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 from mjo
.eja
.eja_operator
import FiniteDimensionalEJAOperator
6 from mjo
.eja
.eja_utils
import _mat2vec
8 class FiniteDimensionalEJAElement(IndexedFreeModuleElement
):
10 An element of a Euclidean Jordan algebra.
15 Oh man, I should not be doing this. This hides the "disabled"
16 methods ``left_matrix`` and ``matrix`` from introspection;
17 in particular it removes them from tab-completion.
19 return filter(lambda s
: s
not in ['left_matrix', 'matrix'],
27 Return ``self`` raised to the power ``n``.
29 Jordan algebras are always power-associative; see for
30 example Faraut and Korányi, Proposition II.1.2 (ii).
32 We have to override this because our superclass uses row
33 vectors instead of column vectors! We, on the other hand,
34 assume column vectors everywhere.
38 sage: from mjo.eja.eja_algebra import random_eja
42 The definition of `x^2` is the unambiguous `x*x`::
44 sage: set_random_seed()
45 sage: x = random_eja().random_element()
49 A few examples of power-associativity::
51 sage: set_random_seed()
52 sage: x = random_eja().random_element()
53 sage: x*(x*x)*(x*x) == x^5
55 sage: (x*x)*(x*x*x) == x^5
58 We also know that powers operator-commute (Koecher, Chapter
61 sage: set_random_seed()
62 sage: x = random_eja().random_element()
63 sage: m = ZZ.random_element(0,10)
64 sage: n = ZZ.random_element(0,10)
65 sage: Lxm = (x^m).operator()
66 sage: Lxn = (x^n).operator()
67 sage: Lxm*Lxn == Lxn*Lxm
72 return self
.parent().one()
76 return (self
**(n
-1))*self
79 def apply_univariate_polynomial(self
, p
):
81 Apply the univariate polynomial ``p`` to this element.
83 A priori, SageMath won't allow us to apply a univariate
84 polynomial to an element of an EJA, because we don't know
85 that EJAs are rings (they are usually not associative). Of
86 course, we know that EJAs are power-associative, so the
87 operation is ultimately kosher. This function sidesteps
88 the CAS to get the answer we want and expect.
92 sage: from mjo.eja.eja_algebra import (HadamardEJA,
97 sage: R = PolynomialRing(QQ, 't')
99 sage: p = t^4 - t^3 + 5*t - 2
100 sage: J = HadamardEJA(5)
101 sage: J.one().apply_univariate_polynomial(p) == 3*J.one()
106 We should always get back an element of the algebra::
108 sage: set_random_seed()
109 sage: p = PolynomialRing(AA, 't').random_element()
110 sage: J = random_eja()
111 sage: x = J.random_element()
112 sage: x.apply_univariate_polynomial(p) in J
116 if len(p
.variables()) > 1:
117 raise ValueError("not a univariate polynomial")
120 # Convert the coeficcients to the parent's base ring,
121 # because a priori they might live in an (unnecessarily)
122 # larger ring for which P.sum() would fail below.
123 cs
= [ R(c
) for c
in p
.coefficients(sparse
=False) ]
124 return P
.sum( cs
[k
]*(self
**k
) for k
in range(len(cs
)) )
127 def characteristic_polynomial(self
):
129 Return the characteristic polynomial of this element.
133 sage: from mjo.eja.eja_algebra import HadamardEJA
137 The rank of `R^3` is three, and the minimal polynomial of
138 the identity element is `(t-1)` from which it follows that
139 the characteristic polynomial should be `(t-1)^3`::
141 sage: J = HadamardEJA(3)
142 sage: J.one().characteristic_polynomial()
143 t^3 - 3*t^2 + 3*t - 1
145 Likewise, the characteristic of the zero element in the
146 rank-three algebra `R^{n}` should be `t^{3}`::
148 sage: J = HadamardEJA(3)
149 sage: J.zero().characteristic_polynomial()
154 The characteristic polynomial of an element should evaluate
155 to zero on that element::
157 sage: set_random_seed()
158 sage: x = HadamardEJA(3).random_element()
159 sage: p = x.characteristic_polynomial()
160 sage: x.apply_univariate_polynomial(p)
163 The characteristic polynomials of the zero and unit elements
164 should be what we think they are in a subalgebra, too::
166 sage: J = HadamardEJA(3)
167 sage: p1 = J.one().characteristic_polynomial()
168 sage: q1 = J.zero().characteristic_polynomial()
169 sage: e0,e1,e2 = J.gens()
170 sage: A = (e0 + 2*e1 + 3*e2).subalgebra_generated_by() # dim 3
171 sage: p2 = A.one().characteristic_polynomial()
172 sage: q2 = A.zero().characteristic_polynomial()
179 p
= self
.parent().characteristic_polynomial_of()
180 return p(*self
.to_vector())
183 def inner_product(self
, other
):
185 Return the parent algebra's inner product of myself and ``other``.
189 sage: from mjo.eja.eja_algebra import (
190 ....: ComplexHermitianEJA,
192 ....: QuaternionHermitianEJA,
193 ....: RealSymmetricEJA,
198 The inner product in the Jordan spin algebra is the usual
199 inner product on `R^n` (this example only works because the
200 basis for the Jordan algebra is the standard basis in `R^n`)::
202 sage: J = JordanSpinEJA(3)
203 sage: x = vector(QQ,[1,2,3])
204 sage: y = vector(QQ,[4,5,6])
205 sage: x.inner_product(y)
207 sage: J.from_vector(x).inner_product(J.from_vector(y))
210 The inner product on `S^n` is `<X,Y> = trace(X*Y)`, where
211 multiplication is the usual matrix multiplication in `S^n`,
212 so the inner product of the identity matrix with itself
215 sage: J = RealSymmetricEJA(3)
216 sage: J.one().inner_product(J.one())
219 Likewise, the inner product on `C^n` is `<X,Y> =
220 Re(trace(X*Y))`, where we must necessarily take the real
221 part because the product of Hermitian matrices may not be
224 sage: J = ComplexHermitianEJA(3)
225 sage: J.one().inner_product(J.one())
228 Ditto for the quaternions::
230 sage: J = QuaternionHermitianEJA(2)
231 sage: J.one().inner_product(J.one())
236 Ensure that we can always compute an inner product, and that
237 it gives us back a real number::
239 sage: set_random_seed()
240 sage: J = random_eja()
241 sage: x,y = J.random_elements(2)
242 sage: x.inner_product(y) in RLF
248 raise TypeError("'other' must live in the same algebra")
250 return P
.inner_product(self
, other
)
253 def operator_commutes_with(self
, other
):
255 Return whether or not this element operator-commutes
260 sage: from mjo.eja.eja_algebra import random_eja
264 The definition of a Jordan algebra says that any element
265 operator-commutes with its square::
267 sage: set_random_seed()
268 sage: x = random_eja().random_element()
269 sage: x.operator_commutes_with(x^2)
274 Test Lemma 1 from Chapter III of Koecher::
276 sage: set_random_seed()
277 sage: u,v = random_eja().random_elements(2)
278 sage: lhs = u.operator_commutes_with(u*v)
279 sage: rhs = v.operator_commutes_with(u^2)
283 Test the first polarization identity from my notes, Koecher
284 Chapter III, or from Baes (2.3)::
286 sage: set_random_seed()
287 sage: x,y = random_eja().random_elements(2)
288 sage: Lx = x.operator()
289 sage: Ly = y.operator()
290 sage: Lxx = (x*x).operator()
291 sage: Lxy = (x*y).operator()
292 sage: bool(2*Lx*Lxy + Ly*Lxx == 2*Lxy*Lx + Lxx*Ly)
295 Test the second polarization identity from my notes or from
298 sage: set_random_seed()
299 sage: x,y,z = random_eja().random_elements(3)
300 sage: Lx = x.operator()
301 sage: Ly = y.operator()
302 sage: Lz = z.operator()
303 sage: Lzy = (z*y).operator()
304 sage: Lxy = (x*y).operator()
305 sage: Lxz = (x*z).operator()
306 sage: bool(Lx*Lzy + Lz*Lxy + Ly*Lxz == Lzy*Lx + Lxy*Lz + Lxz*Ly)
309 Test the third polarization identity from my notes or from
312 sage: set_random_seed()
313 sage: u,y,z = random_eja().random_elements(3)
314 sage: Lu = u.operator()
315 sage: Ly = y.operator()
316 sage: Lz = z.operator()
317 sage: Lzy = (z*y).operator()
318 sage: Luy = (u*y).operator()
319 sage: Luz = (u*z).operator()
320 sage: Luyz = (u*(y*z)).operator()
321 sage: lhs = Lu*Lzy + Lz*Luy + Ly*Luz
322 sage: rhs = Luyz + Ly*Lu*Lz + Lz*Lu*Ly
323 sage: bool(lhs == rhs)
327 if not other
in self
.parent():
328 raise TypeError("'other' must live in the same algebra")
337 Return my determinant, the product of my eigenvalues.
341 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
343 ....: RealSymmetricEJA,
344 ....: ComplexHermitianEJA,
349 sage: J = JordanSpinEJA(2)
350 sage: e0,e1 = J.gens()
351 sage: x = sum( J.gens() )
357 sage: J = JordanSpinEJA(3)
358 sage: e0,e1,e2 = J.gens()
359 sage: x = sum( J.gens() )
363 The determinant of the sole element in the rank-zero trivial
364 algebra is ``1``, by three paths of reasoning. First, its
365 characteristic polynomial is a constant ``1``, so the constant
366 term in that polynomial is ``1``. Second, the characteristic
367 polynomial evaluated at zero is again ``1``. And finally, the
368 (empty) product of its eigenvalues is likewise just unity::
370 sage: J = TrivialEJA()
376 An element is invertible if and only if its determinant is
379 sage: set_random_seed()
380 sage: x = random_eja().random_element()
381 sage: x.is_invertible() == (x.det() != 0)
384 Ensure that the determinant is multiplicative on an associative
385 subalgebra as in Faraut and Korányi's Proposition II.2.2::
387 sage: set_random_seed()
388 sage: J = random_eja().random_element().subalgebra_generated_by()
389 sage: x,y = J.random_elements(2)
390 sage: (x*y).det() == x.det()*y.det()
393 The determinant in matrix algebras is just the usual determinant::
395 sage: set_random_seed()
396 sage: X = matrix.random(QQ,3)
398 sage: J1 = RealSymmetricEJA(3)
399 sage: J2 = RealSymmetricEJA(3,field=QQ,orthonormalize=False)
400 sage: expected = X.det()
401 sage: actual1 = J1(X).det()
402 sage: actual2 = J2(X).det()
403 sage: actual1 == expected
405 sage: actual2 == expected
410 sage: set_random_seed()
411 sage: J1 = ComplexHermitianEJA(2)
412 sage: J2 = ComplexHermitianEJA(2,field=QQ,orthonormalize=False)
413 sage: X = matrix.random(GaussianIntegers(), 2)
415 sage: expected = AA(X.det())
416 sage: actual1 = J1(J1.real_embed(X)).det()
417 sage: actual2 = J2(J2.real_embed(X)).det()
418 sage: expected == actual1
420 sage: expected == actual2
428 # Special case, since we don't get the a0=1
429 # coefficient when the rank of the algebra
431 return P
.base_ring().one()
433 p
= P
._charpoly
_coefficients
()[0]
434 # The _charpoly_coeff function already adds the factor of -1
435 # to ensure that _charpoly_coefficients()[0] is really what
436 # appears in front of t^{0} in the charpoly. However, we want
437 # (-1)^r times THAT for the determinant.
438 return ((-1)**r
)*p(*self
.to_vector())
443 Return the Jordan-multiplicative inverse of this element.
447 In general we appeal to the quadratic representation as in
448 Koecher's Theorem 12 in Chapter III, Section 5. But if the
449 parent algebra's "characteristic polynomial of" coefficients
450 happen to be cached, then we use Proposition II.2.4 in Faraut
451 and Korányi which gives a formula for the inverse based on the
452 characteristic polynomial and the Cayley-Hamilton theorem for
453 Euclidean Jordan algebras::
457 sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
463 The inverse in the spin factor algebra is given in Alizadeh's
466 sage: set_random_seed()
467 sage: J = JordanSpinEJA.random_instance()
468 sage: x = J.random_element()
469 sage: while not x.is_invertible():
470 ....: x = J.random_element()
471 sage: x_vec = x.to_vector()
473 sage: x_bar = x_vec[1:]
474 sage: coeff = x0.inner_product(x0) - x_bar.inner_product(x_bar)
475 sage: x_inverse = x_vec.parent()(x0.list() + (-x_bar).list())
476 sage: if not coeff.is_zero(): x_inverse = x_inverse/coeff
477 sage: x.inverse() == J.from_vector(x_inverse)
480 Trying to invert a non-invertible element throws an error:
482 sage: JordanSpinEJA(3).zero().inverse()
483 Traceback (most recent call last):
485 ValueError: element is not invertible
489 The identity element is its own inverse::
491 sage: set_random_seed()
492 sage: J = random_eja()
493 sage: J.one().inverse() == J.one()
496 If an element has an inverse, it acts like one::
498 sage: set_random_seed()
499 sage: J = random_eja()
500 sage: x = J.random_element()
501 sage: (not x.is_invertible()) or (x.inverse()*x == J.one())
504 The inverse of the inverse is what we started with::
506 sage: set_random_seed()
507 sage: J = random_eja()
508 sage: x = J.random_element()
509 sage: (not x.is_invertible()) or (x.inverse().inverse() == x)
512 Proposition II.2.3 in Faraut and Korányi says that the inverse
513 of an element is the inverse of its left-multiplication operator
514 applied to the algebra's identity, when that inverse exists::
516 sage: set_random_seed()
517 sage: J = random_eja()
518 sage: x = J.random_element()
519 sage: (not x.operator().is_invertible()) or (
520 ....: x.operator().inverse()(J.one()) == x.inverse() )
523 Check that the fast (cached) and slow algorithms give the same
526 sage: set_random_seed() # long time
527 sage: J = random_eja(field=QQ, orthonormalize=False) # long time
528 sage: x = J.random_element() # long time
529 sage: while not x.is_invertible(): # long time
530 ....: x = J.random_element() # long time
531 sage: slow = x.inverse() # long time
532 sage: _ = J._charpoly_coefficients() # long time
533 sage: fast = x.inverse() # long time
534 sage: slow == fast # long time
537 if not self
.is_invertible():
538 raise ValueError("element is not invertible")
540 if self
.parent()._charpoly
_coefficients
.is_in_cache():
541 # We can invert using our charpoly if it will be fast to
542 # compute. If the coefficients are cached, our rank had
544 r
= self
.parent().rank()
545 a
= self
.characteristic_polynomial().coefficients(sparse
=False)
546 return (-1)**(r
+1)*sum(a
[i
+1]*self
**i
for i
in range(r
))/self
.det()
548 return (~self
.quadratic_representation())(self
)
551 def is_invertible(self
):
553 Return whether or not this element is invertible.
557 The usual way to do this is to check if the determinant is
558 zero, but we need the characteristic polynomial for the
559 determinant. The minimal polynomial is a lot easier to get,
560 so we use Corollary 2 in Chapter V of Koecher to check
561 whether or not the parent algebra's zero element is a root
562 of this element's minimal polynomial.
564 That is... unless the coefficients of our algebra's
565 "characteristic polynomial of" function are already cached!
566 In that case, we just use the determinant (which will be fast
569 Beware that we can't use the superclass method, because it
570 relies on the algebra being associative.
574 sage: from mjo.eja.eja_algebra import random_eja
578 The identity element is always invertible::
580 sage: set_random_seed()
581 sage: J = random_eja()
582 sage: J.one().is_invertible()
585 The zero element is never invertible in a non-trivial algebra::
587 sage: set_random_seed()
588 sage: J = random_eja()
589 sage: (not J.is_trivial()) and J.zero().is_invertible()
592 Test that the fast (cached) and slow algorithms give the same
595 sage: set_random_seed() # long time
596 sage: J = random_eja(field=QQ, orthonormalize=False) # long time
597 sage: x = J.random_element() # long time
598 sage: slow = x.is_invertible() # long time
599 sage: _ = J._charpoly_coefficients() # long time
600 sage: fast = x.is_invertible() # long time
601 sage: slow == fast # long time
606 if self
.parent().is_trivial():
611 if self
.parent()._charpoly
_coefficients
.is_in_cache():
612 # The determinant will be quicker than computing the minimal
613 # polynomial from scratch, most likely.
614 return (not self
.det().is_zero())
616 # In fact, we only need to know if the constant term is non-zero,
617 # so we can pass in the field's zero element instead.
618 zero
= self
.base_ring().zero()
619 p
= self
.minimal_polynomial()
620 return not (p(zero
) == zero
)
623 def is_primitive_idempotent(self
):
625 Return whether or not this element is a primitive (or minimal)
628 A primitive idempotent is a non-zero idempotent that is not
629 the sum of two other non-zero idempotents. Remark 2.7.15 in
630 Baes shows that this is what he refers to as a "minimal
633 An element of a Euclidean Jordan algebra is a minimal idempotent
634 if it :meth:`is_idempotent` and if its Peirce subalgebra
635 corresponding to the eigenvalue ``1`` has dimension ``1`` (Baes,
640 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
641 ....: RealSymmetricEJA,
647 This method is sloooooow.
651 The spectral decomposition of a non-regular element should always
652 contain at least one non-minimal idempotent::
654 sage: J = RealSymmetricEJA(3)
655 sage: x = sum(J.gens())
658 sage: [ c.is_primitive_idempotent()
659 ....: for (l,c) in x.spectral_decomposition() ]
662 On the other hand, the spectral decomposition of a regular
663 element should always be in terms of minimal idempotents::
665 sage: J = JordanSpinEJA(4)
666 sage: x = sum( i*J.gens()[i] for i in range(len(J.gens())) )
669 sage: [ c.is_primitive_idempotent()
670 ....: for (l,c) in x.spectral_decomposition() ]
675 The identity element is minimal only in an EJA of rank one::
677 sage: set_random_seed()
678 sage: J = random_eja()
679 sage: J.rank() == 1 or not J.one().is_primitive_idempotent()
682 A non-idempotent cannot be a minimal idempotent::
684 sage: set_random_seed()
685 sage: J = JordanSpinEJA(4)
686 sage: x = J.random_element()
687 sage: (not x.is_idempotent()) and x.is_primitive_idempotent()
690 Proposition 2.7.19 in Baes says that an element is a minimal
691 idempotent if and only if it's idempotent with trace equal to
694 sage: set_random_seed()
695 sage: J = JordanSpinEJA(4)
696 sage: x = J.random_element()
697 sage: expected = (x.is_idempotent() and x.trace() == 1)
698 sage: actual = x.is_primitive_idempotent()
699 sage: actual == expected
702 Primitive idempotents must be non-zero::
704 sage: set_random_seed()
705 sage: J = random_eja()
706 sage: J.zero().is_idempotent()
708 sage: J.zero().is_primitive_idempotent()
711 As a consequence of the fact that primitive idempotents must
712 be non-zero, there are no primitive idempotents in a trivial
713 Euclidean Jordan algebra::
715 sage: J = TrivialEJA()
716 sage: J.one().is_idempotent()
718 sage: J.one().is_primitive_idempotent()
722 if not self
.is_idempotent():
728 (_
,_
,J1
) = self
.parent().peirce_decomposition(self
)
729 return (J1
.dimension() == 1)
732 def is_nilpotent(self
):
734 Return whether or not some power of this element is zero.
738 We use Theorem 5 in Chapter III of Koecher, which says that
739 an element ``x`` is nilpotent if and only if ``x.operator()``
740 is nilpotent. And it is a basic fact of linear algebra that
741 an operator on an `n`-dimensional space is nilpotent if and
742 only if, when raised to the `n`th power, it equals the zero
743 operator (for example, see Axler Corollary 8.8).
747 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
752 sage: J = JordanSpinEJA(3)
753 sage: x = sum(J.gens())
754 sage: x.is_nilpotent()
759 The identity element is never nilpotent, except in a trivial EJA::
761 sage: set_random_seed()
762 sage: J = random_eja()
763 sage: J.one().is_nilpotent() and not J.is_trivial()
766 The additive identity is always nilpotent::
768 sage: set_random_seed()
769 sage: random_eja().zero().is_nilpotent()
774 zero_operator
= P
.zero().operator()
775 return self
.operator()**P
.dimension() == zero_operator
778 def is_regular(self
):
780 Return whether or not this is a regular element.
784 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
789 The identity element always has degree one, but any element
790 linearly-independent from it is regular::
792 sage: J = JordanSpinEJA(5)
793 sage: J.one().is_regular()
795 sage: e0, e1, e2, e3, e4 = J.gens() # e0 is the identity
796 sage: for x in J.gens():
797 ....: (J.one() + x).is_regular()
806 The zero element should never be regular, unless the parent
807 algebra has dimension less than or equal to one::
809 sage: set_random_seed()
810 sage: J = random_eja()
811 sage: J.dimension() <= 1 or not J.zero().is_regular()
814 The unit element isn't regular unless the algebra happens to
815 consist of only its scalar multiples::
817 sage: set_random_seed()
818 sage: J = random_eja()
819 sage: J.dimension() <= 1 or not J.one().is_regular()
823 return self
.degree() == self
.parent().rank()
828 Return the degree of this element, which is defined to be
829 the degree of its minimal polynomial.
833 For now, we skip the messy minimal polynomial computation
834 and instead return the dimension of the vector space spanned
835 by the powers of this element. The latter is a bit more
836 straightforward to compute.
840 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
845 sage: J = JordanSpinEJA(4)
846 sage: J.one().degree()
848 sage: e0,e1,e2,e3 = J.gens()
849 sage: (e0 - e1).degree()
852 In the spin factor algebra (of rank two), all elements that
853 aren't multiples of the identity are regular::
855 sage: set_random_seed()
856 sage: J = JordanSpinEJA.random_instance()
857 sage: n = J.dimension()
858 sage: x = J.random_element()
859 sage: x.degree() == min(n,2) or (x == x.coefficient(0)*J.one())
864 The zero and unit elements are both of degree one in nontrivial
867 sage: set_random_seed()
868 sage: J = random_eja()
869 sage: d = J.zero().degree()
870 sage: (J.is_trivial() and d == 0) or d == 1
872 sage: d = J.one().degree()
873 sage: (J.is_trivial() and d == 0) or d == 1
876 Our implementation agrees with the definition::
878 sage: set_random_seed()
879 sage: x = random_eja().random_element()
880 sage: x.degree() == x.minimal_polynomial().degree()
884 if self
.is_zero() and not self
.parent().is_trivial():
885 # The minimal polynomial of zero in a nontrivial algebra
886 # is "t"; in a trivial algebra it's "1" by convention
887 # (it's an empty product).
889 return self
.subalgebra_generated_by().dimension()
892 def left_matrix(self
):
894 Our parent class defines ``left_matrix`` and ``matrix``
895 methods whose names are misleading. We don't want them.
897 raise NotImplementedError("use operator().matrix() instead")
902 def minimal_polynomial(self
):
904 Return the minimal polynomial of this element,
905 as a function of the variable `t`.
909 We restrict ourselves to the associative subalgebra
910 generated by this element, and then return the minimal
911 polynomial of this element's operator matrix (in that
912 subalgebra). This works by Baes Proposition 2.3.16.
916 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
917 ....: RealSymmetricEJA,
923 Keeping in mind that the polynomial ``1`` evaluates the identity
924 element (also the zero element) of the trivial algebra, it is clear
925 that the polynomial ``1`` is the minimal polynomial of the only
926 element in a trivial algebra::
928 sage: J = TrivialEJA()
929 sage: J.one().minimal_polynomial()
931 sage: J.zero().minimal_polynomial()
936 The minimal polynomial of the identity and zero elements are
937 always the same, except in trivial algebras where the minimal
938 polynomial of the unit/zero element is ``1``::
940 sage: set_random_seed()
941 sage: J = random_eja()
942 sage: mu = J.one().minimal_polynomial()
943 sage: t = mu.parent().gen()
944 sage: mu + int(J.is_trivial())*(t-2)
946 sage: mu = J.zero().minimal_polynomial()
947 sage: t = mu.parent().gen()
948 sage: mu + int(J.is_trivial())*(t-1)
951 The degree of an element is (by one definition) the degree
952 of its minimal polynomial::
954 sage: set_random_seed()
955 sage: x = random_eja().random_element()
956 sage: x.degree() == x.minimal_polynomial().degree()
959 The minimal polynomial and the characteristic polynomial coincide
960 and are known (see Alizadeh, Example 11.11) for all elements of
961 the spin factor algebra that aren't scalar multiples of the
962 identity. We require the dimension of the algebra to be at least
963 two here so that said elements actually exist::
965 sage: set_random_seed()
966 sage: n_max = max(2, JordanSpinEJA._max_random_instance_size())
967 sage: n = ZZ.random_element(2, n_max)
968 sage: J = JordanSpinEJA(n)
969 sage: y = J.random_element()
970 sage: while y == y.coefficient(0)*J.one():
971 ....: y = J.random_element()
972 sage: y0 = y.to_vector()[0]
973 sage: y_bar = y.to_vector()[1:]
974 sage: actual = y.minimal_polynomial()
975 sage: t = PolynomialRing(J.base_ring(),'t').gen(0)
976 sage: expected = t^2 - 2*y0*t + (y0^2 - norm(y_bar)^2)
977 sage: bool(actual == expected)
980 The minimal polynomial should always kill its element::
982 sage: set_random_seed()
983 sage: x = random_eja().random_element()
984 sage: p = x.minimal_polynomial()
985 sage: x.apply_univariate_polynomial(p)
988 The minimal polynomial is invariant under a change of basis,
989 and in particular, a re-scaling of the basis::
991 sage: set_random_seed()
992 sage: n_max = RealSymmetricEJA._max_random_instance_size()
993 sage: n = ZZ.random_element(1, n_max)
994 sage: J1 = RealSymmetricEJA(n)
995 sage: J2 = RealSymmetricEJA(n,orthonormalize=False)
996 sage: X = random_matrix(AA,n)
997 sage: X = X*X.transpose()
1000 sage: x1.minimal_polynomial() == x2.minimal_polynomial()
1005 # We would generate a zero-dimensional subalgebra
1006 # where the minimal polynomial would be constant.
1007 # That might be correct, but only if *this* algebra
1009 if not self
.parent().is_trivial():
1010 # Pretty sure we know what the minimal polynomial of
1011 # the zero operator is going to be. This ensures
1012 # consistency of e.g. the polynomial variable returned
1013 # in the "normal" case without us having to think about it.
1014 return self
.operator().minimal_polynomial()
1016 A
= self
.subalgebra_generated_by(orthonormalize_basis
=False)
1017 return A(self
).operator().minimal_polynomial()
1021 def to_matrix(self
):
1023 Return an (often more natural) representation of this element as a
1026 Every finite-dimensional Euclidean Jordan Algebra is a direct
1027 sum of five simple algebras, four of which comprise Hermitian
1028 matrices. This method returns a "natural" matrix
1029 representation of this element as either a Hermitian matrix or
1034 sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
1035 ....: QuaternionHermitianEJA)
1039 sage: J = ComplexHermitianEJA(3)
1042 sage: J.one().to_matrix()
1052 sage: J = QuaternionHermitianEJA(2)
1055 sage: J.one().to_matrix()
1066 B
= self
.parent().matrix_basis()
1067 W
= self
.parent().matrix_space()
1069 # This is just a manual "from_vector()", but of course
1070 # matrix spaces aren't vector spaces in sage, so they
1071 # don't have a from_vector() method.
1072 return W
.linear_combination( zip(B
, self
.to_vector()) )
1077 The norm of this element with respect to :meth:`inner_product`.
1081 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
1086 sage: J = HadamardEJA(2)
1087 sage: x = sum(J.gens())
1093 sage: J = JordanSpinEJA(4)
1094 sage: x = sum(J.gens())
1099 return self
.inner_product(self
).sqrt()
1104 Return the left-multiplication-by-this-element
1105 operator on the ambient algebra.
1109 sage: from mjo.eja.eja_algebra import random_eja
1113 sage: set_random_seed()
1114 sage: J = random_eja()
1115 sage: x,y = J.random_elements(2)
1116 sage: x.operator()(y) == x*y
1118 sage: y.operator()(x) == x*y
1123 left_mult_by_self
= lambda y
: self
*y
1124 L
= P
.module_morphism(function
=left_mult_by_self
, codomain
=P
)
1125 return FiniteDimensionalEJAOperator(P
, P
, L
.matrix() )
1128 def quadratic_representation(self
, other
=None):
1130 Return the quadratic representation of this element.
1134 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
1139 The explicit form in the spin factor algebra is given by
1140 Alizadeh's Example 11.12::
1142 sage: set_random_seed()
1143 sage: x = JordanSpinEJA.random_instance().random_element()
1144 sage: x_vec = x.to_vector()
1145 sage: Q = matrix.identity(x.base_ring(), 0)
1146 sage: n = x_vec.degree()
1149 ....: x_bar = x_vec[1:]
1150 ....: A = matrix(x.base_ring(), 1, [x_vec.inner_product(x_vec)])
1151 ....: B = 2*x0*x_bar.row()
1152 ....: C = 2*x0*x_bar.column()
1153 ....: D = matrix.identity(x.base_ring(), n-1)
1154 ....: D = (x0^2 - x_bar.inner_product(x_bar))*D
1155 ....: D = D + 2*x_bar.tensor_product(x_bar)
1156 ....: Q = matrix.block(2,2,[A,B,C,D])
1157 sage: Q == x.quadratic_representation().matrix()
1160 Test all of the properties from Theorem 11.2 in Alizadeh::
1162 sage: set_random_seed()
1163 sage: J = random_eja()
1164 sage: x,y = J.random_elements(2)
1165 sage: Lx = x.operator()
1166 sage: Lxx = (x*x).operator()
1167 sage: Qx = x.quadratic_representation()
1168 sage: Qy = y.quadratic_representation()
1169 sage: Qxy = x.quadratic_representation(y)
1170 sage: Qex = J.one().quadratic_representation(x)
1171 sage: n = ZZ.random_element(10)
1172 sage: Qxn = (x^n).quadratic_representation()
1176 sage: 2*Qxy == (x+y).quadratic_representation() - Qx - Qy
1179 Property 2 (multiply on the right for :trac:`28272`):
1181 sage: alpha = J.base_ring().random_element()
1182 sage: (alpha*x).quadratic_representation() == Qx*(alpha^2)
1187 sage: not x.is_invertible() or ( Qx(x.inverse()) == x )
1190 sage: not x.is_invertible() or (
1193 ....: x.inverse().quadratic_representation() )
1196 sage: Qxy(J.one()) == x*y
1201 sage: not x.is_invertible() or (
1202 ....: x.quadratic_representation(x.inverse())*Qx
1203 ....: == Qx*x.quadratic_representation(x.inverse()) )
1206 sage: not x.is_invertible() or (
1207 ....: x.quadratic_representation(x.inverse())*Qx
1209 ....: 2*Lx*Qex - Qx )
1212 sage: 2*Lx*Qex - Qx == Lxx
1217 sage: Qy(x).quadratic_representation() == Qy*Qx*Qy
1227 sage: not x.is_invertible() or (
1228 ....: Qx*x.inverse().operator() == Lx )
1233 sage: not x.operator_commutes_with(y) or (
1234 ....: Qx(y)^n == Qxn(y^n) )
1240 elif not other
in self
.parent():
1241 raise TypeError("'other' must live in the same algebra")
1244 M
= other
.operator()
1245 return ( L
*M
+ M
*L
- (self
*other
).operator() )
1249 def spectral_decomposition(self
):
1251 Return the unique spectral decomposition of this element.
1255 Following Faraut and Korányi's Theorem III.1.1, we restrict this
1256 element's left-multiplication-by operator to the subalgebra it
1257 generates. We then compute the spectral decomposition of that
1258 operator, and the spectral projectors we get back must be the
1259 left-multiplication-by operators for the idempotents we
1260 seek. Thus applying them to the identity element gives us those
1263 Since the eigenvalues are required to be distinct, we take
1264 the spectral decomposition of the zero element to be zero
1265 times the identity element of the algebra (which is idempotent,
1270 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
1274 The spectral decomposition of the identity is ``1`` times itself,
1275 and the spectral decomposition of zero is ``0`` times the identity::
1277 sage: J = RealSymmetricEJA(3)
1280 sage: J.one().spectral_decomposition()
1282 sage: J.zero().spectral_decomposition()
1287 sage: J = RealSymmetricEJA(4)
1288 sage: x = sum(J.gens())
1289 sage: sd = x.spectral_decomposition()
1294 sage: c0.inner_product(c1) == 0
1296 sage: c0.is_idempotent()
1298 sage: c1.is_idempotent()
1300 sage: c0 + c1 == J.one()
1302 sage: l0*c0 + l1*c1 == x
1305 The spectral decomposition should work in subalgebras, too::
1307 sage: J = RealSymmetricEJA(4)
1308 sage: (e0, e1, e2, e3, e4, e5, e6, e7, e8, e9) = J.gens()
1309 sage: A = 2*e5 - 2*e8
1310 sage: (lambda1, c1) = A.spectral_decomposition()[1]
1311 sage: (J0, J5, J1) = J.peirce_decomposition(c1)
1312 sage: (f0, f1, f2) = J1.gens()
1313 sage: f0.spectral_decomposition()
1317 A
= self
.subalgebra_generated_by(orthonormalize_basis
=True)
1319 for (evalue
, proj
) in A(self
).operator().spectral_decomposition():
1320 result
.append( (evalue
, proj(A
.one()).superalgebra_element()) )
1323 def subalgebra_generated_by(self
, orthonormalize_basis
=False):
1325 Return the associative subalgebra of the parent EJA generated
1328 Since our parent algebra is unital, we want "subalgebra" to mean
1329 "unital subalgebra" as well; thus the subalgebra that an element
1330 generates will itself be a Euclidean Jordan algebra after
1331 restricting the algebra operations appropriately. This is the
1332 subalgebra that Faraut and Korányi work with in section II.2, for
1337 sage: from mjo.eja.eja_algebra import random_eja
1341 This subalgebra, being composed of only powers, is associative::
1343 sage: set_random_seed()
1344 sage: x0 = random_eja().random_element()
1345 sage: A = x0.subalgebra_generated_by()
1346 sage: x,y,z = A.random_elements(3)
1347 sage: (x*y)*z == x*(y*z)
1350 Squaring in the subalgebra should work the same as in
1353 sage: set_random_seed()
1354 sage: x = random_eja().random_element()
1355 sage: A = x.subalgebra_generated_by()
1356 sage: A(x^2) == A(x)*A(x)
1359 By definition, the subalgebra generated by the zero element is
1360 the one-dimensional algebra generated by the identity
1361 element... unless the original algebra was trivial, in which
1362 case the subalgebra is trivial too::
1364 sage: set_random_seed()
1365 sage: A = random_eja().zero().subalgebra_generated_by()
1366 sage: (A.is_trivial() and A.dimension() == 0) or A.dimension() == 1
1370 from mjo
.eja
.eja_element_subalgebra
import FiniteDimensionalEJAElementSubalgebra
1371 return FiniteDimensionalEJAElementSubalgebra(self
, orthonormalize_basis
)
1374 def subalgebra_idempotent(self
):
1376 Find an idempotent in the associative subalgebra I generate
1377 using Proposition 2.3.5 in Baes.
1381 sage: from mjo.eja.eja_algebra import random_eja
1385 Ensure that we can find an idempotent in a non-trivial algebra
1386 where there are non-nilpotent elements, or that we get the dumb
1387 solution in the trivial algebra::
1389 sage: set_random_seed()
1390 sage: J = random_eja()
1391 sage: x = J.random_element()
1392 sage: while x.is_nilpotent() and not J.is_trivial():
1393 ....: x = J.random_element()
1394 sage: c = x.subalgebra_idempotent()
1399 if self
.parent().is_trivial():
1402 if self
.is_nilpotent():
1403 raise ValueError("this only works with non-nilpotent elements!")
1405 J
= self
.subalgebra_generated_by()
1408 # The image of the matrix of left-u^m-multiplication
1409 # will be minimal for some natural number s...
1411 minimal_dim
= J
.dimension()
1412 for i
in range(1, minimal_dim
):
1413 this_dim
= (u
**i
).operator().matrix().image().dimension()
1414 if this_dim
< minimal_dim
:
1415 minimal_dim
= this_dim
1418 # Now minimal_matrix should correspond to the smallest
1419 # non-zero subspace in Baes's (or really, Koecher's)
1422 # However, we need to restrict the matrix to work on the
1423 # subspace... or do we? Can't we just solve, knowing that
1424 # A(c) = u^(s+1) should have a solution in the big space,
1427 # Beware, solve_right() means that we're using COLUMN vectors.
1428 # Our FiniteDimensionalAlgebraElement superclass uses rows.
1430 A
= u_next
.operator().matrix()
1431 c
= J
.from_vector(A
.solve_right(u_next
.to_vector()))
1433 # Now c is the idempotent we want, but it still lives in the subalgebra.
1434 return c
.superalgebra_element()
1439 Return my trace, the sum of my eigenvalues.
1441 In a trivial algebra, however you want to look at it, the trace is
1442 an empty sum for which we declare the result to be zero.
1446 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
1453 sage: J = TrivialEJA()
1454 sage: J.zero().trace()
1458 sage: J = JordanSpinEJA(3)
1459 sage: x = sum(J.gens())
1465 sage: J = HadamardEJA(5)
1466 sage: J.one().trace()
1471 The trace of an element is a real number::
1473 sage: set_random_seed()
1474 sage: J = random_eja()
1475 sage: J.random_element().trace() in RLF
1483 # Special case for the trivial algebra where
1484 # the trace is an empty sum.
1485 return P
.base_ring().zero()
1487 p
= P
._charpoly
_coefficients
()[r
-1]
1488 # The _charpoly_coeff function already adds the factor of
1489 # -1 to ensure that _charpoly_coeff(r-1) is really what
1490 # appears in front of t^{r-1} in the charpoly. However,
1491 # we want the negative of THAT for the trace.
1492 return -p(*self
.to_vector())
1495 def trace_inner_product(self
, other
):
1497 Return the trace inner product of myself and ``other``.
1501 sage: from mjo.eja.eja_algebra import random_eja
1505 The trace inner product is commutative, bilinear, and associative::
1507 sage: set_random_seed()
1508 sage: J = random_eja()
1509 sage: x,y,z = J.random_elements(3)
1511 sage: x.trace_inner_product(y) == y.trace_inner_product(x)
1514 sage: a = J.base_ring().random_element();
1515 sage: actual = (a*(x+z)).trace_inner_product(y)
1516 sage: expected = ( a*x.trace_inner_product(y) +
1517 ....: a*z.trace_inner_product(y) )
1518 sage: actual == expected
1520 sage: actual = x.trace_inner_product(a*(y+z))
1521 sage: expected = ( a*x.trace_inner_product(y) +
1522 ....: a*x.trace_inner_product(z) )
1523 sage: actual == expected
1526 sage: (x*y).trace_inner_product(z) == y.trace_inner_product(x*z)
1530 if not other
in self
.parent():
1531 raise TypeError("'other' must live in the same algebra")
1533 return (self
*other
).trace()
1536 def trace_norm(self
):
1538 The norm of this element with respect to :meth:`trace_inner_product`.
1542 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
1547 sage: J = HadamardEJA(2)
1548 sage: x = sum(J.gens())
1549 sage: x.trace_norm()
1554 sage: J = JordanSpinEJA(4)
1555 sage: x = sum(J.gens())
1556 sage: x.trace_norm()
1560 return self
.trace_inner_product(self
).sqrt()