1 from sage
.matrix
.constructor
import matrix
2 from sage
.misc
.cachefunc
import cached_method
3 from sage
.modules
.free_module
import VectorSpace
4 from sage
.modules
.with_basis
.indexed_element
import IndexedFreeModuleElement
6 from mjo
.eja
.eja_operator
import FiniteDimensionalEJAOperator
7 from mjo
.eja
.eja_utils
import _scale
10 class FiniteDimensionalEJAElement(IndexedFreeModuleElement
):
12 An element of a Euclidean Jordan algebra.
17 Oh man, I should not be doing this. This hides the "disabled"
18 methods ``left_matrix`` and ``matrix`` from introspection;
19 in particular it removes them from tab-completion.
21 return filter(lambda s
: s
not in ['left_matrix', 'matrix'],
29 Return ``self`` raised to the power ``n``.
31 Jordan algebras are always power-associative; see for
32 example Faraut and Korányi, Proposition II.1.2 (ii).
34 We have to override this because our superclass uses row
35 vectors instead of column vectors! We, on the other hand,
36 assume column vectors everywhere.
40 sage: from mjo.eja.eja_algebra import random_eja
44 The definition of `x^2` is the unambiguous `x*x`::
46 sage: x = random_eja().random_element()
50 A few examples of power-associativity::
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: x = random_eja().random_element()
62 sage: m = ZZ.random_element(0,10)
63 sage: n = ZZ.random_element(0,10)
64 sage: Lxm = (x^m).operator()
65 sage: Lxn = (x^n).operator()
66 sage: Lxm*Lxn == Lxn*Lxm
71 return self
.parent().one()
75 return (self
**(n
-1))*self
78 def apply_univariate_polynomial(self
, p
):
80 Apply the univariate polynomial ``p`` to this element.
82 A priori, SageMath won't allow us to apply a univariate
83 polynomial to an element of an EJA, because we don't know
84 that EJAs are rings (they are usually not associative). Of
85 course, we know that EJAs are power-associative, so the
86 operation is ultimately kosher. This function sidesteps
87 the CAS to get the answer we want and expect.
91 sage: from mjo.eja.eja_algebra import (HadamardEJA,
96 sage: R = PolynomialRing(QQ, 't')
98 sage: p = t^4 - t^3 + 5*t - 2
99 sage: J = HadamardEJA(5)
100 sage: J.one().apply_univariate_polynomial(p) == 3*J.one()
105 We should always get back an element of the algebra::
107 sage: p = PolynomialRing(AA, 't').random_element()
108 sage: J = random_eja()
109 sage: x = J.random_element()
110 sage: x.apply_univariate_polynomial(p) in J
114 if len(p
.variables()) > 1:
115 raise ValueError("not a univariate polynomial")
118 # Convert the coeficcients to the parent's base ring,
119 # because a priori they might live in an (unnecessarily)
120 # larger ring for which P.sum() would fail below.
121 cs
= [ R(c
) for c
in p
.coefficients(sparse
=False) ]
122 return P
.sum( cs
[k
]*(self
**k
) for k
in range(len(cs
)) )
125 def characteristic_polynomial(self
):
127 Return the characteristic polynomial of this element.
131 sage: from mjo.eja.eja_algebra import (random_eja,
136 The rank of `R^3` is three, and the minimal polynomial of
137 the identity element is `(t-1)` from which it follows that
138 the characteristic polynomial should be `(t-1)^3`::
140 sage: J = HadamardEJA(3)
141 sage: J.one().characteristic_polynomial()
142 t^3 - 3*t^2 + 3*t - 1
144 Likewise, the characteristic of the zero element in the
145 rank-three algebra `R^{n}` should be `t^{3}`::
147 sage: J = HadamardEJA(3)
148 sage: J.zero().characteristic_polynomial()
153 The characteristic polynomial of an element should evaluate
154 to zero on that element::
156 sage: x = random_eja().random_element()
157 sage: p = x.characteristic_polynomial()
158 sage: x.apply_univariate_polynomial(p).is_zero()
161 The characteristic polynomials of the zero and unit elements
162 should be what we think they are in a subalgebra, too::
164 sage: J = HadamardEJA(3)
165 sage: p1 = J.one().characteristic_polynomial()
166 sage: q1 = J.zero().characteristic_polynomial()
167 sage: b0,b1,b2 = J.gens()
168 sage: A = (b0 + 2*b1 + 3*b2).subalgebra_generated_by() # dim 3
169 sage: p2 = A.one().characteristic_polynomial()
170 sage: q2 = A.zero().characteristic_polynomial()
177 p
= self
.parent().characteristic_polynomial_of()
178 return p(*self
.to_vector())
181 def inner_product(self
, other
):
183 Return the parent algebra's inner product of myself and ``other``.
187 sage: from mjo.eja.eja_algebra import (
188 ....: ComplexHermitianEJA,
190 ....: QuaternionHermitianEJA,
191 ....: RealSymmetricEJA,
196 The inner product in the Jordan spin algebra is the usual
197 inner product on `R^n` (this example only works because the
198 basis for the Jordan algebra is the standard basis in `R^n`)::
200 sage: J = JordanSpinEJA(3)
201 sage: x = vector(QQ,[1,2,3])
202 sage: y = vector(QQ,[4,5,6])
203 sage: x.inner_product(y)
205 sage: J.from_vector(x).inner_product(J.from_vector(y))
208 The inner product on `S^n` is `<X,Y> = trace(X*Y)`, where
209 multiplication is the usual matrix multiplication in `S^n`,
210 so the inner product of the identity matrix with itself
213 sage: J = RealSymmetricEJA(3)
214 sage: J.one().inner_product(J.one())
217 Likewise, the inner product on `C^n` is `<X,Y> =
218 Re(trace(X*Y))`, where we must necessarily take the real
219 part because the product of Hermitian matrices may not be
222 sage: J = ComplexHermitianEJA(3)
223 sage: J.one().inner_product(J.one())
226 Ditto for the quaternions::
228 sage: J = QuaternionHermitianEJA(2)
229 sage: J.one().inner_product(J.one())
234 Ensure that we can always compute an inner product, and that
235 it gives us back a real number::
237 sage: J = random_eja()
238 sage: x,y = J.random_elements(2)
239 sage: x.inner_product(y) in RLF
245 raise TypeError("'other' must live in the same algebra")
247 return P
.inner_product(self
, other
)
250 def operator_commutes_with(self
, other
):
252 Return whether or not this element operator-commutes
257 sage: from mjo.eja.eja_algebra import random_eja
261 The definition of a Jordan algebra says that any element
262 operator-commutes with its square::
264 sage: x = random_eja().random_element()
265 sage: x.operator_commutes_with(x^2)
270 Test Lemma 1 from Chapter III of Koecher::
272 sage: u,v = random_eja().random_elements(2)
273 sage: lhs = u.operator_commutes_with(u*v)
274 sage: rhs = v.operator_commutes_with(u^2)
278 Test the first polarization identity from my notes, Koecher
279 Chapter III, or from Baes (2.3)::
281 sage: x,y = random_eja().random_elements(2)
282 sage: Lx = x.operator()
283 sage: Ly = y.operator()
284 sage: Lxx = (x*x).operator()
285 sage: Lxy = (x*y).operator()
286 sage: bool(2*Lx*Lxy + Ly*Lxx == 2*Lxy*Lx + Lxx*Ly)
289 Test the second polarization identity from my notes or from
292 sage: x,y,z = random_eja().random_elements(3) # long time
293 sage: Lx = x.operator() # long time
294 sage: Ly = y.operator() # long time
295 sage: Lz = z.operator() # long time
296 sage: Lzy = (z*y).operator() # long time
297 sage: Lxy = (x*y).operator() # long time
298 sage: Lxz = (x*z).operator() # long time
299 sage: lhs = Lx*Lzy + Lz*Lxy + Ly*Lxz # long time
300 sage: rhs = Lzy*Lx + Lxy*Lz + Lxz*Ly # long time
301 sage: bool(lhs == rhs) # long time
304 Test the third polarization identity from my notes or from
307 sage: u,y,z = random_eja().random_elements(3) # long time
308 sage: Lu = u.operator() # long time
309 sage: Ly = y.operator() # long time
310 sage: Lz = z.operator() # long time
311 sage: Lzy = (z*y).operator() # long time
312 sage: Luy = (u*y).operator() # long time
313 sage: Luz = (u*z).operator() # long time
314 sage: Luyz = (u*(y*z)).operator() # long time
315 sage: lhs = Lu*Lzy + Lz*Luy + Ly*Luz # long time
316 sage: rhs = Luyz + Ly*Lu*Lz + Lz*Lu*Ly # long time
317 sage: bool(lhs == rhs) # long time
321 if not other
in self
.parent():
322 raise TypeError("'other' must live in the same algebra")
331 Return my determinant, the product of my eigenvalues.
335 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
337 ....: RealSymmetricEJA,
338 ....: ComplexHermitianEJA,
343 sage: J = JordanSpinEJA(2)
344 sage: x = sum( J.gens() )
350 sage: J = JordanSpinEJA(3)
351 sage: x = sum( J.gens() )
355 The determinant of the sole element in the rank-zero trivial
356 algebra is ``1``, by three paths of reasoning. First, its
357 characteristic polynomial is a constant ``1``, so the constant
358 term in that polynomial is ``1``. Second, the characteristic
359 polynomial evaluated at zero is again ``1``. And finally, the
360 (empty) product of its eigenvalues is likewise just unity::
362 sage: J = TrivialEJA()
368 An element is invertible if and only if its determinant is
371 sage: x = random_eja().random_element()
372 sage: x.is_invertible() == (x.det() != 0)
375 Ensure that the determinant is multiplicative on an associative
376 subalgebra as in Faraut and Korányi's Proposition II.2.2::
378 sage: x0 = random_eja().random_element()
379 sage: J = x0.subalgebra_generated_by(orthonormalize=False)
380 sage: x,y = J.random_elements(2)
381 sage: (x*y).det() == x.det()*y.det()
384 The determinant in real matrix algebras is the usual determinant::
386 sage: X = matrix.random(QQ,3)
388 sage: J1 = RealSymmetricEJA(3)
389 sage: J2 = RealSymmetricEJA(3,field=QQ,orthonormalize=False)
390 sage: expected = X.det()
391 sage: actual1 = J1(X).det()
392 sage: actual2 = J2(X).det()
393 sage: actual1 == expected
395 sage: actual2 == expected
403 # Special case, since we don't get the a0=1
404 # coefficient when the rank of the algebra
406 return P
.base_ring().one()
408 p
= P
._charpoly
_coefficients
()[0]
409 # The _charpoly_coeff function already adds the factor of -1
410 # to ensure that _charpoly_coefficients()[0] is really what
411 # appears in front of t^{0} in the charpoly. However, we want
412 # (-1)^r times THAT for the determinant.
413 return ((-1)**r
)*p(*self
.to_vector())
419 Return the Jordan-multiplicative inverse of this element.
423 In general we appeal to the quadratic representation as in
424 Koecher's Theorem 12 in Chapter III, Section 5. But if the
425 parent algebra's "characteristic polynomial of" coefficients
426 happen to be cached, then we use Proposition II.2.4 in Faraut
427 and Korányi which gives a formula for the inverse based on the
428 characteristic polynomial and the Cayley-Hamilton theorem for
429 Euclidean Jordan algebras::
433 sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
439 The inverse in the spin factor algebra is given in Alizadeh's
442 sage: J = JordanSpinEJA.random_instance()
443 sage: x = J.random_element()
444 sage: while not x.is_invertible():
445 ....: x = J.random_element()
446 sage: x_vec = x.to_vector()
448 sage: x_bar = x_vec[1:]
449 sage: coeff = x0.inner_product(x0) - x_bar.inner_product(x_bar)
450 sage: x_inverse = x_vec.parent()(x0.list() + (-x_bar).list())
451 sage: if not coeff.is_zero(): x_inverse = x_inverse/coeff
452 sage: x.inverse() == J.from_vector(x_inverse)
455 Trying to invert a non-invertible element throws an error:
457 sage: JordanSpinEJA(3).zero().inverse()
458 Traceback (most recent call last):
460 ZeroDivisionError: element is not invertible
464 The identity element is its own inverse::
466 sage: J = random_eja()
467 sage: J.one().inverse() == J.one()
470 If an element has an inverse, it acts like one::
472 sage: J = random_eja()
473 sage: x = J.random_element()
474 sage: (not x.is_invertible()) or (x.inverse()*x == J.one())
477 The inverse of the inverse is what we started with::
479 sage: J = random_eja()
480 sage: x = J.random_element()
481 sage: (not x.is_invertible()) or (x.inverse().inverse() == x)
484 Proposition II.2.3 in Faraut and Korányi says that the inverse
485 of an element is the inverse of its left-multiplication operator
486 applied to the algebra's identity, when that inverse exists::
488 sage: J = random_eja() # long time
489 sage: x = J.random_element() # long time
490 sage: (not x.operator().is_invertible()) or ( # long time
491 ....: x.operator().inverse()(J.one()) # long time
493 ....: x.inverse() ) # long time
496 Check that the fast (cached) and slow algorithms give the same
499 sage: J = random_eja(field=QQ, orthonormalize=False) # long time
500 sage: x = J.random_element() # long time
501 sage: while not x.is_invertible(): # long time
502 ....: x = J.random_element() # long time
503 sage: slow = x.inverse() # long time
504 sage: _ = J._charpoly_coefficients() # long time
505 sage: fast = x.inverse() # long time
506 sage: slow == fast # long time
509 not_invertible_msg
= "element is not invertible"
511 algebra
= self
.parent()
512 if algebra
._charpoly
_coefficients
.is_in_cache():
513 # We can invert using our charpoly if it will be fast to
514 # compute. If the coefficients are cached, our rank had
516 if self
.det().is_zero():
517 raise ZeroDivisionError(not_invertible_msg
)
519 a
= self
.characteristic_polynomial().coefficients(sparse
=False)
520 return (-1)**(r
+1)*algebra
.sum(a
[i
+1]*self
**i
521 for i
in range(r
))/self
.det()
524 inv
= (~self
.quadratic_representation())(self
)
525 self
.is_invertible
.set_cache(True)
527 except ZeroDivisionError:
528 self
.is_invertible
.set_cache(False)
529 raise ZeroDivisionError(not_invertible_msg
)
533 def is_invertible(self
):
535 Return whether or not this element is invertible.
539 If computing my determinant will be fast, we do so and compare
540 with zero (Proposition II.2.4 in Faraut and
541 Koranyi). Otherwise, Proposition II.3.2 in Faraut and Koranyi
542 reduces the problem to the invertibility of my quadratic
547 sage: from mjo.eja.eja_algebra import random_eja
551 The identity element is always invertible::
553 sage: J = random_eja()
554 sage: J.one().is_invertible()
557 The zero element is never invertible in a non-trivial algebra::
559 sage: J = random_eja()
560 sage: (not J.is_trivial()) and J.zero().is_invertible()
563 Test that the fast (cached) and slow algorithms give the same
566 sage: J = random_eja(field=QQ, orthonormalize=False) # long time
567 sage: x = J.random_element() # long time
568 sage: slow = x.is_invertible() # long time
569 sage: _ = J._charpoly_coefficients() # long time
570 sage: fast = x.is_invertible() # long time
571 sage: slow == fast # long time
575 if self
.parent().is_trivial():
580 if self
.parent()._charpoly
_coefficients
.is_in_cache():
581 # The determinant will be quicker than inverting the
582 # quadratic representation, most likely.
583 return (not self
.det().is_zero())
585 # The easiest way to determine if I'm invertible is to try.
587 inv
= (~self
.quadratic_representation())(self
)
588 self
.inverse
.set_cache(inv
)
590 except ZeroDivisionError:
594 def is_primitive_idempotent(self
):
596 Return whether or not this element is a primitive (or minimal)
599 A primitive idempotent is a non-zero idempotent that is not
600 the sum of two other non-zero idempotents. Remark 2.7.15 in
601 Baes shows that this is what he refers to as a "minimal
604 An element of a Euclidean Jordan algebra is a minimal idempotent
605 if it :meth:`is_idempotent` and if its Peirce subalgebra
606 corresponding to the eigenvalue ``1`` has dimension ``1`` (Baes,
611 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
612 ....: RealSymmetricEJA,
618 This method is sloooooow.
622 The spectral decomposition of a non-regular element should always
623 contain at least one non-minimal idempotent::
625 sage: J = RealSymmetricEJA(3)
626 sage: x = sum(J.gens())
629 sage: [ c.is_primitive_idempotent()
630 ....: for (l,c) in x.spectral_decomposition() ]
633 On the other hand, the spectral decomposition of a regular
634 element should always be in terms of minimal idempotents::
636 sage: J = JordanSpinEJA(4)
637 sage: x = sum( i*J.monomial(i) for i in range(len(J.gens())) )
640 sage: [ c.is_primitive_idempotent()
641 ....: for (l,c) in x.spectral_decomposition() ]
646 The identity element is minimal only in an EJA of rank one::
648 sage: J = random_eja()
649 sage: J.rank() == 1 or not J.one().is_primitive_idempotent()
652 A non-idempotent cannot be a minimal idempotent::
654 sage: J = JordanSpinEJA(4)
655 sage: x = J.random_element()
656 sage: (not x.is_idempotent()) and x.is_primitive_idempotent()
659 Proposition 2.7.19 in Baes says that an element is a minimal
660 idempotent if and only if it's idempotent with trace equal to
663 sage: J = JordanSpinEJA(4)
664 sage: x = J.random_element()
665 sage: expected = (x.is_idempotent() and x.trace() == 1)
666 sage: actual = x.is_primitive_idempotent()
667 sage: actual == expected
670 Primitive idempotents must be non-zero::
672 sage: J = random_eja()
673 sage: J.zero().is_idempotent()
675 sage: J.zero().is_primitive_idempotent()
678 As a consequence of the fact that primitive idempotents must
679 be non-zero, there are no primitive idempotents in a trivial
680 Euclidean Jordan algebra::
682 sage: J = TrivialEJA()
683 sage: J.one().is_idempotent()
685 sage: J.one().is_primitive_idempotent()
689 if not self
.is_idempotent():
695 (_
,_
,J1
) = self
.parent().peirce_decomposition(self
)
696 return (J1
.dimension() == 1)
699 def is_nilpotent(self
):
701 Return whether or not some power of this element is zero.
705 We use Theorem 5 in Chapter III of Koecher, which says that
706 an element ``x`` is nilpotent if and only if ``x.operator()``
707 is nilpotent. And it is a basic fact of linear algebra that
708 an operator on an `n`-dimensional space is nilpotent if and
709 only if, when raised to the `n`th power, it equals the zero
710 operator (for example, see Axler Corollary 8.8).
714 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
719 sage: J = JordanSpinEJA(3)
720 sage: x = sum(J.gens())
721 sage: x.is_nilpotent()
726 The identity element is never nilpotent, except in a trivial EJA::
728 sage: J = random_eja()
729 sage: J.one().is_nilpotent() and not J.is_trivial()
732 The additive identity is always nilpotent::
734 sage: random_eja().zero().is_nilpotent()
739 zero_operator
= P
.zero().operator()
740 return self
.operator()**P
.dimension() == zero_operator
743 def is_regular(self
):
745 Return whether or not this is a regular element.
749 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
754 The identity element always has degree one, but any element
755 linearly-independent from it is regular::
757 sage: J = JordanSpinEJA(5)
758 sage: J.one().is_regular()
760 sage: b0, b1, b2, b3, b4 = J.gens()
763 sage: for x in J.gens():
764 ....: (J.one() + x).is_regular()
773 The zero element should never be regular, unless the parent
774 algebra has dimension less than or equal to one::
776 sage: J = random_eja()
777 sage: J.dimension() <= 1 or not J.zero().is_regular()
780 The unit element isn't regular unless the algebra happens to
781 consist of only its scalar multiples::
783 sage: J = random_eja()
784 sage: J.dimension() <= 1 or not J.one().is_regular()
788 return self
.degree() == self
.parent().rank()
793 Return the degree of this element, which is defined to be
794 the degree of its minimal polynomial.
802 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
807 sage: J = JordanSpinEJA(4)
808 sage: J.one().degree()
810 sage: b0,b1,b2,b3 = J.gens()
811 sage: (b0 - b1).degree()
814 In the spin factor algebra (of rank two), all elements that
815 aren't multiples of the identity are regular::
817 sage: J = JordanSpinEJA.random_instance()
818 sage: n = J.dimension()
819 sage: x = J.random_element()
820 sage: x.degree() == min(n,2) or (x == x.coefficient(0)*J.one())
825 The zero and unit elements are both of degree one in nontrivial
828 sage: J = random_eja()
829 sage: d = J.zero().degree()
830 sage: (J.is_trivial() and d == 0) or d == 1
832 sage: d = J.one().degree()
833 sage: (J.is_trivial() and d == 0) or d == 1
836 Our implementation agrees with the definition::
838 sage: x = random_eja().random_element()
839 sage: x.degree() == x.minimal_polynomial().degree()
843 n
= self
.parent().dimension()
846 # The minimal polynomial is an empty product, i.e. the
847 # constant polynomial "1" having degree zero.
850 # The minimal polynomial of zero in a nontrivial algebra
851 # is "t", and is of degree one.
854 # If this is a nonzero element of a nontrivial algebra, it
855 # has degree at least one. It follows that, in an algebra
856 # of dimension one, the degree must be actually one.
859 # BEWARE: The subalgebra_generated_by() method uses the result
860 # of this method to construct a basis for the subalgebra. That
861 # means, in particular, that we cannot implement this method
862 # as ``self.subalgebra_generated_by().dimension()``.
864 # Algorithm: keep appending (vector representations of) powers
865 # self as rows to a matrix and echelonizing it. When its rank
866 # stops increasing, we've reached a redundancy.
868 # Given the special cases above, we can assume that "self" is
869 # nonzero, the algebra is nontrivial, and that its dimension
871 M
= matrix([(self
.parent().one()).to_vector()])
874 # Specifying the row-reduction algorithm can e.g. help over
875 # AA because it avoids the RecursionError that gets thrown
876 # when we have to look too hard for a root.
878 # Beware: QQ supports an entirely different set of "algorithm"
879 # keywords than do AA and RR.
881 from sage
.rings
.all
import QQ
882 if self
.parent().base_ring() is not QQ
:
883 algo
= "scaled_partial_pivoting"
886 M
= matrix(M
.rows() + [(self
**d
).to_vector()])
889 if new_rank
== old_rank
:
898 def left_matrix(self
):
900 Our parent class defines ``left_matrix`` and ``matrix``
901 methods whose names are misleading. We don't want them.
903 raise NotImplementedError("use operator().matrix() instead")
908 def minimal_polynomial(self
):
910 Return the minimal polynomial of this element,
911 as a function of the variable `t`.
915 We restrict ourselves to the associative subalgebra
916 generated by this element, and then return the minimal
917 polynomial of this element's operator matrix (in that
918 subalgebra). This works by Baes Proposition 2.3.16.
922 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
923 ....: RealSymmetricEJA,
929 Keeping in mind that the polynomial ``1`` evaluates the identity
930 element (also the zero element) of the trivial algebra, it is clear
931 that the polynomial ``1`` is the minimal polynomial of the only
932 element in a trivial algebra::
934 sage: J = TrivialEJA()
935 sage: J.one().minimal_polynomial()
937 sage: J.zero().minimal_polynomial()
942 The minimal polynomial of the identity and zero elements are
943 always the same, except in trivial algebras where the minimal
944 polynomial of the unit/zero element is ``1``::
946 sage: J = random_eja()
947 sage: mu = J.one().minimal_polynomial()
948 sage: t = mu.parent().gen()
949 sage: mu + int(J.is_trivial())*(t-2)
951 sage: mu = J.zero().minimal_polynomial()
952 sage: t = mu.parent().gen()
953 sage: mu + int(J.is_trivial())*(t-1)
956 The degree of an element is (by one definition) the degree
957 of its minimal polynomial::
959 sage: x = random_eja().random_element()
960 sage: x.degree() == x.minimal_polynomial().degree()
963 The minimal polynomial and the characteristic polynomial coincide
964 and are known (see Alizadeh, Example 11.11) for all elements of
965 the spin factor algebra that aren't scalar multiples of the
966 identity. We require the dimension of the algebra to be at least
967 two here so that said elements actually exist::
969 sage: d_max = JordanSpinEJA._max_random_instance_dimension()
970 sage: n = ZZ.random_element(2, max(2,d_max))
971 sage: J = JordanSpinEJA(n)
972 sage: y = J.random_element()
973 sage: while y == y.coefficient(0)*J.one():
974 ....: y = J.random_element()
975 sage: y0 = y.to_vector()[0]
976 sage: y_bar = y.to_vector()[1:]
977 sage: actual = y.minimal_polynomial()
978 sage: t = PolynomialRing(J.base_ring(),'t').gen(0)
979 sage: expected = t^2 - 2*y0*t + (y0^2 - norm(y_bar)^2)
980 sage: bool(actual == expected)
983 The minimal polynomial should always kill its element::
985 sage: x = random_eja().random_element() # long time
986 sage: p = x.minimal_polynomial() # long time
987 sage: x.apply_univariate_polynomial(p) # long time
990 The minimal polynomial is invariant under a change of basis,
991 and in particular, a re-scaling of the basis::
993 sage: d_max = RealSymmetricEJA._max_random_instance_dimension()
994 sage: d = ZZ.random_element(1, d_max)
995 sage: n = RealSymmetricEJA._max_random_instance_size(d)
996 sage: J1 = RealSymmetricEJA(n)
997 sage: J2 = RealSymmetricEJA(n,orthonormalize=False)
998 sage: X = random_matrix(AA,n)
999 sage: X = X*X.transpose()
1002 sage: x1.minimal_polynomial() == x2.minimal_polynomial()
1007 # Pretty sure we know what the minimal polynomial of
1008 # the zero operator is going to be. This ensures
1009 # consistency of e.g. the polynomial variable returned
1010 # in the "normal" case without us having to think about it.
1011 return self
.operator().minimal_polynomial()
1013 # If we don't orthonormalize the subalgebra's basis, then the
1014 # first two monomials in the subalgebra will be self^0 and
1015 # self^1... assuming that self^1 is not a scalar multiple of
1016 # self^0 (the unit element). We special case these to avoid
1017 # having to solve a system to coerce self into the subalgebra.
1018 A
= self
.subalgebra_generated_by(orthonormalize
=False)
1020 if A
.dimension() == 1:
1021 # Does a solve to find the scalar multiple alpha such that
1022 # alpha*unit = self. We have to do this because the basis
1023 # for the subalgebra will be [ self^0 ], and not [ self^1 ]!
1024 unit
= self
.parent().one()
1025 alpha
= self
.to_vector() / unit
.to_vector()
1026 return (unit
.operator()*alpha
).minimal_polynomial()
1028 # If the dimension of the subalgebra is >= 2, then we just
1029 # use the second basis element.
1030 return A
.monomial(1).operator().minimal_polynomial()
1034 def to_matrix(self
):
1036 Return an (often more natural) representation of this element as a
1039 Every finite-dimensional Euclidean Jordan Algebra is a direct
1040 sum of five simple algebras, four of which comprise Hermitian
1041 matrices. This method returns a "natural" matrix
1042 representation of this element as either a Hermitian matrix or
1047 sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
1049 ....: QuaternionHermitianEJA,
1050 ....: RealSymmetricEJA)
1054 sage: J = ComplexHermitianEJA(3)
1057 sage: J.one().to_matrix()
1068 sage: J = QuaternionHermitianEJA(2)
1071 sage: J.one().to_matrix()
1078 This also works in Cartesian product algebras::
1080 sage: J1 = HadamardEJA(1)
1081 sage: J2 = RealSymmetricEJA(2)
1082 sage: J = cartesian_product([J1,J2])
1083 sage: x = sum(J.gens())
1084 sage: x.to_matrix()[0]
1086 sage: x.to_matrix()[1]
1087 [ 1 0.7071067811865475?]
1088 [0.7071067811865475? 1]
1091 B
= self
.parent().matrix_basis()
1092 W
= self
.parent().matrix_space()
1094 # This is just a manual "from_vector()", but of course
1095 # matrix spaces aren't vector spaces in sage, so they
1096 # don't have a from_vector() method.
1097 return W
.linear_combination( zip(B
, self
.to_vector()) )
1103 The norm of this element with respect to :meth:`inner_product`.
1107 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
1112 sage: J = HadamardEJA(2)
1113 sage: x = sum(J.gens())
1119 sage: J = JordanSpinEJA(4)
1120 sage: x = sum(J.gens())
1125 return self
.inner_product(self
).sqrt()
1130 Return the left-multiplication-by-this-element
1131 operator on the ambient algebra.
1135 sage: from mjo.eja.eja_algebra import random_eja
1139 sage: J = random_eja()
1140 sage: x,y = J.random_elements(2)
1141 sage: x.operator()(y) == x*y
1143 sage: y.operator()(x) == x*y
1148 left_mult_by_self
= lambda y
: self
*y
1149 L
= P
.module_morphism(function
=left_mult_by_self
, codomain
=P
)
1150 return FiniteDimensionalEJAOperator(P
, P
, L
.matrix() )
1153 def quadratic_representation(self
, other
=None):
1155 Return the quadratic representation of this element.
1159 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
1164 The explicit form in the spin factor algebra is given by
1165 Alizadeh's Example 11.12::
1167 sage: x = JordanSpinEJA.random_instance().random_element()
1168 sage: x_vec = x.to_vector()
1169 sage: Q = matrix.identity(x.base_ring(), 0)
1170 sage: n = x_vec.degree()
1173 ....: x_bar = x_vec[1:]
1174 ....: A = matrix(x.base_ring(), 1, [x_vec.inner_product(x_vec)])
1175 ....: B = 2*x0*x_bar.row()
1176 ....: C = 2*x0*x_bar.column()
1177 ....: D = matrix.identity(x.base_ring(), n-1)
1178 ....: D = (x0^2 - x_bar.inner_product(x_bar))*D
1179 ....: D = D + 2*x_bar.tensor_product(x_bar)
1180 ....: Q = matrix.block(2,2,[A,B,C,D])
1181 sage: Q == x.quadratic_representation().matrix()
1184 Test all of the properties from Theorem 11.2 in Alizadeh::
1186 sage: J = random_eja()
1187 sage: x,y = J.random_elements(2)
1188 sage: Lx = x.operator()
1189 sage: Lxx = (x*x).operator()
1190 sage: Qx = x.quadratic_representation()
1191 sage: Qy = y.quadratic_representation()
1192 sage: Qxy = x.quadratic_representation(y)
1193 sage: Qex = J.one().quadratic_representation(x)
1194 sage: n = ZZ.random_element(10)
1195 sage: Qxn = (x^n).quadratic_representation()
1199 sage: 2*Qxy == (x+y).quadratic_representation() - Qx - Qy
1202 Property 2 (multiply on the right for :trac:`28272`):
1204 sage: alpha = J.base_ring().random_element()
1205 sage: (alpha*x).quadratic_representation() == Qx*(alpha^2)
1210 sage: not x.is_invertible() or ( Qx(x.inverse()) == x )
1213 sage: not x.is_invertible() or (
1216 ....: x.inverse().quadratic_representation() )
1219 sage: Qxy(J.one()) == x*y
1224 sage: not x.is_invertible() or (
1225 ....: x.quadratic_representation(x.inverse())*Qx
1226 ....: == Qx*x.quadratic_representation(x.inverse()) )
1229 sage: not x.is_invertible() or (
1230 ....: x.quadratic_representation(x.inverse())*Qx
1232 ....: 2*Lx*Qex - Qx )
1235 sage: 2*Lx*Qex - Qx == Lxx
1240 sage: Qy(x).quadratic_representation() == Qy*Qx*Qy
1250 sage: not x.is_invertible() or (
1251 ....: Qx*x.inverse().operator() == Lx )
1256 sage: not x.operator_commutes_with(y) or (
1257 ....: Qx(y)^n == Qxn(y^n) )
1263 elif not other
in self
.parent():
1264 raise TypeError("'other' must live in the same algebra")
1267 M
= other
.operator()
1268 return ( L
*M
+ M
*L
- (self
*other
).operator() )
1272 def spectral_decomposition(self
):
1274 Return the unique spectral decomposition of this element.
1278 Following Faraut and Korányi's Theorem III.1.1, we restrict this
1279 element's left-multiplication-by operator to the subalgebra it
1280 generates. We then compute the spectral decomposition of that
1281 operator, and the spectral projectors we get back must be the
1282 left-multiplication-by operators for the idempotents we
1283 seek. Thus applying them to the identity element gives us those
1286 Since the eigenvalues are required to be distinct, we take
1287 the spectral decomposition of the zero element to be zero
1288 times the identity element of the algebra (which is idempotent,
1293 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
1297 The spectral decomposition of the identity is ``1`` times itself,
1298 and the spectral decomposition of zero is ``0`` times the identity::
1300 sage: J = RealSymmetricEJA(3)
1303 sage: J.one().spectral_decomposition()
1305 sage: J.zero().spectral_decomposition()
1310 sage: J = RealSymmetricEJA(4)
1311 sage: x = sum(J.gens())
1312 sage: sd = x.spectral_decomposition()
1317 sage: c0.inner_product(c1) == 0
1319 sage: c0.is_idempotent()
1321 sage: c1.is_idempotent()
1323 sage: c0 + c1 == J.one()
1325 sage: l0*c0 + l1*c1 == x
1328 The spectral decomposition should work in subalgebras, too::
1330 sage: J = RealSymmetricEJA(4)
1331 sage: (b0, b1, b2, b3, b4, b5, b6, b7, b8, b9) = J.gens()
1332 sage: A = 2*b5 - 2*b8
1333 sage: (lambda1, c1) = A.spectral_decomposition()[1]
1334 sage: (J0, J5, J1) = J.peirce_decomposition(c1)
1335 sage: (f0, f1, f2) = J1.gens()
1336 sage: f0.spectral_decomposition()
1337 [(0, 1.000000000000000?*c2), (1, 1.000000000000000?*c0)]
1340 A
= self
.subalgebra_generated_by(orthonormalize
=True)
1342 for (evalue
, proj
) in A(self
).operator().spectral_decomposition():
1343 result
.append( (evalue
, proj(A
.one()).superalgebra_element()) )
1346 def subalgebra_generated_by(self
, **kwargs
):
1348 Return the associative subalgebra of the parent EJA generated
1351 Since our parent algebra is unital, we want "subalgebra" to mean
1352 "unital subalgebra" as well; thus the subalgebra that an element
1353 generates will itself be a Euclidean Jordan algebra after
1354 restricting the algebra operations appropriately. This is the
1355 subalgebra that Faraut and Korányi work with in section II.2, for
1360 sage: from mjo.eja.eja_algebra import (random_eja,
1362 ....: RealSymmetricEJA)
1366 We can create subalgebras of Cartesian product EJAs that are not
1367 themselves Cartesian product EJAs (they're just "regular" EJAs)::
1369 sage: J1 = HadamardEJA(3)
1370 sage: J2 = RealSymmetricEJA(2)
1371 sage: J = cartesian_product([J1,J2])
1372 sage: J.one().subalgebra_generated_by()
1373 Euclidean Jordan algebra of dimension 1 over Algebraic Real Field
1377 This subalgebra, being composed of only powers, is associative::
1379 sage: x0 = random_eja().random_element()
1380 sage: A = x0.subalgebra_generated_by(orthonormalize=False)
1381 sage: x,y,z = A.random_elements(3)
1382 sage: (x*y)*z == x*(y*z)
1385 Squaring in the subalgebra should work the same as in
1388 sage: x = random_eja().random_element()
1389 sage: A = x.subalgebra_generated_by(orthonormalize=False)
1390 sage: A(x^2) == A(x)*A(x)
1393 By definition, the subalgebra generated by the zero element is
1394 the one-dimensional algebra generated by the identity
1395 element... unless the original algebra was trivial, in which
1396 case the subalgebra is trivial too::
1398 sage: A = random_eja().zero().subalgebra_generated_by()
1399 sage: (A.is_trivial() and A.dimension() == 0) or A.dimension() == 1
1403 powers
= tuple( self
**k
for k
in range(self
.degree()) )
1404 A
= self
.parent().subalgebra(powers
,
1409 A
.one
.set_cache(A(self
.parent().one()))
1413 def subalgebra_idempotent(self
):
1415 Find an idempotent in the associative subalgebra I generate
1416 using Proposition 2.3.5 in Baes.
1420 sage: from mjo.eja.eja_algebra import random_eja
1424 Ensure that we can find an idempotent in a non-trivial algebra
1425 where there are non-nilpotent elements, or that we get the dumb
1426 solution in the trivial algebra::
1428 sage: J = random_eja()
1429 sage: x = J.random_element()
1430 sage: while x.is_nilpotent() and not J.is_trivial():
1431 ....: x = J.random_element()
1432 sage: c = x.subalgebra_idempotent()
1437 if self
.parent().is_trivial():
1440 if self
.is_nilpotent():
1441 raise ValueError("this only works with non-nilpotent elements!")
1443 J
= self
.subalgebra_generated_by()
1446 # The image of the matrix of left-u^m-multiplication
1447 # will be minimal for some natural number s...
1449 minimal_dim
= J
.dimension()
1450 for i
in range(1, minimal_dim
):
1451 this_dim
= (u
**i
).operator().matrix().image().dimension()
1452 if this_dim
< minimal_dim
:
1453 minimal_dim
= this_dim
1456 # Now minimal_matrix should correspond to the smallest
1457 # non-zero subspace in Baes's (or really, Koecher's)
1460 # However, we need to restrict the matrix to work on the
1461 # subspace... or do we? Can't we just solve, knowing that
1462 # A(c) = u^(s+1) should have a solution in the big space,
1465 # Beware, solve_right() means that we're using COLUMN vectors.
1466 # Our FiniteDimensionalAlgebraElement superclass uses rows.
1468 A
= u_next
.operator().matrix()
1469 c
= J
.from_vector(A
.solve_right(u_next
.to_vector()))
1471 # Now c is the idempotent we want, but it still lives in the subalgebra.
1472 return c
.superalgebra_element()
1477 Return my trace, the sum of my eigenvalues.
1479 In a trivial algebra, however you want to look at it, the trace is
1480 an empty sum for which we declare the result to be zero.
1484 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
1491 sage: J = TrivialEJA()
1492 sage: J.zero().trace()
1496 sage: J = JordanSpinEJA(3)
1497 sage: x = sum(J.gens())
1503 sage: J = HadamardEJA(5)
1504 sage: J.one().trace()
1509 The trace of an element is a real number::
1511 sage: J = random_eja()
1512 sage: J.random_element().trace() in RLF
1515 The trace is linear::
1517 sage: J = random_eja()
1518 sage: x,y = J.random_elements(2)
1519 sage: alpha = J.base_ring().random_element()
1520 sage: (alpha*x + y).trace() == alpha*x.trace() + y.trace()
1523 The trace of a square is nonnegative::
1525 sage: x = random_eja().random_element()
1526 sage: (x*x).trace() >= 0
1534 # Special case for the trivial algebra where
1535 # the trace is an empty sum.
1536 return P
.base_ring().zero()
1538 p
= P
._charpoly
_coefficients
()[r
-1]
1539 # The _charpoly_coeff function already adds the factor of
1540 # -1 to ensure that _charpoly_coeff(r-1) is really what
1541 # appears in front of t^{r-1} in the charpoly. However,
1542 # we want the negative of THAT for the trace.
1543 return -p(*self
.to_vector())
1546 def trace_inner_product(self
, other
):
1548 Return the trace inner product of myself and ``other``.
1552 sage: from mjo.eja.eja_algebra import random_eja
1556 The trace inner product is commutative, bilinear, and associative::
1558 sage: J = random_eja()
1559 sage: x,y,z = J.random_elements(3)
1561 sage: x.trace_inner_product(y) == y.trace_inner_product(x)
1564 sage: a = J.base_ring().random_element();
1565 sage: actual = (a*(x+z)).trace_inner_product(y)
1566 sage: expected = ( a*x.trace_inner_product(y) +
1567 ....: a*z.trace_inner_product(y) )
1568 sage: actual == expected
1570 sage: actual = x.trace_inner_product(a*(y+z))
1571 sage: expected = ( a*x.trace_inner_product(y) +
1572 ....: a*x.trace_inner_product(z) )
1573 sage: actual == expected
1576 sage: (x*y).trace_inner_product(z) == y.trace_inner_product(x*z)
1580 if not other
in self
.parent():
1581 raise TypeError("'other' must live in the same algebra")
1583 return (self
*other
).trace()
1586 def trace_norm(self
):
1588 The norm of this element with respect to :meth:`trace_inner_product`.
1592 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
1597 sage: J = HadamardEJA(2)
1598 sage: x = sum(J.gens())
1599 sage: x.trace_norm()
1604 sage: J = JordanSpinEJA(4)
1605 sage: x = sum(J.gens())
1606 sage: x.trace_norm()
1610 return self
.trace_inner_product(self
).sqrt()
1613 class CartesianProductEJAElement(FiniteDimensionalEJAElement
):
1616 Compute the determinant of this product-element using the
1617 determianants of its factors.
1619 This result Follows from the spectral decomposition of (say)
1620 the pair `(x,y)` in terms of the Jordan frame `\left\{ (c_1,
1621 0),(c_2, 0),...,(0,d_1),(0,d_2),... \right\}.
1623 from sage
.misc
.misc_c
import prod
1624 return prod( f
.det() for f
in self
.cartesian_factors() )
1626 def to_matrix(self
):
1627 # An override is necessary to call our custom _scale().
1628 B
= self
.parent().matrix_basis()
1629 W
= self
.parent().matrix_space()
1631 # Aaaaand linear combinations don't work in Cartesian
1632 # product spaces, even though they provide a method with
1633 # that name. This is hidden behind an "if" because the
1634 # _scale() function is slow.
1635 pairs
= zip(B
, self
.to_vector())
1636 return W
.sum( _scale(b
, alpha
) for (b
,alpha
) in pairs
)