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_element_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_minimal_idempotent(self
):
551 Return whether or not this element is a minimal idempotent.
554 An element of a Euclidean Jordan algebra is a minimal idempotent
555 if it :meth:`is_idempotent` and if its Peirce subalgebra
556 corresponding to the eigenvalue ``1`` has dimension ``1`` (Baes,
561 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
562 ....: RealSymmetricEJA,
567 This method is sloooooow.
571 The spectral decomposition of a non-regular element should always
572 contain at least one non-minimal idempotent::
574 sage: J = RealSymmetricEJA(3, AA)
575 sage: x = sum(J.gens())
578 sage: [ c.is_minimal_idempotent()
579 ....: for (l,c) in x.spectral_decomposition() ]
582 On the other hand, the spectral decomposition of a regular
583 element should always be in terms of minimal idempotents::
585 sage: J = JordanSpinEJA(4, AA)
586 sage: x = sum( i*J.gens()[i] for i in range(len(J.gens())) )
589 sage: [ c.is_minimal_idempotent()
590 ....: for (l,c) in x.spectral_decomposition() ]
595 The identity element is minimal only in an EJA of rank one::
597 sage: set_random_seed()
598 sage: J = random_eja()
599 sage: J.rank() == 1 or not J.one().is_minimal_idempotent()
602 A non-idempotent cannot be a minimal idempotent::
604 sage: set_random_seed()
605 sage: J = JordanSpinEJA(4)
606 sage: x = J.random_element()
607 sage: (not x.is_idempotent()) and x.is_minimal_idempotent()
610 Proposition 2.7.19 in Baes says that an element is a minimal
611 idempotent if and only if it's idempotent with trace equal to
614 sage: set_random_seed()
615 sage: J = JordanSpinEJA(4)
616 sage: x = J.random_element()
617 sage: expected = (x.is_idempotent() and x.trace() == 1)
618 sage: actual = x.is_minimal_idempotent()
619 sage: actual == expected
623 # TODO: when the Peirce decomposition is implemented for real,
624 # we can use that instead of finding this eigenspace manually.
626 # Trivial eigenspaces don't appear in the list, so we default to the
627 # trivial one and override it if there's a nontrivial space in the
629 if not self
.is_idempotent():
632 J1
= VectorSpace(self
.parent().base_ring(), 0)
633 for (eigval
, eigspace
) in self
.operator().matrix().left_eigenspaces():
636 return (J1
.dimension() == 1)
639 def is_nilpotent(self
):
641 Return whether or not some power of this element is zero.
645 We use Theorem 5 in Chapter III of Koecher, which says that
646 an element ``x`` is nilpotent if and only if ``x.operator()``
647 is nilpotent. And it is a basic fact of linear algebra that
648 an operator on an `n`-dimensional space is nilpotent if and
649 only if, when raised to the `n`th power, it equals the zero
650 operator (for example, see Axler Corollary 8.8).
654 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
659 sage: J = JordanSpinEJA(3)
660 sage: x = sum(J.gens())
661 sage: x.is_nilpotent()
666 The identity element is never nilpotent, except in a trivial EJA::
668 sage: set_random_seed()
669 sage: J = random_eja()
670 sage: J.one().is_nilpotent() and not J.is_trivial()
673 The additive identity is always nilpotent::
675 sage: set_random_seed()
676 sage: random_eja().zero().is_nilpotent()
681 zero_operator
= P
.zero().operator()
682 return self
.operator()**P
.dimension() == zero_operator
685 def is_regular(self
):
687 Return whether or not this is a regular element.
691 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
696 The identity element always has degree one, but any element
697 linearly-independent from it is regular::
699 sage: J = JordanSpinEJA(5)
700 sage: J.one().is_regular()
702 sage: e0, e1, e2, e3, e4 = J.gens() # e0 is the identity
703 sage: for x in J.gens():
704 ....: (J.one() + x).is_regular()
713 The zero element should never be regular, unless the parent
714 algebra has dimension less than or equal to one::
716 sage: set_random_seed()
717 sage: J = random_eja()
718 sage: J.dimension() <= 1 or not J.zero().is_regular()
721 The unit element isn't regular unless the algebra happens to
722 consist of only its scalar multiples::
724 sage: set_random_seed()
725 sage: J = random_eja()
726 sage: J.dimension() <= 1 or not J.one().is_regular()
730 return self
.degree() == self
.parent().rank()
735 Return the degree of this element, which is defined to be
736 the degree of its minimal polynomial.
740 For now, we skip the messy minimal polynomial computation
741 and instead return the dimension of the vector space spanned
742 by the powers of this element. The latter is a bit more
743 straightforward to compute.
747 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
752 sage: J = JordanSpinEJA(4)
753 sage: J.one().degree()
755 sage: e0,e1,e2,e3 = J.gens()
756 sage: (e0 - e1).degree()
759 In the spin factor algebra (of rank two), all elements that
760 aren't multiples of the identity are regular::
762 sage: set_random_seed()
763 sage: J = JordanSpinEJA.random_instance()
764 sage: x = J.random_element()
765 sage: x == x.coefficient(0)*J.one() or x.degree() == 2
770 The zero and unit elements are both of degree one in nontrivial
773 sage: set_random_seed()
774 sage: J = random_eja()
775 sage: d = J.zero().degree()
776 sage: (J.is_trivial() and d == 0) or d == 1
778 sage: d = J.one().degree()
779 sage: (J.is_trivial() and d == 0) or d == 1
782 Our implementation agrees with the definition::
784 sage: set_random_seed()
785 sage: x = random_eja().random_element()
786 sage: x.degree() == x.minimal_polynomial().degree()
790 if self
.is_zero() and not self
.parent().is_trivial():
791 # The minimal polynomial of zero in a nontrivial algebra
792 # is "t"; in a trivial algebra it's "1" by convention
793 # (it's an empty product).
795 return self
.subalgebra_generated_by().dimension()
798 def left_matrix(self
):
800 Our parent class defines ``left_matrix`` and ``matrix``
801 methods whose names are misleading. We don't want them.
803 raise NotImplementedError("use operator().matrix() instead")
808 def minimal_polynomial(self
):
810 Return the minimal polynomial of this element,
811 as a function of the variable `t`.
815 We restrict ourselves to the associative subalgebra
816 generated by this element, and then return the minimal
817 polynomial of this element's operator matrix (in that
818 subalgebra). This works by Baes Proposition 2.3.16.
822 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
823 ....: RealSymmetricEJA,
829 Keeping in mind that the polynomial ``1`` evaluates the identity
830 element (also the zero element) of the trivial algebra, it is clear
831 that the polynomial ``1`` is the minimal polynomial of the only
832 element in a trivial algebra::
834 sage: J = TrivialEJA()
835 sage: J.one().minimal_polynomial()
837 sage: J.zero().minimal_polynomial()
842 The minimal polynomial of the identity and zero elements are
845 sage: set_random_seed()
846 sage: J = random_eja(nontrivial=True)
847 sage: J.one().minimal_polynomial()
849 sage: J.zero().minimal_polynomial()
852 The degree of an element is (by one definition) the degree
853 of its minimal polynomial::
855 sage: set_random_seed()
856 sage: x = random_eja().random_element()
857 sage: x.degree() == x.minimal_polynomial().degree()
860 The minimal polynomial and the characteristic polynomial coincide
861 and are known (see Alizadeh, Example 11.11) for all elements of
862 the spin factor algebra that aren't scalar multiples of the
863 identity. We require the dimension of the algebra to be at least
864 two here so that said elements actually exist::
866 sage: set_random_seed()
867 sage: n_max = max(2, JordanSpinEJA._max_test_case_size())
868 sage: n = ZZ.random_element(2, n_max)
869 sage: J = JordanSpinEJA(n)
870 sage: y = J.random_element()
871 sage: while y == y.coefficient(0)*J.one():
872 ....: y = J.random_element()
873 sage: y0 = y.to_vector()[0]
874 sage: y_bar = y.to_vector()[1:]
875 sage: actual = y.minimal_polynomial()
876 sage: t = PolynomialRing(J.base_ring(),'t').gen(0)
877 sage: expected = t^2 - 2*y0*t + (y0^2 - norm(y_bar)^2)
878 sage: bool(actual == expected)
881 The minimal polynomial should always kill its element::
883 sage: set_random_seed()
884 sage: x = random_eja().random_element()
885 sage: p = x.minimal_polynomial()
886 sage: x.apply_univariate_polynomial(p)
889 The minimal polynomial is invariant under a change of basis,
890 and in particular, a re-scaling of the basis::
892 sage: set_random_seed()
893 sage: n_max = RealSymmetricEJA._max_test_case_size()
894 sage: n = ZZ.random_element(1, n_max)
895 sage: J1 = RealSymmetricEJA(n,QQ)
896 sage: J2 = RealSymmetricEJA(n,QQ,normalize_basis=False)
897 sage: X = random_matrix(QQ,n)
898 sage: X = X*X.transpose()
901 sage: x1.minimal_polynomial() == x2.minimal_polynomial()
906 # We would generate a zero-dimensional subalgebra
907 # where the minimal polynomial would be constant.
908 # That might be correct, but only if *this* algebra
910 if not self
.parent().is_trivial():
911 # Pretty sure we know what the minimal polynomial of
912 # the zero operator is going to be. This ensures
913 # consistency of e.g. the polynomial variable returned
914 # in the "normal" case without us having to think about it.
915 return self
.operator().minimal_polynomial()
917 A
= self
.subalgebra_generated_by()
918 return A(self
).operator().minimal_polynomial()
922 def natural_representation(self
):
924 Return a more-natural representation of this element.
926 Every finite-dimensional Euclidean Jordan Algebra is a
927 direct sum of five simple algebras, four of which comprise
928 Hermitian matrices. This method returns the original
929 "natural" representation of this element as a Hermitian
930 matrix, if it has one. If not, you get the usual representation.
934 sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
935 ....: QuaternionHermitianEJA)
939 sage: J = ComplexHermitianEJA(3)
942 sage: J.one().natural_representation()
952 sage: J = QuaternionHermitianEJA(3)
955 sage: J.one().natural_representation()
956 [1 0 0 0 0 0 0 0 0 0 0 0]
957 [0 1 0 0 0 0 0 0 0 0 0 0]
958 [0 0 1 0 0 0 0 0 0 0 0 0]
959 [0 0 0 1 0 0 0 0 0 0 0 0]
960 [0 0 0 0 1 0 0 0 0 0 0 0]
961 [0 0 0 0 0 1 0 0 0 0 0 0]
962 [0 0 0 0 0 0 1 0 0 0 0 0]
963 [0 0 0 0 0 0 0 1 0 0 0 0]
964 [0 0 0 0 0 0 0 0 1 0 0 0]
965 [0 0 0 0 0 0 0 0 0 1 0 0]
966 [0 0 0 0 0 0 0 0 0 0 1 0]
967 [0 0 0 0 0 0 0 0 0 0 0 1]
970 B
= self
.parent().natural_basis()
971 W
= self
.parent().natural_basis_space()
972 return W
.linear_combination(izip(B
,self
.to_vector()))
977 The norm of this element with respect to :meth:`inner_product`.
981 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
982 ....: RealCartesianProductEJA)
986 sage: J = RealCartesianProductEJA(2)
987 sage: x = sum(J.gens())
993 sage: J = JordanSpinEJA(4)
994 sage: x = sum(J.gens())
999 return self
.inner_product(self
).sqrt()
1004 Return the left-multiplication-by-this-element
1005 operator on the ambient algebra.
1009 sage: from mjo.eja.eja_algebra import random_eja
1013 sage: set_random_seed()
1014 sage: J = random_eja()
1015 sage: x,y = J.random_elements(2)
1016 sage: x.operator()(y) == x*y
1018 sage: y.operator()(x) == x*y
1023 left_mult_by_self
= lambda y
: self
*y
1024 L
= P
.module_morphism(function
=left_mult_by_self
, codomain
=P
)
1025 return FiniteDimensionalEuclideanJordanAlgebraOperator(
1031 def quadratic_representation(self
, other
=None):
1033 Return the quadratic representation of this element.
1037 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
1042 The explicit form in the spin factor algebra is given by
1043 Alizadeh's Example 11.12::
1045 sage: set_random_seed()
1046 sage: x = JordanSpinEJA.random_instance().random_element()
1047 sage: x_vec = x.to_vector()
1048 sage: n = x_vec.degree()
1050 sage: x_bar = x_vec[1:]
1051 sage: A = matrix(QQ, 1, [x_vec.inner_product(x_vec)])
1052 sage: B = 2*x0*x_bar.row()
1053 sage: C = 2*x0*x_bar.column()
1054 sage: D = matrix.identity(QQ, n-1)
1055 sage: D = (x0^2 - x_bar.inner_product(x_bar))*D
1056 sage: D = D + 2*x_bar.tensor_product(x_bar)
1057 sage: Q = matrix.block(2,2,[A,B,C,D])
1058 sage: Q == x.quadratic_representation().matrix()
1061 Test all of the properties from Theorem 11.2 in Alizadeh::
1063 sage: set_random_seed()
1064 sage: J = random_eja()
1065 sage: x,y = J.random_elements(2)
1066 sage: Lx = x.operator()
1067 sage: Lxx = (x*x).operator()
1068 sage: Qx = x.quadratic_representation()
1069 sage: Qy = y.quadratic_representation()
1070 sage: Qxy = x.quadratic_representation(y)
1071 sage: Qex = J.one().quadratic_representation(x)
1072 sage: n = ZZ.random_element(10)
1073 sage: Qxn = (x^n).quadratic_representation()
1077 sage: 2*Qxy == (x+y).quadratic_representation() - Qx - Qy
1080 Property 2 (multiply on the right for :trac:`28272`):
1082 sage: alpha = J.base_ring().random_element()
1083 sage: (alpha*x).quadratic_representation() == Qx*(alpha^2)
1088 sage: not x.is_invertible() or ( Qx(x.inverse()) == x )
1091 sage: not x.is_invertible() or (
1094 ....: x.inverse().quadratic_representation() )
1097 sage: Qxy(J.one()) == x*y
1102 sage: not x.is_invertible() or (
1103 ....: x.quadratic_representation(x.inverse())*Qx
1104 ....: == Qx*x.quadratic_representation(x.inverse()) )
1107 sage: not x.is_invertible() or (
1108 ....: x.quadratic_representation(x.inverse())*Qx
1110 ....: 2*Lx*Qex - Qx )
1113 sage: 2*Lx*Qex - Qx == Lxx
1118 sage: Qy(x).quadratic_representation() == Qy*Qx*Qy
1128 sage: not x.is_invertible() or (
1129 ....: Qx*x.inverse().operator() == Lx )
1134 sage: not x.operator_commutes_with(y) or (
1135 ....: Qx(y)^n == Qxn(y^n) )
1141 elif not other
in self
.parent():
1142 raise TypeError("'other' must live in the same algebra")
1145 M
= other
.operator()
1146 return ( L
*M
+ M
*L
- (self
*other
).operator() )
1150 def spectral_decomposition(self
):
1152 Return the unique spectral decomposition of this element.
1156 Following Faraut and Korányi's Theorem III.1.1, we restrict this
1157 element's left-multiplication-by operator to the subalgebra it
1158 generates. We then compute the spectral decomposition of that
1159 operator, and the spectral projectors we get back must be the
1160 left-multiplication-by operators for the idempotents we
1161 seek. Thus applying them to the identity element gives us those
1164 Since the eigenvalues are required to be distinct, we take
1165 the spectral decomposition of the zero element to be zero
1166 times the identity element of the algebra (which is idempotent,
1171 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
1175 The spectral decomposition of the identity is ``1`` times itself,
1176 and the spectral decomposition of zero is ``0`` times the identity::
1178 sage: J = RealSymmetricEJA(3,AA)
1181 sage: J.one().spectral_decomposition()
1183 sage: J.zero().spectral_decomposition()
1188 sage: J = RealSymmetricEJA(4,AA)
1189 sage: x = sum(J.gens())
1190 sage: sd = x.spectral_decomposition()
1195 sage: c0.inner_product(c1) == 0
1197 sage: c0.is_idempotent()
1199 sage: c1.is_idempotent()
1201 sage: c0 + c1 == J.one()
1203 sage: l0*c0 + l1*c1 == x
1208 A
= self
.subalgebra_generated_by(orthonormalize_basis
=True)
1210 for (evalue
, proj
) in A(self
).operator().spectral_decomposition():
1211 result
.append( (evalue
, proj(A
.one()).superalgebra_element()) )
1214 def subalgebra_generated_by(self
, orthonormalize_basis
=False):
1216 Return the associative subalgebra of the parent EJA generated
1219 Since our parent algebra is unital, we want "subalgebra" to mean
1220 "unital subalgebra" as well; thus the subalgebra that an element
1221 generates will itself be a Euclidean Jordan algebra after
1222 restricting the algebra operations appropriately. This is the
1223 subalgebra that Faraut and Korányi work with in section II.2, for
1228 sage: from mjo.eja.eja_algebra import random_eja
1232 This subalgebra, being composed of only powers, is associative::
1234 sage: set_random_seed()
1235 sage: x0 = random_eja().random_element()
1236 sage: A = x0.subalgebra_generated_by()
1237 sage: x,y,z = A.random_elements(3)
1238 sage: (x*y)*z == x*(y*z)
1241 Squaring in the subalgebra should work the same as in
1244 sage: set_random_seed()
1245 sage: x = random_eja().random_element()
1246 sage: A = x.subalgebra_generated_by()
1247 sage: A(x^2) == A(x)*A(x)
1250 By definition, the subalgebra generated by the zero element is
1251 the one-dimensional algebra generated by the identity
1252 element... unless the original algebra was trivial, in which
1253 case the subalgebra is trivial too::
1255 sage: set_random_seed()
1256 sage: A = random_eja().zero().subalgebra_generated_by()
1257 sage: (A.is_trivial() and A.dimension() == 0) or A.dimension() == 1
1261 return FiniteDimensionalEuclideanJordanElementSubalgebra(self
, orthonormalize_basis
)
1264 def subalgebra_idempotent(self
):
1266 Find an idempotent in the associative subalgebra I generate
1267 using Proposition 2.3.5 in Baes.
1271 sage: from mjo.eja.eja_algebra import random_eja
1275 sage: set_random_seed()
1276 sage: J = random_eja()
1277 sage: x = J.random_element()
1278 sage: while x.is_nilpotent():
1279 ....: x = J.random_element()
1280 sage: c = x.subalgebra_idempotent()
1285 if self
.is_nilpotent():
1286 raise ValueError("this only works with non-nilpotent elements!")
1288 J
= self
.subalgebra_generated_by()
1291 # The image of the matrix of left-u^m-multiplication
1292 # will be minimal for some natural number s...
1294 minimal_dim
= J
.dimension()
1295 for i
in xrange(1, minimal_dim
):
1296 this_dim
= (u
**i
).operator().matrix().image().dimension()
1297 if this_dim
< minimal_dim
:
1298 minimal_dim
= this_dim
1301 # Now minimal_matrix should correspond to the smallest
1302 # non-zero subspace in Baes's (or really, Koecher's)
1305 # However, we need to restrict the matrix to work on the
1306 # subspace... or do we? Can't we just solve, knowing that
1307 # A(c) = u^(s+1) should have a solution in the big space,
1310 # Beware, solve_right() means that we're using COLUMN vectors.
1311 # Our FiniteDimensionalAlgebraElement superclass uses rows.
1313 A
= u_next
.operator().matrix()
1314 c
= J
.from_vector(A
.solve_right(u_next
.to_vector()))
1316 # Now c is the idempotent we want, but it still lives in the subalgebra.
1317 return c
.superalgebra_element()
1322 Return my trace, the sum of my eigenvalues.
1324 In a trivial algebra, however you want to look at it, the trace is
1325 an empty sum for which we declare the result to be zero.
1329 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
1330 ....: RealCartesianProductEJA,
1336 sage: J = TrivialEJA()
1337 sage: J.zero().trace()
1341 sage: J = JordanSpinEJA(3)
1342 sage: x = sum(J.gens())
1348 sage: J = RealCartesianProductEJA(5)
1349 sage: J.one().trace()
1354 The trace of an element is a real number::
1356 sage: set_random_seed()
1357 sage: J = random_eja()
1358 sage: J.random_element().trace() in RLF
1366 # Special case for the trivial algebra where
1367 # the trace is an empty sum.
1368 return P
.base_ring().zero()
1370 p
= P
._charpoly
_coeff
(r
-1)
1371 # The _charpoly_coeff function already adds the factor of
1372 # -1 to ensure that _charpoly_coeff(r-1) is really what
1373 # appears in front of t^{r-1} in the charpoly. However,
1374 # we want the negative of THAT for the trace.
1375 return -p(*self
.to_vector())
1378 def trace_inner_product(self
, other
):
1380 Return the trace inner product of myself and ``other``.
1384 sage: from mjo.eja.eja_algebra import random_eja
1388 The trace inner product is commutative, bilinear, and associative::
1390 sage: set_random_seed()
1391 sage: J = random_eja()
1392 sage: x,y,z = J.random_elements(3)
1394 sage: x.trace_inner_product(y) == y.trace_inner_product(x)
1397 sage: a = J.base_ring().random_element();
1398 sage: actual = (a*(x+z)).trace_inner_product(y)
1399 sage: expected = ( a*x.trace_inner_product(y) +
1400 ....: a*z.trace_inner_product(y) )
1401 sage: actual == expected
1403 sage: actual = x.trace_inner_product(a*(y+z))
1404 sage: expected = ( a*x.trace_inner_product(y) +
1405 ....: a*x.trace_inner_product(z) )
1406 sage: actual == expected
1409 sage: (x*y).trace_inner_product(z) == y.trace_inner_product(x*z)
1413 if not other
in self
.parent():
1414 raise TypeError("'other' must live in the same algebra")
1416 return (self
*other
).trace()
1419 def trace_norm(self
):
1421 The norm of this element with respect to :meth:`trace_inner_product`.
1425 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
1426 ....: RealCartesianProductEJA)
1430 sage: J = RealCartesianProductEJA(2)
1431 sage: x = sum(J.gens())
1432 sage: x.trace_norm()
1437 sage: J = JordanSpinEJA(4)
1438 sage: x = sum(J.gens())
1439 sage: x.trace_norm()
1443 return self
.trace_inner_product(self
).sqrt()