + 2*x_bar.tensor_product(x_bar)
967 sage: Q = matrix.block(2,2,[A,B,C,D])
968 sage: Q == x.quadratic_representation().matrix()
971 Test all of the properties from Theorem 11.2 in Alizadeh::
973 sage: set_random_seed()
974 sage: J = random_eja()
975 sage: x,y = J.random_elements(2)
976 sage: Lx = x.operator()
977 sage: Lxx = (x*x).operator()
978 sage: Qx = x.quadratic_representation()
979 sage: Qy = y.quadratic_representation()
980 sage: Qxy = x.quadratic_representation(y)
981 sage: Qex = J.one().quadratic_representation(x)
982 sage: n = ZZ.random_element(10)
983 sage: Qxn = (x^n).quadratic_representation()
987 sage: 2*Qxy == (x+y).quadratic_representation() - Qx - Qy
990 Property 2 (multiply on the right for :trac:`28272`):
992 sage: alpha = J.base_ring().random_element()
993 sage: (alpha*x).quadratic_representation() == Qx*(alpha^2)
998 sage: not x.is_invertible() or ( Qx(x.inverse()) == x )
1001 sage: not x.is_invertible() or (
1004 ....: x.inverse().quadratic_representation() )
1007 sage: Qxy(J.one()) == x*y
1012 sage: not x.is_invertible() or (
1013 ....: x.quadratic_representation(x.inverse())*Qx
1014 ....: == Qx*x.quadratic_representation(x.inverse()) )
1017 sage: not x.is_invertible() or (
1018 ....: x.quadratic_representation(x.inverse())*Qx
1020 ....: 2*Lx*Qex - Qx )
1023 sage: 2*Lx*Qex - Qx == Lxx
1028 sage: Qy(x).quadratic_representation() == Qy*Qx*Qy
1038 sage: not x.is_invertible() or (
1039 ....: Qx*x.inverse().operator() == Lx )
1044 sage: not x.operator_commutes_with(y) or (
1045 ....: Qx(y)^n == Qxn(y^n) )
1051 elif not other
in self
.parent():
1052 raise TypeError("'other' must live in the same algebra")
1055 M
= other
.operator()
1056 return ( L
*M
+ M
*L
- (self
*other
).operator() )
1060 def spectral_decomposition(self
):
1062 Return the unique spectral decomposition of this element.
1066 Following Faraut and Korányi's Theorem III.1.1, we restrict this
1067 element's left-multiplication-by operator to the subalgebra it
1068 generates. We then compute the spectral decomposition of that
1069 operator, and the spectral projectors we get back must be the
1070 left-multiplication-by operators for the idempotents we
1071 seek. Thus applying them to the identity element gives us those
1074 Since the eigenvalues are required to be distinct, we take
1075 the spectral decomposition of the zero element to be zero
1076 times the identity element of the algebra (which is idempotent,
1081 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
1085 The spectral decomposition of the identity is ``1`` times itself,
1086 and the spectral decomposition of zero is ``0`` times the identity::
1088 sage: J = RealSymmetricEJA(3,AA)
1091 sage: J.one().spectral_decomposition()
1093 sage: J.zero().spectral_decomposition()
1098 sage: J = RealSymmetricEJA(4,AA)
1099 sage: x = sum(J.gens())
1100 sage: sd = x.spectral_decomposition()
1105 sage: c0.inner_product(c1) == 0
1107 sage: c0.is_idempotent()
1109 sage: c1.is_idempotent()
1111 sage: c0 + c1 == J.one()
1113 sage: l0*c0 + l1*c1 == x
1118 A
= self
.subalgebra_generated_by(orthonormalize_basis
=True)
1120 for (evalue
, proj
) in A(self
).operator().spectral_decomposition():
1121 result
.append( (evalue
, proj(A
.one()).superalgebra_element()) )
1124 def subalgebra_generated_by(self
, orthonormalize_basis
=False):
1126 Return the associative subalgebra of the parent EJA generated
1129 Since our parent algebra is unital, we want "subalgebra" to mean
1130 "unital subalgebra" as well; thus the subalgebra that an element
1131 generates will itself be a Euclidean Jordan algebra after
1132 restricting the algebra operations appropriately. This is the
1133 subalgebra that Faraut and Korányi work with in section II.2, for
1138 sage: from mjo.eja.eja_algebra import random_eja
1142 This subalgebra, being composed of only powers, is associative::
1144 sage: set_random_seed()
1145 sage: x0 = random_eja().random_element()
1146 sage: A = x0.subalgebra_generated_by()
1147 sage: x,y,z = A.random_elements(3)
1148 sage: (x*y)*z == x*(y*z)
1151 Squaring in the subalgebra should work the same as in
1154 sage: set_random_seed()
1155 sage: x = random_eja().random_element()
1156 sage: A = x.subalgebra_generated_by()
1157 sage: A(x^2) == A(x)*A(x)
1160 By definition, the subalgebra generated by the zero element is
1161 the one-dimensional algebra generated by the identity
1162 element... unless the original algebra was trivial, in which
1163 case the subalgebra is trivial too::
1165 sage: set_random_seed()
1166 sage: A = random_eja().zero().subalgebra_generated_by()
1167 sage: (A.is_trivial() and A.dimension() == 0) or A.dimension() == 1
1171 return FiniteDimensionalEuclideanJordanElementSubalgebra(self
, orthonormalize_basis
)
1174 def subalgebra_idempotent(self
):
1176 Find an idempotent in the associative subalgebra I generate
1177 using Proposition 2.3.5 in Baes.
1181 sage: from mjo.eja.eja_algebra import random_eja
1185 sage: set_random_seed()
1186 sage: J = random_eja()
1187 sage: x = J.random_element()
1188 sage: while x.is_nilpotent():
1189 ....: x = J.random_element()
1190 sage: c = x.subalgebra_idempotent()
1195 if self
.is_nilpotent():
1196 raise ValueError("this only works with non-nilpotent elements!")
1198 J
= self
.subalgebra_generated_by()
1201 # The image of the matrix of left-u^m-multiplication
1202 # will be minimal for some natural number s...
1204 minimal_dim
= J
.dimension()
1205 for i
in xrange(1, minimal_dim
):
1206 this_dim
= (u
**i
).operator().matrix().image().dimension()
1207 if this_dim
< minimal_dim
:
1208 minimal_dim
= this_dim
1211 # Now minimal_matrix should correspond to the smallest
1212 # non-zero subspace in Baes's (or really, Koecher's)
1215 # However, we need to restrict the matrix to work on the
1216 # subspace... or do we? Can't we just solve, knowing that
1217 # A(c) = u^(s+1) should have a solution in the big space,
1220 # Beware, solve_right() means that we're using COLUMN vectors.
1221 # Our FiniteDimensionalAlgebraElement superclass uses rows.
1223 A
= u_next
.operator().matrix()
1224 c
= J
.from_vector(A
.solve_right(u_next
.to_vector()))
1226 # Now c is the idempotent we want, but it still lives in the subalgebra.
1227 return c
.superalgebra_element()
1232 Return my trace, the sum of my eigenvalues.
1234 In a trivial algebra, however you want to look at it, the trace is
1235 an empty sum for which we declare the result to be zero.
1239 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
1240 ....: RealCartesianProductEJA,
1246 sage: J = TrivialEJA()
1247 sage: J.zero().trace()
1251 sage: J = JordanSpinEJA(3)
1252 sage: x = sum(J.gens())
1258 sage: J = RealCartesianProductEJA(5)
1259 sage: J.one().trace()
1264 The trace of an element is a real number::
1266 sage: set_random_seed()
1267 sage: J = random_eja()
1268 sage: J.random_element().trace() in RLF
1276 # Special case for the trivial algebra where
1277 # the trace is an empty sum.
1278 return P
.base_ring().zero()
1280 p
= P
._charpoly
_coeff
(r
-1)
1281 # The _charpoly_coeff function already adds the factor of
1282 # -1 to ensure that _charpoly_coeff(r-1) is really what
1283 # appears in front of t^{r-1} in the charpoly. However,
1284 # we want the negative of THAT for the trace.
1285 return -p(*self
.to_vector())
1288 def trace_inner_product(self
, other
):
1290 Return the trace inner product of myself and ``other``.
1294 sage: from mjo.eja.eja_algebra import random_eja
1298 The trace inner product is commutative, bilinear, and associative::
1300 sage: set_random_seed()
1301 sage: J = random_eja()
1302 sage: x,y,z = J.random_elements(3)
1304 sage: x.trace_inner_product(y) == y.trace_inner_product(x)
1307 sage: a = J.base_ring().random_element();
1308 sage: actual = (a*(x+z)).trace_inner_product(y)
1309 sage: expected = ( a*x.trace_inner_product(y) +
1310 ....: a*z.trace_inner_product(y) )
1311 sage: actual == expected
1313 sage: actual = x.trace_inner_product(a*(y+z))
1314 sage: expected = ( a*x.trace_inner_product(y) +
1315 ....: a*x.trace_inner_product(z) )
1316 sage: actual == expected
1319 sage: (x*y).trace_inner_product(z) == y.trace_inner_product(x*z)
1323 if not other
in self
.parent():
1324 raise TypeError("'other' must live in the same algebra")
1326 return (self
*other
).trace()
1329 def trace_norm(self
):
1331 The norm of this element with respect to :meth:`trace_inner_product`.
1335 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
1336 ....: RealCartesianProductEJA)
1340 sage: J = RealCartesianProductEJA(2)
1341 sage: x = sum(J.gens())
1342 sage: x.trace_norm()
1347 sage: J = JordanSpinEJA(4)
1348 sage: x = sum(J.gens())
1349 sage: x.trace_norm()
1353 return self
.trace_inner_product(self
).sqrt()