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eja: finally enable tests for the trace inner product.
[sage.d.git] / mjo / eja / euclidean_jordan_algebra.py
1 """
2 Euclidean Jordan Algebras. These are formally-real Jordan Algebras;
3 specifically those where u^2 + v^2 = 0 implies that u = v = 0. They
4 are used in optimization, and have some additional nice methods beyond
5 what can be supported in a general Jordan Algebra.
6 """
7
8 from sage.categories.magmatic_algebras import MagmaticAlgebras
9 from sage.structure.element import is_Matrix
10 from sage.structure.category_object import normalize_names
11
12 from sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra import FiniteDimensionalAlgebra
13 from sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra_element import FiniteDimensionalAlgebraElement
14
15 class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
16 @staticmethod
17 def __classcall_private__(cls,
18 field,
19 mult_table,
20 names='e',
21 assume_associative=False,
22 category=None,
23 rank=None,
24 natural_basis=None):
25 n = len(mult_table)
26 mult_table = [b.base_extend(field) for b in mult_table]
27 for b in mult_table:
28 b.set_immutable()
29 if not (is_Matrix(b) and b.dimensions() == (n, n)):
30 raise ValueError("input is not a multiplication table")
31 mult_table = tuple(mult_table)
32
33 cat = MagmaticAlgebras(field).FiniteDimensional().WithBasis()
34 cat.or_subcategory(category)
35 if assume_associative:
36 cat = cat.Associative()
37
38 names = normalize_names(n, names)
39
40 fda = super(FiniteDimensionalEuclideanJordanAlgebra, cls)
41 return fda.__classcall__(cls,
42 field,
43 mult_table,
44 assume_associative=assume_associative,
45 names=names,
46 category=cat,
47 rank=rank,
48 natural_basis=natural_basis)
49
50
51 def __init__(self,
52 field,
53 mult_table,
54 names='e',
55 assume_associative=False,
56 category=None,
57 rank=None,
58 natural_basis=None):
59 """
60 EXAMPLES:
61
62 By definition, Jordan multiplication commutes::
63
64 sage: set_random_seed()
65 sage: J = random_eja()
66 sage: x = J.random_element()
67 sage: y = J.random_element()
68 sage: x*y == y*x
69 True
70
71 """
72 self._rank = rank
73 self._natural_basis = natural_basis
74 self._multiplication_table = mult_table
75 fda = super(FiniteDimensionalEuclideanJordanAlgebra, self)
76 fda.__init__(field,
77 mult_table,
78 names=names,
79 category=category)
80
81
82 def _repr_(self):
83 """
84 Return a string representation of ``self``.
85 """
86 fmt = "Euclidean Jordan algebra of degree {} over {}"
87 return fmt.format(self.degree(), self.base_ring())
88
89
90
91 @cached_method
92 def _charpoly_coeff(self, i):
93 """
94 Return the coefficient polynomial "a_{i}" of this algebra's
95 general characteristic polynomial.
96
97 Having this be a separate cached method lets us compute and
98 store the trace/determinant (a_{r-1} and a_{0} respectively)
99 separate from the entire characteristic polynomial.
100 """
101 (A_of_x, x) = self._charpoly_matrix()
102 R = A_of_x.base_ring()
103 A_cols = A_of_x.columns()
104 A_cols[i] = (x**self.rank()).vector()
105 numerator = column_matrix(A_of_x.base_ring(), A_cols).det()
106 denominator = A_of_x.det()
107
108 # We're relying on the theory here to ensure that each a_i is
109 # indeed back in R, and the added negative signs are to make
110 # the whole charpoly expression sum to zero.
111 return R(-numerator/denominator)
112
113
114 @cached_method
115 def _charpoly_matrix(self):
116 """
117 Compute the matrix whose entries A_ij are polynomials in
118 X1,...,XN. This same matrix is used in more than one method and
119 it's not so fast to construct.
120 """
121 r = self.rank()
122 n = self.dimension()
123
124 # Construct a new algebra over a multivariate polynomial ring...
125 names = ['X' + str(i) for i in range(1,n+1)]
126 R = PolynomialRing(self.base_ring(), names)
127 J = FiniteDimensionalEuclideanJordanAlgebra(R,
128 self._multiplication_table,
129 rank=r)
130
131 idmat = identity_matrix(J.base_ring(), n)
132
133 x = J(vector(R, R.gens()))
134 l1 = [column_matrix((x**k).vector()) for k in range(r)]
135 l2 = [idmat.column(k-1).column() for k in range(r+1, n+1)]
136 A_of_x = block_matrix(R, 1, n, (l1 + l2))
137 return (A_of_x, x)
138
139
140 @cached_method
141 def characteristic_polynomial(self):
142 """
143 EXAMPLES:
144
145 The characteristic polynomial in the spin algebra is given in
146 Alizadeh, Example 11.11::
147
148 sage: J = JordanSpinEJA(3)
149 sage: p = J.characteristic_polynomial(); p
150 X1^2 - X2^2 - X3^2 + (-2*t)*X1 + t^2
151 sage: xvec = J.one().vector()
152 sage: p(*xvec)
153 t^2 - 2*t + 1
154
155 """
156 r = self.rank()
157 n = self.dimension()
158
159 # The list of coefficient polynomials a_1, a_2, ..., a_n.
160 a = [ self._charpoly_coeff(i) for i in range(n) ]
161
162 # We go to a bit of trouble here to reorder the
163 # indeterminates, so that it's easier to evaluate the
164 # characteristic polynomial at x's coordinates and get back
165 # something in terms of t, which is what we want.
166 R = a[0].parent()
167 S = PolynomialRing(self.base_ring(),'t')
168 t = S.gen(0)
169 S = PolynomialRing(S, R.variable_names())
170 t = S(t)
171
172 # Note: all entries past the rth should be zero. The
173 # coefficient of the highest power (x^r) is 1, but it doesn't
174 # appear in the solution vector which contains coefficients
175 # for the other powers (to make them sum to x^r).
176 if (r < n):
177 a[r] = 1 # corresponds to x^r
178 else:
179 # When the rank is equal to the dimension, trying to
180 # assign a[r] goes out-of-bounds.
181 a.append(1) # corresponds to x^r
182
183 return sum( a[k]*(t**k) for k in range(len(a)) )
184
185
186 def inner_product(self, x, y):
187 """
188 The inner product associated with this Euclidean Jordan algebra.
189
190 Defaults to the trace inner product, but can be overridden by
191 subclasses if they are sure that the necessary properties are
192 satisfied.
193
194 EXAMPLES:
195
196 The inner product must satisfy its axiom for this algebra to truly
197 be a Euclidean Jordan Algebra::
198
199 sage: set_random_seed()
200 sage: J = random_eja()
201 sage: x = J.random_element()
202 sage: y = J.random_element()
203 sage: z = J.random_element()
204 sage: (x*y).inner_product(z) == y.inner_product(x*z)
205 True
206
207 """
208 if (not x in self) or (not y in self):
209 raise TypeError("arguments must live in this algebra")
210 return x.trace_inner_product(y)
211
212
213 def natural_basis(self):
214 """
215 Return a more-natural representation of this algebra's basis.
216
217 Every finite-dimensional Euclidean Jordan Algebra is a direct
218 sum of five simple algebras, four of which comprise Hermitian
219 matrices. This method returns the original "natural" basis
220 for our underlying vector space. (Typically, the natural basis
221 is used to construct the multiplication table in the first place.)
222
223 Note that this will always return a matrix. The standard basis
224 in `R^n` will be returned as `n`-by-`1` column matrices.
225
226 EXAMPLES::
227
228 sage: J = RealSymmetricEJA(2)
229 sage: J.basis()
230 Family (e0, e1, e2)
231 sage: J.natural_basis()
232 (
233 [1 0] [0 1] [0 0]
234 [0 0], [1 0], [0 1]
235 )
236
237 ::
238
239 sage: J = JordanSpinEJA(2)
240 sage: J.basis()
241 Family (e0, e1)
242 sage: J.natural_basis()
243 (
244 [1] [0]
245 [0], [1]
246 )
247
248 """
249 if self._natural_basis is None:
250 return tuple( b.vector().column() for b in self.basis() )
251 else:
252 return self._natural_basis
253
254
255 def rank(self):
256 """
257 Return the rank of this EJA.
258 """
259 if self._rank is None:
260 raise ValueError("no rank specified at genesis")
261 else:
262 return self._rank
263
264
265 class Element(FiniteDimensionalAlgebraElement):
266 """
267 An element of a Euclidean Jordan algebra.
268 """
269
270 def __init__(self, A, elt=None):
271 """
272 EXAMPLES:
273
274 The identity in `S^n` is converted to the identity in the EJA::
275
276 sage: J = RealSymmetricEJA(3)
277 sage: I = identity_matrix(QQ,3)
278 sage: J(I) == J.one()
279 True
280
281 This skew-symmetric matrix can't be represented in the EJA::
282
283 sage: J = RealSymmetricEJA(3)
284 sage: A = matrix(QQ,3, lambda i,j: i-j)
285 sage: J(A)
286 Traceback (most recent call last):
287 ...
288 ArithmeticError: vector is not in free module
289
290 """
291 # Goal: if we're given a matrix, and if it lives in our
292 # parent algebra's "natural ambient space," convert it
293 # into an algebra element.
294 #
295 # The catch is, we make a recursive call after converting
296 # the given matrix into a vector that lives in the algebra.
297 # This we need to try the parent class initializer first,
298 # to avoid recursing forever if we're given something that
299 # already fits into the algebra, but also happens to live
300 # in the parent's "natural ambient space" (this happens with
301 # vectors in R^n).
302 try:
303 FiniteDimensionalAlgebraElement.__init__(self, A, elt)
304 except ValueError:
305 natural_basis = A.natural_basis()
306 if elt in natural_basis[0].matrix_space():
307 # Thanks for nothing! Matrix spaces aren't vector
308 # spaces in Sage, so we have to figure out its
309 # natural-basis coordinates ourselves.
310 V = VectorSpace(elt.base_ring(), elt.nrows()**2)
311 W = V.span( _mat2vec(s) for s in natural_basis )
312 coords = W.coordinates(_mat2vec(elt))
313 FiniteDimensionalAlgebraElement.__init__(self, A, coords)
314
315 def __pow__(self, n):
316 """
317 Return ``self`` raised to the power ``n``.
318
319 Jordan algebras are always power-associative; see for
320 example Faraut and Koranyi, Proposition II.1.2 (ii).
321
322 .. WARNING:
323
324 We have to override this because our superclass uses row vectors
325 instead of column vectors! We, on the other hand, assume column
326 vectors everywhere.
327
328 EXAMPLES::
329
330 sage: set_random_seed()
331 sage: x = random_eja().random_element()
332 sage: x.operator_matrix()*x.vector() == (x^2).vector()
333 True
334
335 A few examples of power-associativity::
336
337 sage: set_random_seed()
338 sage: x = random_eja().random_element()
339 sage: x*(x*x)*(x*x) == x^5
340 True
341 sage: (x*x)*(x*x*x) == x^5
342 True
343
344 We also know that powers operator-commute (Koecher, Chapter
345 III, Corollary 1)::
346
347 sage: set_random_seed()
348 sage: x = random_eja().random_element()
349 sage: m = ZZ.random_element(0,10)
350 sage: n = ZZ.random_element(0,10)
351 sage: Lxm = (x^m).operator_matrix()
352 sage: Lxn = (x^n).operator_matrix()
353 sage: Lxm*Lxn == Lxn*Lxm
354 True
355
356 """
357 A = self.parent()
358 if n == 0:
359 return A.one()
360 elif n == 1:
361 return self
362 else:
363 return A( (self.operator_matrix()**(n-1))*self.vector() )
364
365
366 def apply_univariate_polynomial(self, p):
367 """
368 Apply the univariate polynomial ``p`` to this element.
369
370 A priori, SageMath won't allow us to apply a univariate
371 polynomial to an element of an EJA, because we don't know
372 that EJAs are rings (they are usually not associative). Of
373 course, we know that EJAs are power-associative, so the
374 operation is ultimately kosher. This function sidesteps
375 the CAS to get the answer we want and expect.
376
377 EXAMPLES::
378
379 sage: R = PolynomialRing(QQ, 't')
380 sage: t = R.gen(0)
381 sage: p = t^4 - t^3 + 5*t - 2
382 sage: J = RealCartesianProductEJA(5)
383 sage: J.one().apply_univariate_polynomial(p) == 3*J.one()
384 True
385
386 TESTS:
387
388 We should always get back an element of the algebra::
389
390 sage: set_random_seed()
391 sage: p = PolynomialRing(QQ, 't').random_element()
392 sage: J = random_eja()
393 sage: x = J.random_element()
394 sage: x.apply_univariate_polynomial(p) in J
395 True
396
397 """
398 if len(p.variables()) > 1:
399 raise ValueError("not a univariate polynomial")
400 P = self.parent()
401 R = P.base_ring()
402 # Convert the coeficcients to the parent's base ring,
403 # because a priori they might live in an (unnecessarily)
404 # larger ring for which P.sum() would fail below.
405 cs = [ R(c) for c in p.coefficients(sparse=False) ]
406 return P.sum( cs[k]*(self**k) for k in range(len(cs)) )
407
408
409 def characteristic_polynomial(self):
410 """
411 Return the characteristic polynomial of this element.
412
413 EXAMPLES:
414
415 The rank of `R^3` is three, and the minimal polynomial of
416 the identity element is `(t-1)` from which it follows that
417 the characteristic polynomial should be `(t-1)^3`::
418
419 sage: J = RealCartesianProductEJA(3)
420 sage: J.one().characteristic_polynomial()
421 t^3 - 3*t^2 + 3*t - 1
422
423 Likewise, the characteristic of the zero element in the
424 rank-three algebra `R^{n}` should be `t^{3}`::
425
426 sage: J = RealCartesianProductEJA(3)
427 sage: J.zero().characteristic_polynomial()
428 t^3
429
430 The characteristic polynomial of an element should evaluate
431 to zero on that element::
432
433 sage: set_random_seed()
434 sage: x = RealCartesianProductEJA(3).random_element()
435 sage: p = x.characteristic_polynomial()
436 sage: x.apply_univariate_polynomial(p)
437 0
438
439 """
440 p = self.parent().characteristic_polynomial()
441 return p(*self.vector())
442
443
444 def inner_product(self, other):
445 """
446 Return the parent algebra's inner product of myself and ``other``.
447
448 EXAMPLES:
449
450 The inner product in the Jordan spin algebra is the usual
451 inner product on `R^n` (this example only works because the
452 basis for the Jordan algebra is the standard basis in `R^n`)::
453
454 sage: J = JordanSpinEJA(3)
455 sage: x = vector(QQ,[1,2,3])
456 sage: y = vector(QQ,[4,5,6])
457 sage: x.inner_product(y)
458 32
459 sage: J(x).inner_product(J(y))
460 32
461
462 The inner product on `S^n` is `<X,Y> = trace(X*Y)`, where
463 multiplication is the usual matrix multiplication in `S^n`,
464 so the inner product of the identity matrix with itself
465 should be the `n`::
466
467 sage: J = RealSymmetricEJA(3)
468 sage: J.one().inner_product(J.one())
469 3
470
471 Likewise, the inner product on `C^n` is `<X,Y> =
472 Re(trace(X*Y))`, where we must necessarily take the real
473 part because the product of Hermitian matrices may not be
474 Hermitian::
475
476 sage: J = ComplexHermitianEJA(3)
477 sage: J.one().inner_product(J.one())
478 3
479
480 Ditto for the quaternions::
481
482 sage: J = QuaternionHermitianEJA(3)
483 sage: J.one().inner_product(J.one())
484 3
485
486 TESTS:
487
488 Ensure that we can always compute an inner product, and that
489 it gives us back a real number::
490
491 sage: set_random_seed()
492 sage: J = random_eja()
493 sage: x = J.random_element()
494 sage: y = J.random_element()
495 sage: x.inner_product(y) in RR
496 True
497
498 """
499 P = self.parent()
500 if not other in P:
501 raise TypeError("'other' must live in the same algebra")
502
503 return P.inner_product(self, other)
504
505
506 def operator_commutes_with(self, other):
507 """
508 Return whether or not this element operator-commutes
509 with ``other``.
510
511 EXAMPLES:
512
513 The definition of a Jordan algebra says that any element
514 operator-commutes with its square::
515
516 sage: set_random_seed()
517 sage: x = random_eja().random_element()
518 sage: x.operator_commutes_with(x^2)
519 True
520
521 TESTS:
522
523 Test Lemma 1 from Chapter III of Koecher::
524
525 sage: set_random_seed()
526 sage: J = random_eja()
527 sage: u = J.random_element()
528 sage: v = J.random_element()
529 sage: lhs = u.operator_commutes_with(u*v)
530 sage: rhs = v.operator_commutes_with(u^2)
531 sage: lhs == rhs
532 True
533
534 """
535 if not other in self.parent():
536 raise TypeError("'other' must live in the same algebra")
537
538 A = self.operator_matrix()
539 B = other.operator_matrix()
540 return (A*B == B*A)
541
542
543 def det(self):
544 """
545 Return my determinant, the product of my eigenvalues.
546
547 EXAMPLES::
548
549 sage: J = JordanSpinEJA(2)
550 sage: e0,e1 = J.gens()
551 sage: x = sum( J.gens() )
552 sage: x.det()
553 0
554
555 ::
556
557 sage: J = JordanSpinEJA(3)
558 sage: e0,e1,e2 = J.gens()
559 sage: x = sum( J.gens() )
560 sage: x.det()
561 -1
562
563 TESTS:
564
565 An element is invertible if and only if its determinant is
566 non-zero::
567
568 sage: set_random_seed()
569 sage: x = random_eja().random_element()
570 sage: x.is_invertible() == (x.det() != 0)
571 True
572
573 """
574 P = self.parent()
575 r = P.rank()
576 p = P._charpoly_coeff(0)
577 # The _charpoly_coeff function already adds the factor of
578 # -1 to ensure that _charpoly_coeff(0) is really what
579 # appears in front of t^{0} in the charpoly. However,
580 # we want (-1)^r times THAT for the determinant.
581 return ((-1)**r)*p(*self.vector())
582
583
584 def inverse(self):
585 """
586 Return the Jordan-multiplicative inverse of this element.
587
588 We can't use the superclass method because it relies on the
589 algebra being associative.
590
591 EXAMPLES:
592
593 The inverse in the spin factor algebra is given in Alizadeh's
594 Example 11.11::
595
596 sage: set_random_seed()
597 sage: n = ZZ.random_element(1,10)
598 sage: J = JordanSpinEJA(n)
599 sage: x = J.random_element()
600 sage: while x.is_zero():
601 ....: x = J.random_element()
602 sage: x_vec = x.vector()
603 sage: x0 = x_vec[0]
604 sage: x_bar = x_vec[1:]
605 sage: coeff = 1/(x0^2 - x_bar.inner_product(x_bar))
606 sage: inv_vec = x_vec.parent()([x0] + (-x_bar).list())
607 sage: x_inverse = coeff*inv_vec
608 sage: x.inverse() == J(x_inverse)
609 True
610
611 TESTS:
612
613 The identity element is its own inverse::
614
615 sage: set_random_seed()
616 sage: J = random_eja()
617 sage: J.one().inverse() == J.one()
618 True
619
620 If an element has an inverse, it acts like one::
621
622 sage: set_random_seed()
623 sage: J = random_eja()
624 sage: x = J.random_element()
625 sage: (not x.is_invertible()) or (x.inverse()*x == J.one())
626 True
627
628 """
629 if self.parent().is_associative():
630 elt = FiniteDimensionalAlgebraElement(self.parent(), self)
631 return elt.inverse()
632
633 # TODO: we can do better once the call to is_invertible()
634 # doesn't crash on irregular elements.
635 #if not self.is_invertible():
636 # raise ValueError('element is not invertible')
637
638 # We do this a little different than the usual recursive
639 # call to a finite-dimensional algebra element, because we
640 # wind up with an inverse that lives in the subalgebra and
641 # we need information about the parent to convert it back.
642 V = self.span_of_powers()
643 assoc_subalg = self.subalgebra_generated_by()
644 # Mis-design warning: the basis used for span_of_powers()
645 # and subalgebra_generated_by() must be the same, and in
646 # the same order!
647 elt = assoc_subalg(V.coordinates(self.vector()))
648
649 # This will be in the subalgebra's coordinates...
650 fda_elt = FiniteDimensionalAlgebraElement(assoc_subalg, elt)
651 subalg_inverse = fda_elt.inverse()
652
653 # So we have to convert back...
654 basis = [ self.parent(v) for v in V.basis() ]
655 pairs = zip(subalg_inverse.vector(), basis)
656 return self.parent().linear_combination(pairs)
657
658
659 def is_invertible(self):
660 """
661 Return whether or not this element is invertible.
662
663 We can't use the superclass method because it relies on
664 the algebra being associative.
665
666 ALGORITHM:
667
668 The usual way to do this is to check if the determinant is
669 zero, but we need the characteristic polynomial for the
670 determinant. The minimal polynomial is a lot easier to get,
671 so we use Corollary 2 in Chapter V of Koecher to check
672 whether or not the paren't algebra's zero element is a root
673 of this element's minimal polynomial.
674
675 TESTS:
676
677 The identity element is always invertible::
678
679 sage: set_random_seed()
680 sage: J = random_eja()
681 sage: J.one().is_invertible()
682 True
683
684 The zero element is never invertible::
685
686 sage: set_random_seed()
687 sage: J = random_eja()
688 sage: J.zero().is_invertible()
689 False
690
691 """
692 zero = self.parent().zero()
693 p = self.minimal_polynomial()
694 return not (p(zero) == zero)
695
696
697 def is_nilpotent(self):
698 """
699 Return whether or not some power of this element is zero.
700
701 The superclass method won't work unless we're in an
702 associative algebra, and we aren't. However, we generate
703 an assocoative subalgebra and we're nilpotent there if and
704 only if we're nilpotent here (probably).
705
706 TESTS:
707
708 The identity element is never nilpotent::
709
710 sage: set_random_seed()
711 sage: random_eja().one().is_nilpotent()
712 False
713
714 The additive identity is always nilpotent::
715
716 sage: set_random_seed()
717 sage: random_eja().zero().is_nilpotent()
718 True
719
720 """
721 # The element we're going to call "is_nilpotent()" on.
722 # Either myself, interpreted as an element of a finite-
723 # dimensional algebra, or an element of an associative
724 # subalgebra.
725 elt = None
726
727 if self.parent().is_associative():
728 elt = FiniteDimensionalAlgebraElement(self.parent(), self)
729 else:
730 V = self.span_of_powers()
731 assoc_subalg = self.subalgebra_generated_by()
732 # Mis-design warning: the basis used for span_of_powers()
733 # and subalgebra_generated_by() must be the same, and in
734 # the same order!
735 elt = assoc_subalg(V.coordinates(self.vector()))
736
737 # Recursive call, but should work since elt lives in an
738 # associative algebra.
739 return elt.is_nilpotent()
740
741
742 def is_regular(self):
743 """
744 Return whether or not this is a regular element.
745
746 EXAMPLES:
747
748 The identity element always has degree one, but any element
749 linearly-independent from it is regular::
750
751 sage: J = JordanSpinEJA(5)
752 sage: J.one().is_regular()
753 False
754 sage: e0, e1, e2, e3, e4 = J.gens() # e0 is the identity
755 sage: for x in J.gens():
756 ....: (J.one() + x).is_regular()
757 False
758 True
759 True
760 True
761 True
762
763 """
764 return self.degree() == self.parent().rank()
765
766
767 def degree(self):
768 """
769 Compute the degree of this element the straightforward way
770 according to the definition; by appending powers to a list
771 and figuring out its dimension (that is, whether or not
772 they're linearly dependent).
773
774 EXAMPLES::
775
776 sage: J = JordanSpinEJA(4)
777 sage: J.one().degree()
778 1
779 sage: e0,e1,e2,e3 = J.gens()
780 sage: (e0 - e1).degree()
781 2
782
783 In the spin factor algebra (of rank two), all elements that
784 aren't multiples of the identity are regular::
785
786 sage: set_random_seed()
787 sage: n = ZZ.random_element(1,10)
788 sage: J = JordanSpinEJA(n)
789 sage: x = J.random_element()
790 sage: x == x.coefficient(0)*J.one() or x.degree() == 2
791 True
792
793 """
794 return self.span_of_powers().dimension()
795
796
797 def minimal_polynomial(self):
798 """
799 Return the minimal polynomial of this element,
800 as a function of the variable `t`.
801
802 ALGORITHM:
803
804 We restrict ourselves to the associative subalgebra
805 generated by this element, and then return the minimal
806 polynomial of this element's operator matrix (in that
807 subalgebra). This works by Baes Proposition 2.3.16.
808
809 TESTS:
810
811 The minimal polynomial of the identity and zero elements are
812 always the same::
813
814 sage: set_random_seed()
815 sage: J = random_eja()
816 sage: J.one().minimal_polynomial()
817 t - 1
818 sage: J.zero().minimal_polynomial()
819 t
820
821 The degree of an element is (by one definition) the degree
822 of its minimal polynomial::
823
824 sage: set_random_seed()
825 sage: x = random_eja().random_element()
826 sage: x.degree() == x.minimal_polynomial().degree()
827 True
828
829 The minimal polynomial and the characteristic polynomial coincide
830 and are known (see Alizadeh, Example 11.11) for all elements of
831 the spin factor algebra that aren't scalar multiples of the
832 identity::
833
834 sage: set_random_seed()
835 sage: n = ZZ.random_element(2,10)
836 sage: J = JordanSpinEJA(n)
837 sage: y = J.random_element()
838 sage: while y == y.coefficient(0)*J.one():
839 ....: y = J.random_element()
840 sage: y0 = y.vector()[0]
841 sage: y_bar = y.vector()[1:]
842 sage: actual = y.minimal_polynomial()
843 sage: t = PolynomialRing(J.base_ring(),'t').gen(0)
844 sage: expected = t^2 - 2*y0*t + (y0^2 - norm(y_bar)^2)
845 sage: bool(actual == expected)
846 True
847
848 The minimal polynomial should always kill its element::
849
850 sage: set_random_seed()
851 sage: x = random_eja().random_element()
852 sage: p = x.minimal_polynomial()
853 sage: x.apply_univariate_polynomial(p)
854 0
855
856 """
857 V = self.span_of_powers()
858 assoc_subalg = self.subalgebra_generated_by()
859 # Mis-design warning: the basis used for span_of_powers()
860 # and subalgebra_generated_by() must be the same, and in
861 # the same order!
862 elt = assoc_subalg(V.coordinates(self.vector()))
863
864 # We get back a symbolic polynomial in 'x' but want a real
865 # polynomial in 't'.
866 p_of_x = elt.operator_matrix().minimal_polynomial()
867 return p_of_x.change_variable_name('t')
868
869
870 def natural_representation(self):
871 """
872 Return a more-natural representation of this element.
873
874 Every finite-dimensional Euclidean Jordan Algebra is a
875 direct sum of five simple algebras, four of which comprise
876 Hermitian matrices. This method returns the original
877 "natural" representation of this element as a Hermitian
878 matrix, if it has one. If not, you get the usual representation.
879
880 EXAMPLES::
881
882 sage: J = ComplexHermitianEJA(3)
883 sage: J.one()
884 e0 + e5 + e8
885 sage: J.one().natural_representation()
886 [1 0 0 0 0 0]
887 [0 1 0 0 0 0]
888 [0 0 1 0 0 0]
889 [0 0 0 1 0 0]
890 [0 0 0 0 1 0]
891 [0 0 0 0 0 1]
892
893 ::
894
895 sage: J = QuaternionHermitianEJA(3)
896 sage: J.one()
897 e0 + e9 + e14
898 sage: J.one().natural_representation()
899 [1 0 0 0 0 0 0 0 0 0 0 0]
900 [0 1 0 0 0 0 0 0 0 0 0 0]
901 [0 0 1 0 0 0 0 0 0 0 0 0]
902 [0 0 0 1 0 0 0 0 0 0 0 0]
903 [0 0 0 0 1 0 0 0 0 0 0 0]
904 [0 0 0 0 0 1 0 0 0 0 0 0]
905 [0 0 0 0 0 0 1 0 0 0 0 0]
906 [0 0 0 0 0 0 0 1 0 0 0 0]
907 [0 0 0 0 0 0 0 0 1 0 0 0]
908 [0 0 0 0 0 0 0 0 0 1 0 0]
909 [0 0 0 0 0 0 0 0 0 0 1 0]
910 [0 0 0 0 0 0 0 0 0 0 0 1]
911
912 """
913 B = self.parent().natural_basis()
914 W = B[0].matrix_space()
915 return W.linear_combination(zip(self.vector(), B))
916
917
918 def operator_matrix(self):
919 """
920 Return the matrix that represents left- (or right-)
921 multiplication by this element in the parent algebra.
922
923 We have to override this because the superclass method
924 returns a matrix that acts on row vectors (that is, on
925 the right).
926
927 EXAMPLES:
928
929 Test the first polarization identity from my notes, Koecher Chapter
930 III, or from Baes (2.3)::
931
932 sage: set_random_seed()
933 sage: J = random_eja()
934 sage: x = J.random_element()
935 sage: y = J.random_element()
936 sage: Lx = x.operator_matrix()
937 sage: Ly = y.operator_matrix()
938 sage: Lxx = (x*x).operator_matrix()
939 sage: Lxy = (x*y).operator_matrix()
940 sage: bool(2*Lx*Lxy + Ly*Lxx == 2*Lxy*Lx + Lxx*Ly)
941 True
942
943 Test the second polarization identity from my notes or from
944 Baes (2.4)::
945
946 sage: set_random_seed()
947 sage: J = random_eja()
948 sage: x = J.random_element()
949 sage: y = J.random_element()
950 sage: z = J.random_element()
951 sage: Lx = x.operator_matrix()
952 sage: Ly = y.operator_matrix()
953 sage: Lz = z.operator_matrix()
954 sage: Lzy = (z*y).operator_matrix()
955 sage: Lxy = (x*y).operator_matrix()
956 sage: Lxz = (x*z).operator_matrix()
957 sage: bool(Lx*Lzy + Lz*Lxy + Ly*Lxz == Lzy*Lx + Lxy*Lz + Lxz*Ly)
958 True
959
960 Test the third polarization identity from my notes or from
961 Baes (2.5)::
962
963 sage: set_random_seed()
964 sage: J = random_eja()
965 sage: u = J.random_element()
966 sage: y = J.random_element()
967 sage: z = J.random_element()
968 sage: Lu = u.operator_matrix()
969 sage: Ly = y.operator_matrix()
970 sage: Lz = z.operator_matrix()
971 sage: Lzy = (z*y).operator_matrix()
972 sage: Luy = (u*y).operator_matrix()
973 sage: Luz = (u*z).operator_matrix()
974 sage: Luyz = (u*(y*z)).operator_matrix()
975 sage: lhs = Lu*Lzy + Lz*Luy + Ly*Luz
976 sage: rhs = Luyz + Ly*Lu*Lz + Lz*Lu*Ly
977 sage: bool(lhs == rhs)
978 True
979
980 """
981 fda_elt = FiniteDimensionalAlgebraElement(self.parent(), self)
982 return fda_elt.matrix().transpose()
983
984
985 def quadratic_representation(self, other=None):
986 """
987 Return the quadratic representation of this element.
988
989 EXAMPLES:
990
991 The explicit form in the spin factor algebra is given by
992 Alizadeh's Example 11.12::
993
994 sage: set_random_seed()
995 sage: n = ZZ.random_element(1,10)
996 sage: J = JordanSpinEJA(n)
997 sage: x = J.random_element()
998 sage: x_vec = x.vector()
999 sage: x0 = x_vec[0]
1000 sage: x_bar = x_vec[1:]
1001 sage: A = matrix(QQ, 1, [x_vec.inner_product(x_vec)])
1002 sage: B = 2*x0*x_bar.row()
1003 sage: C = 2*x0*x_bar.column()
1004 sage: D = identity_matrix(QQ, n-1)
1005 sage: D = (x0^2 - x_bar.inner_product(x_bar))*D
1006 sage: D = D + 2*x_bar.tensor_product(x_bar)
1007 sage: Q = block_matrix(2,2,[A,B,C,D])
1008 sage: Q == x.quadratic_representation()
1009 True
1010
1011 Test all of the properties from Theorem 11.2 in Alizadeh::
1012
1013 sage: set_random_seed()
1014 sage: J = random_eja()
1015 sage: x = J.random_element()
1016 sage: y = J.random_element()
1017
1018 Property 1:
1019
1020 sage: actual = x.quadratic_representation(y)
1021 sage: expected = ( (x+y).quadratic_representation()
1022 ....: -x.quadratic_representation()
1023 ....: -y.quadratic_representation() ) / 2
1024 sage: actual == expected
1025 True
1026
1027 Property 2:
1028
1029 sage: alpha = QQ.random_element()
1030 sage: actual = (alpha*x).quadratic_representation()
1031 sage: expected = (alpha^2)*x.quadratic_representation()
1032 sage: actual == expected
1033 True
1034
1035 Property 5:
1036
1037 sage: Qy = y.quadratic_representation()
1038 sage: actual = J(Qy*x.vector()).quadratic_representation()
1039 sage: expected = Qy*x.quadratic_representation()*Qy
1040 sage: actual == expected
1041 True
1042
1043 Property 6:
1044
1045 sage: k = ZZ.random_element(1,10)
1046 sage: actual = (x^k).quadratic_representation()
1047 sage: expected = (x.quadratic_representation())^k
1048 sage: actual == expected
1049 True
1050
1051 """
1052 if other is None:
1053 other=self
1054 elif not other in self.parent():
1055 raise TypeError("'other' must live in the same algebra")
1056
1057 L = self.operator_matrix()
1058 M = other.operator_matrix()
1059 return ( L*M + M*L - (self*other).operator_matrix() )
1060
1061
1062 def span_of_powers(self):
1063 """
1064 Return the vector space spanned by successive powers of
1065 this element.
1066 """
1067 # The dimension of the subalgebra can't be greater than
1068 # the big algebra, so just put everything into a list
1069 # and let span() get rid of the excess.
1070 #
1071 # We do the extra ambient_vector_space() in case we're messing
1072 # with polynomials and the direct parent is a module.
1073 V = self.vector().parent().ambient_vector_space()
1074 return V.span( (self**d).vector() for d in xrange(V.dimension()) )
1075
1076
1077 def subalgebra_generated_by(self):
1078 """
1079 Return the associative subalgebra of the parent EJA generated
1080 by this element.
1081
1082 TESTS::
1083
1084 sage: set_random_seed()
1085 sage: x = random_eja().random_element()
1086 sage: x.subalgebra_generated_by().is_associative()
1087 True
1088
1089 Squaring in the subalgebra should be the same thing as
1090 squaring in the superalgebra::
1091
1092 sage: set_random_seed()
1093 sage: x = random_eja().random_element()
1094 sage: u = x.subalgebra_generated_by().random_element()
1095 sage: u.operator_matrix()*u.vector() == (u**2).vector()
1096 True
1097
1098 """
1099 # First get the subspace spanned by the powers of myself...
1100 V = self.span_of_powers()
1101 F = self.base_ring()
1102
1103 # Now figure out the entries of the right-multiplication
1104 # matrix for the successive basis elements b0, b1,... of
1105 # that subspace.
1106 mats = []
1107 for b_right in V.basis():
1108 eja_b_right = self.parent()(b_right)
1109 b_right_rows = []
1110 # The first row of the right-multiplication matrix by
1111 # b1 is what we get if we apply that matrix to b1. The
1112 # second row of the right multiplication matrix by b1
1113 # is what we get when we apply that matrix to b2...
1114 #
1115 # IMPORTANT: this assumes that all vectors are COLUMN
1116 # vectors, unlike our superclass (which uses row vectors).
1117 for b_left in V.basis():
1118 eja_b_left = self.parent()(b_left)
1119 # Multiply in the original EJA, but then get the
1120 # coordinates from the subalgebra in terms of its
1121 # basis.
1122 this_row = V.coordinates((eja_b_left*eja_b_right).vector())
1123 b_right_rows.append(this_row)
1124 b_right_matrix = matrix(F, b_right_rows)
1125 mats.append(b_right_matrix)
1126
1127 # It's an algebra of polynomials in one element, and EJAs
1128 # are power-associative.
1129 #
1130 # TODO: choose generator names intelligently.
1131 return FiniteDimensionalEuclideanJordanAlgebra(F, mats, assume_associative=True, names='f')
1132
1133
1134 def subalgebra_idempotent(self):
1135 """
1136 Find an idempotent in the associative subalgebra I generate
1137 using Proposition 2.3.5 in Baes.
1138
1139 TESTS::
1140
1141 sage: set_random_seed()
1142 sage: J = RealCartesianProductEJA(5)
1143 sage: c = J.random_element().subalgebra_idempotent()
1144 sage: c^2 == c
1145 True
1146 sage: J = JordanSpinEJA(5)
1147 sage: c = J.random_element().subalgebra_idempotent()
1148 sage: c^2 == c
1149 True
1150
1151 """
1152 if self.is_nilpotent():
1153 raise ValueError("this only works with non-nilpotent elements!")
1154
1155 V = self.span_of_powers()
1156 J = self.subalgebra_generated_by()
1157 # Mis-design warning: the basis used for span_of_powers()
1158 # and subalgebra_generated_by() must be the same, and in
1159 # the same order!
1160 u = J(V.coordinates(self.vector()))
1161
1162 # The image of the matrix of left-u^m-multiplication
1163 # will be minimal for some natural number s...
1164 s = 0
1165 minimal_dim = V.dimension()
1166 for i in xrange(1, V.dimension()):
1167 this_dim = (u**i).operator_matrix().image().dimension()
1168 if this_dim < minimal_dim:
1169 minimal_dim = this_dim
1170 s = i
1171
1172 # Now minimal_matrix should correspond to the smallest
1173 # non-zero subspace in Baes's (or really, Koecher's)
1174 # proposition.
1175 #
1176 # However, we need to restrict the matrix to work on the
1177 # subspace... or do we? Can't we just solve, knowing that
1178 # A(c) = u^(s+1) should have a solution in the big space,
1179 # too?
1180 #
1181 # Beware, solve_right() means that we're using COLUMN vectors.
1182 # Our FiniteDimensionalAlgebraElement superclass uses rows.
1183 u_next = u**(s+1)
1184 A = u_next.operator_matrix()
1185 c_coordinates = A.solve_right(u_next.vector())
1186
1187 # Now c_coordinates is the idempotent we want, but it's in
1188 # the coordinate system of the subalgebra.
1189 #
1190 # We need the basis for J, but as elements of the parent algebra.
1191 #
1192 basis = [self.parent(v) for v in V.basis()]
1193 return self.parent().linear_combination(zip(c_coordinates, basis))
1194
1195
1196 def trace(self):
1197 """
1198 Return my trace, the sum of my eigenvalues.
1199
1200 EXAMPLES::
1201
1202 sage: J = JordanSpinEJA(3)
1203 sage: x = sum(J.gens())
1204 sage: x.trace()
1205 2
1206
1207 ::
1208
1209 sage: J = RealCartesianProductEJA(5)
1210 sage: J.one().trace()
1211 5
1212
1213 TESTS:
1214
1215 The trace of an element is a real number::
1216
1217 sage: set_random_seed()
1218 sage: J = random_eja()
1219 sage: J.random_element().trace() in J.base_ring()
1220 True
1221
1222 """
1223 P = self.parent()
1224 r = P.rank()
1225 p = P._charpoly_coeff(r-1)
1226 # The _charpoly_coeff function already adds the factor of
1227 # -1 to ensure that _charpoly_coeff(r-1) is really what
1228 # appears in front of t^{r-1} in the charpoly. However,
1229 # we want the negative of THAT for the trace.
1230 return -p(*self.vector())
1231
1232
1233 def trace_inner_product(self, other):
1234 """
1235 Return the trace inner product of myself and ``other``.
1236
1237 TESTS:
1238
1239 The trace inner product is commutative::
1240
1241 sage: set_random_seed()
1242 sage: J = random_eja()
1243 sage: x = J.random_element(); y = J.random_element()
1244 sage: x.trace_inner_product(y) == y.trace_inner_product(x)
1245 True
1246
1247 The trace inner product is bilinear::
1248
1249 sage: set_random_seed()
1250 sage: J = random_eja()
1251 sage: x = J.random_element()
1252 sage: y = J.random_element()
1253 sage: z = J.random_element()
1254 sage: a = QQ.random_element();
1255 sage: actual = (a*(x+z)).trace_inner_product(y)
1256 sage: expected = ( a*x.trace_inner_product(y) +
1257 ....: a*z.trace_inner_product(y) )
1258 sage: actual == expected
1259 True
1260 sage: actual = x.trace_inner_product(a*(y+z))
1261 sage: expected = ( a*x.trace_inner_product(y) +
1262 ....: a*x.trace_inner_product(z) )
1263 sage: actual == expected
1264 True
1265
1266 The trace inner product satisfies the compatibility
1267 condition in the definition of a Euclidean Jordan algebra::
1268
1269 sage: set_random_seed()
1270 sage: J = random_eja()
1271 sage: x = J.random_element()
1272 sage: y = J.random_element()
1273 sage: z = J.random_element()
1274 sage: (x*y).trace_inner_product(z) == y.trace_inner_product(x*z)
1275 True
1276
1277 """
1278 if not other in self.parent():
1279 raise TypeError("'other' must live in the same algebra")
1280
1281 return (self*other).trace()
1282
1283
1284 class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra):
1285 """
1286 Return the Euclidean Jordan Algebra corresponding to the set
1287 `R^n` under the Hadamard product.
1288
1289 Note: this is nothing more than the Cartesian product of ``n``
1290 copies of the spin algebra. Once Cartesian product algebras
1291 are implemented, this can go.
1292
1293 EXAMPLES:
1294
1295 This multiplication table can be verified by hand::
1296
1297 sage: J = RealCartesianProductEJA(3)
1298 sage: e0,e1,e2 = J.gens()
1299 sage: e0*e0
1300 e0
1301 sage: e0*e1
1302 0
1303 sage: e0*e2
1304 0
1305 sage: e1*e1
1306 e1
1307 sage: e1*e2
1308 0
1309 sage: e2*e2
1310 e2
1311
1312 """
1313 @staticmethod
1314 def __classcall_private__(cls, n, field=QQ):
1315 # The FiniteDimensionalAlgebra constructor takes a list of
1316 # matrices, the ith representing right multiplication by the ith
1317 # basis element in the vector space. So if e_1 = (1,0,0), then
1318 # right (Hadamard) multiplication of x by e_1 picks out the first
1319 # component of x; and likewise for the ith basis element e_i.
1320 Qs = [ matrix(field, n, n, lambda k,j: 1*(k == j == i))
1321 for i in xrange(n) ]
1322
1323 fdeja = super(RealCartesianProductEJA, cls)
1324 return fdeja.__classcall_private__(cls, field, Qs, rank=n)
1325
1326 def inner_product(self, x, y):
1327 return _usual_ip(x,y)
1328
1329
1330 def random_eja():
1331 """
1332 Return a "random" finite-dimensional Euclidean Jordan Algebra.
1333
1334 ALGORITHM:
1335
1336 For now, we choose a random natural number ``n`` (greater than zero)
1337 and then give you back one of the following:
1338
1339 * The cartesian product of the rational numbers ``n`` times; this is
1340 ``QQ^n`` with the Hadamard product.
1341
1342 * The Jordan spin algebra on ``QQ^n``.
1343
1344 * The ``n``-by-``n`` rational symmetric matrices with the symmetric
1345 product.
1346
1347 * The ``n``-by-``n`` complex-rational Hermitian matrices embedded
1348 in the space of ``2n``-by-``2n`` real symmetric matrices.
1349
1350 * The ``n``-by-``n`` quaternion-rational Hermitian matrices embedded
1351 in the space of ``4n``-by-``4n`` real symmetric matrices.
1352
1353 Later this might be extended to return Cartesian products of the
1354 EJAs above.
1355
1356 TESTS::
1357
1358 sage: random_eja()
1359 Euclidean Jordan algebra of degree...
1360
1361 """
1362
1363 # The max_n component lets us choose different upper bounds on the
1364 # value "n" that gets passed to the constructor. This is needed
1365 # because e.g. R^{10} is reasonable to test, while the Hermitian
1366 # 10-by-10 quaternion matrices are not.
1367 (constructor, max_n) = choice([(RealCartesianProductEJA, 6),
1368 (JordanSpinEJA, 6),
1369 (RealSymmetricEJA, 5),
1370 (ComplexHermitianEJA, 4),
1371 (QuaternionHermitianEJA, 3)])
1372 n = ZZ.random_element(1, max_n)
1373 return constructor(n, field=QQ)
1374
1375
1376
1377 def _real_symmetric_basis(n, field=QQ):
1378 """
1379 Return a basis for the space of real symmetric n-by-n matrices.
1380 """
1381 # The basis of symmetric matrices, as matrices, in their R^(n-by-n)
1382 # coordinates.
1383 S = []
1384 for i in xrange(n):
1385 for j in xrange(i+1):
1386 Eij = matrix(field, n, lambda k,l: k==i and l==j)
1387 if i == j:
1388 Sij = Eij
1389 else:
1390 # Beware, orthogonal but not normalized!
1391 Sij = Eij + Eij.transpose()
1392 S.append(Sij)
1393 return tuple(S)
1394
1395
1396 def _complex_hermitian_basis(n, field=QQ):
1397 """
1398 Returns a basis for the space of complex Hermitian n-by-n matrices.
1399
1400 TESTS::
1401
1402 sage: set_random_seed()
1403 sage: n = ZZ.random_element(1,5)
1404 sage: all( M.is_symmetric() for M in _complex_hermitian_basis(n) )
1405 True
1406
1407 """
1408 F = QuadraticField(-1, 'I')
1409 I = F.gen()
1410
1411 # This is like the symmetric case, but we need to be careful:
1412 #
1413 # * We want conjugate-symmetry, not just symmetry.
1414 # * The diagonal will (as a result) be real.
1415 #
1416 S = []
1417 for i in xrange(n):
1418 for j in xrange(i+1):
1419 Eij = matrix(field, n, lambda k,l: k==i and l==j)
1420 if i == j:
1421 Sij = _embed_complex_matrix(Eij)
1422 S.append(Sij)
1423 else:
1424 # Beware, orthogonal but not normalized! The second one
1425 # has a minus because it's conjugated.
1426 Sij_real = _embed_complex_matrix(Eij + Eij.transpose())
1427 S.append(Sij_real)
1428 Sij_imag = _embed_complex_matrix(I*Eij - I*Eij.transpose())
1429 S.append(Sij_imag)
1430 return tuple(S)
1431
1432
1433 def _quaternion_hermitian_basis(n, field=QQ):
1434 """
1435 Returns a basis for the space of quaternion Hermitian n-by-n matrices.
1436
1437 TESTS::
1438
1439 sage: set_random_seed()
1440 sage: n = ZZ.random_element(1,5)
1441 sage: all( M.is_symmetric() for M in _quaternion_hermitian_basis(n) )
1442 True
1443
1444 """
1445 Q = QuaternionAlgebra(QQ,-1,-1)
1446 I,J,K = Q.gens()
1447
1448 # This is like the symmetric case, but we need to be careful:
1449 #
1450 # * We want conjugate-symmetry, not just symmetry.
1451 # * The diagonal will (as a result) be real.
1452 #
1453 S = []
1454 for i in xrange(n):
1455 for j in xrange(i+1):
1456 Eij = matrix(Q, n, lambda k,l: k==i and l==j)
1457 if i == j:
1458 Sij = _embed_quaternion_matrix(Eij)
1459 S.append(Sij)
1460 else:
1461 # Beware, orthogonal but not normalized! The second,
1462 # third, and fourth ones have a minus because they're
1463 # conjugated.
1464 Sij_real = _embed_quaternion_matrix(Eij + Eij.transpose())
1465 S.append(Sij_real)
1466 Sij_I = _embed_quaternion_matrix(I*Eij - I*Eij.transpose())
1467 S.append(Sij_I)
1468 Sij_J = _embed_quaternion_matrix(J*Eij - J*Eij.transpose())
1469 S.append(Sij_J)
1470 Sij_K = _embed_quaternion_matrix(K*Eij - K*Eij.transpose())
1471 S.append(Sij_K)
1472 return tuple(S)
1473
1474
1475 def _mat2vec(m):
1476 return vector(m.base_ring(), m.list())
1477
1478 def _vec2mat(v):
1479 return matrix(v.base_ring(), sqrt(v.degree()), v.list())
1480
1481 def _multiplication_table_from_matrix_basis(basis):
1482 """
1483 At least three of the five simple Euclidean Jordan algebras have the
1484 symmetric multiplication (A,B) |-> (AB + BA)/2, where the
1485 multiplication on the right is matrix multiplication. Given a basis
1486 for the underlying matrix space, this function returns a
1487 multiplication table (obtained by looping through the basis
1488 elements) for an algebra of those matrices. A reordered copy
1489 of the basis is also returned to work around the fact that
1490 the ``span()`` in this function will change the order of the basis
1491 from what we think it is, to... something else.
1492 """
1493 # In S^2, for example, we nominally have four coordinates even
1494 # though the space is of dimension three only. The vector space V
1495 # is supposed to hold the entire long vector, and the subspace W
1496 # of V will be spanned by the vectors that arise from symmetric
1497 # matrices. Thus for S^2, dim(V) == 4 and dim(W) == 3.
1498 field = basis[0].base_ring()
1499 dimension = basis[0].nrows()
1500
1501 V = VectorSpace(field, dimension**2)
1502 W = V.span( _mat2vec(s) for s in basis )
1503
1504 # Taking the span above reorders our basis (thanks, jerk!) so we
1505 # need to put our "matrix basis" in the same order as the
1506 # (reordered) vector basis.
1507 S = tuple( _vec2mat(b) for b in W.basis() )
1508
1509 Qs = []
1510 for s in S:
1511 # Brute force the multiplication-by-s matrix by looping
1512 # through all elements of the basis and doing the computation
1513 # to find out what the corresponding row should be. BEWARE:
1514 # these multiplication tables won't be symmetric! It therefore
1515 # becomes REALLY IMPORTANT that the underlying algebra
1516 # constructor uses ROW vectors and not COLUMN vectors. That's
1517 # why we're computing rows here and not columns.
1518 Q_rows = []
1519 for t in S:
1520 this_row = _mat2vec((s*t + t*s)/2)
1521 Q_rows.append(W.coordinates(this_row))
1522 Q = matrix(field, W.dimension(), Q_rows)
1523 Qs.append(Q)
1524
1525 return (Qs, S)
1526
1527
1528 def _embed_complex_matrix(M):
1529 """
1530 Embed the n-by-n complex matrix ``M`` into the space of real
1531 matrices of size 2n-by-2n via the map the sends each entry `z = a +
1532 bi` to the block matrix ``[[a,b],[-b,a]]``.
1533
1534 EXAMPLES::
1535
1536 sage: F = QuadraticField(-1,'i')
1537 sage: x1 = F(4 - 2*i)
1538 sage: x2 = F(1 + 2*i)
1539 sage: x3 = F(-i)
1540 sage: x4 = F(6)
1541 sage: M = matrix(F,2,[[x1,x2],[x3,x4]])
1542 sage: _embed_complex_matrix(M)
1543 [ 4 -2| 1 2]
1544 [ 2 4|-2 1]
1545 [-----+-----]
1546 [ 0 -1| 6 0]
1547 [ 1 0| 0 6]
1548
1549 TESTS:
1550
1551 Embedding is a homomorphism (isomorphism, in fact)::
1552
1553 sage: set_random_seed()
1554 sage: n = ZZ.random_element(5)
1555 sage: F = QuadraticField(-1, 'i')
1556 sage: X = random_matrix(F, n)
1557 sage: Y = random_matrix(F, n)
1558 sage: actual = _embed_complex_matrix(X) * _embed_complex_matrix(Y)
1559 sage: expected = _embed_complex_matrix(X*Y)
1560 sage: actual == expected
1561 True
1562
1563 """
1564 n = M.nrows()
1565 if M.ncols() != n:
1566 raise ValueError("the matrix 'M' must be square")
1567 field = M.base_ring()
1568 blocks = []
1569 for z in M.list():
1570 a = z.real()
1571 b = z.imag()
1572 blocks.append(matrix(field, 2, [[a,b],[-b,a]]))
1573
1574 # We can drop the imaginaries here.
1575 return block_matrix(field.base_ring(), n, blocks)
1576
1577
1578 def _unembed_complex_matrix(M):
1579 """
1580 The inverse of _embed_complex_matrix().
1581
1582 EXAMPLES::
1583
1584 sage: A = matrix(QQ,[ [ 1, 2, 3, 4],
1585 ....: [-2, 1, -4, 3],
1586 ....: [ 9, 10, 11, 12],
1587 ....: [-10, 9, -12, 11] ])
1588 sage: _unembed_complex_matrix(A)
1589 [ 2*i + 1 4*i + 3]
1590 [ 10*i + 9 12*i + 11]
1591
1592 TESTS:
1593
1594 Unembedding is the inverse of embedding::
1595
1596 sage: set_random_seed()
1597 sage: F = QuadraticField(-1, 'i')
1598 sage: M = random_matrix(F, 3)
1599 sage: _unembed_complex_matrix(_embed_complex_matrix(M)) == M
1600 True
1601
1602 """
1603 n = ZZ(M.nrows())
1604 if M.ncols() != n:
1605 raise ValueError("the matrix 'M' must be square")
1606 if not n.mod(2).is_zero():
1607 raise ValueError("the matrix 'M' must be a complex embedding")
1608
1609 F = QuadraticField(-1, 'i')
1610 i = F.gen()
1611
1612 # Go top-left to bottom-right (reading order), converting every
1613 # 2-by-2 block we see to a single complex element.
1614 elements = []
1615 for k in xrange(n/2):
1616 for j in xrange(n/2):
1617 submat = M[2*k:2*k+2,2*j:2*j+2]
1618 if submat[0,0] != submat[1,1]:
1619 raise ValueError('bad on-diagonal submatrix')
1620 if submat[0,1] != -submat[1,0]:
1621 raise ValueError('bad off-diagonal submatrix')
1622 z = submat[0,0] + submat[0,1]*i
1623 elements.append(z)
1624
1625 return matrix(F, n/2, elements)
1626
1627
1628 def _embed_quaternion_matrix(M):
1629 """
1630 Embed the n-by-n quaternion matrix ``M`` into the space of real
1631 matrices of size 4n-by-4n by first sending each quaternion entry
1632 `z = a + bi + cj + dk` to the block-complex matrix
1633 ``[[a + bi, c+di],[-c + di, a-bi]]`, and then embedding those into
1634 a real matrix.
1635
1636 EXAMPLES::
1637
1638 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1639 sage: i,j,k = Q.gens()
1640 sage: x = 1 + 2*i + 3*j + 4*k
1641 sage: M = matrix(Q, 1, [[x]])
1642 sage: _embed_quaternion_matrix(M)
1643 [ 1 2 3 4]
1644 [-2 1 -4 3]
1645 [-3 4 1 -2]
1646 [-4 -3 2 1]
1647
1648 Embedding is a homomorphism (isomorphism, in fact)::
1649
1650 sage: set_random_seed()
1651 sage: n = ZZ.random_element(5)
1652 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1653 sage: X = random_matrix(Q, n)
1654 sage: Y = random_matrix(Q, n)
1655 sage: actual = _embed_quaternion_matrix(X)*_embed_quaternion_matrix(Y)
1656 sage: expected = _embed_quaternion_matrix(X*Y)
1657 sage: actual == expected
1658 True
1659
1660 """
1661 quaternions = M.base_ring()
1662 n = M.nrows()
1663 if M.ncols() != n:
1664 raise ValueError("the matrix 'M' must be square")
1665
1666 F = QuadraticField(-1, 'i')
1667 i = F.gen()
1668
1669 blocks = []
1670 for z in M.list():
1671 t = z.coefficient_tuple()
1672 a = t[0]
1673 b = t[1]
1674 c = t[2]
1675 d = t[3]
1676 cplx_matrix = matrix(F, 2, [[ a + b*i, c + d*i],
1677 [-c + d*i, a - b*i]])
1678 blocks.append(_embed_complex_matrix(cplx_matrix))
1679
1680 # We should have real entries by now, so use the realest field
1681 # we've got for the return value.
1682 return block_matrix(quaternions.base_ring(), n, blocks)
1683
1684
1685 def _unembed_quaternion_matrix(M):
1686 """
1687 The inverse of _embed_quaternion_matrix().
1688
1689 EXAMPLES::
1690
1691 sage: M = matrix(QQ, [[ 1, 2, 3, 4],
1692 ....: [-2, 1, -4, 3],
1693 ....: [-3, 4, 1, -2],
1694 ....: [-4, -3, 2, 1]])
1695 sage: _unembed_quaternion_matrix(M)
1696 [1 + 2*i + 3*j + 4*k]
1697
1698 TESTS:
1699
1700 Unembedding is the inverse of embedding::
1701
1702 sage: set_random_seed()
1703 sage: Q = QuaternionAlgebra(QQ, -1, -1)
1704 sage: M = random_matrix(Q, 3)
1705 sage: _unembed_quaternion_matrix(_embed_quaternion_matrix(M)) == M
1706 True
1707
1708 """
1709 n = ZZ(M.nrows())
1710 if M.ncols() != n:
1711 raise ValueError("the matrix 'M' must be square")
1712 if not n.mod(4).is_zero():
1713 raise ValueError("the matrix 'M' must be a complex embedding")
1714
1715 Q = QuaternionAlgebra(QQ,-1,-1)
1716 i,j,k = Q.gens()
1717
1718 # Go top-left to bottom-right (reading order), converting every
1719 # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1
1720 # quaternion block.
1721 elements = []
1722 for l in xrange(n/4):
1723 for m in xrange(n/4):
1724 submat = _unembed_complex_matrix(M[4*l:4*l+4,4*m:4*m+4])
1725 if submat[0,0] != submat[1,1].conjugate():
1726 raise ValueError('bad on-diagonal submatrix')
1727 if submat[0,1] != -submat[1,0].conjugate():
1728 raise ValueError('bad off-diagonal submatrix')
1729 z = submat[0,0].real() + submat[0,0].imag()*i
1730 z += submat[0,1].real()*j + submat[0,1].imag()*k
1731 elements.append(z)
1732
1733 return matrix(Q, n/4, elements)
1734
1735
1736 # The usual inner product on R^n.
1737 def _usual_ip(x,y):
1738 return x.vector().inner_product(y.vector())
1739
1740 # The inner product used for the real symmetric simple EJA.
1741 # We keep it as a separate function because e.g. the complex
1742 # algebra uses the same inner product, except divided by 2.
1743 def _matrix_ip(X,Y):
1744 X_mat = X.natural_representation()
1745 Y_mat = Y.natural_representation()
1746 return (X_mat*Y_mat).trace()
1747
1748
1749 class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra):
1750 """
1751 The rank-n simple EJA consisting of real symmetric n-by-n
1752 matrices, the usual symmetric Jordan product, and the trace inner
1753 product. It has dimension `(n^2 + n)/2` over the reals.
1754
1755 EXAMPLES::
1756
1757 sage: J = RealSymmetricEJA(2)
1758 sage: e0, e1, e2 = J.gens()
1759 sage: e0*e0
1760 e0
1761 sage: e1*e1
1762 e0 + e2
1763 sage: e2*e2
1764 e2
1765
1766 TESTS:
1767
1768 The degree of this algebra is `(n^2 + n) / 2`::
1769
1770 sage: set_random_seed()
1771 sage: n = ZZ.random_element(1,5)
1772 sage: J = RealSymmetricEJA(n)
1773 sage: J.degree() == (n^2 + n)/2
1774 True
1775
1776 The Jordan multiplication is what we think it is::
1777
1778 sage: set_random_seed()
1779 sage: n = ZZ.random_element(1,5)
1780 sage: J = RealSymmetricEJA(n)
1781 sage: x = J.random_element()
1782 sage: y = J.random_element()
1783 sage: actual = (x*y).natural_representation()
1784 sage: X = x.natural_representation()
1785 sage: Y = y.natural_representation()
1786 sage: expected = (X*Y + Y*X)/2
1787 sage: actual == expected
1788 True
1789 sage: J(expected) == x*y
1790 True
1791
1792 """
1793 @staticmethod
1794 def __classcall_private__(cls, n, field=QQ):
1795 S = _real_symmetric_basis(n, field=field)
1796 (Qs, T) = _multiplication_table_from_matrix_basis(S)
1797
1798 fdeja = super(RealSymmetricEJA, cls)
1799 return fdeja.__classcall_private__(cls,
1800 field,
1801 Qs,
1802 rank=n,
1803 natural_basis=T)
1804
1805 def inner_product(self, x, y):
1806 return _matrix_ip(x,y)
1807
1808
1809 class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra):
1810 """
1811 The rank-n simple EJA consisting of complex Hermitian n-by-n
1812 matrices over the real numbers, the usual symmetric Jordan product,
1813 and the real-part-of-trace inner product. It has dimension `n^2` over
1814 the reals.
1815
1816 TESTS:
1817
1818 The degree of this algebra is `n^2`::
1819
1820 sage: set_random_seed()
1821 sage: n = ZZ.random_element(1,5)
1822 sage: J = ComplexHermitianEJA(n)
1823 sage: J.degree() == n^2
1824 True
1825
1826 The Jordan multiplication is what we think it is::
1827
1828 sage: set_random_seed()
1829 sage: n = ZZ.random_element(1,5)
1830 sage: J = ComplexHermitianEJA(n)
1831 sage: x = J.random_element()
1832 sage: y = J.random_element()
1833 sage: actual = (x*y).natural_representation()
1834 sage: X = x.natural_representation()
1835 sage: Y = y.natural_representation()
1836 sage: expected = (X*Y + Y*X)/2
1837 sage: actual == expected
1838 True
1839 sage: J(expected) == x*y
1840 True
1841
1842 """
1843 @staticmethod
1844 def __classcall_private__(cls, n, field=QQ):
1845 S = _complex_hermitian_basis(n)
1846 (Qs, T) = _multiplication_table_from_matrix_basis(S)
1847
1848 fdeja = super(ComplexHermitianEJA, cls)
1849 return fdeja.__classcall_private__(cls,
1850 field,
1851 Qs,
1852 rank=n,
1853 natural_basis=T)
1854
1855 def inner_product(self, x, y):
1856 # Since a+bi on the diagonal is represented as
1857 #
1858 # a + bi = [ a b ]
1859 # [ -b a ],
1860 #
1861 # we'll double-count the "a" entries if we take the trace of
1862 # the embedding.
1863 return _matrix_ip(x,y)/2
1864
1865
1866 class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra):
1867 """
1868 The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
1869 matrices, the usual symmetric Jordan product, and the
1870 real-part-of-trace inner product. It has dimension `2n^2 - n` over
1871 the reals.
1872
1873 TESTS:
1874
1875 The degree of this algebra is `n^2`::
1876
1877 sage: set_random_seed()
1878 sage: n = ZZ.random_element(1,5)
1879 sage: J = QuaternionHermitianEJA(n)
1880 sage: J.degree() == 2*(n^2) - n
1881 True
1882
1883 The Jordan multiplication is what we think it is::
1884
1885 sage: set_random_seed()
1886 sage: n = ZZ.random_element(1,5)
1887 sage: J = QuaternionHermitianEJA(n)
1888 sage: x = J.random_element()
1889 sage: y = J.random_element()
1890 sage: actual = (x*y).natural_representation()
1891 sage: X = x.natural_representation()
1892 sage: Y = y.natural_representation()
1893 sage: expected = (X*Y + Y*X)/2
1894 sage: actual == expected
1895 True
1896 sage: J(expected) == x*y
1897 True
1898
1899 """
1900 @staticmethod
1901 def __classcall_private__(cls, n, field=QQ):
1902 S = _quaternion_hermitian_basis(n)
1903 (Qs, T) = _multiplication_table_from_matrix_basis(S)
1904
1905 fdeja = super(QuaternionHermitianEJA, cls)
1906 return fdeja.__classcall_private__(cls,
1907 field,
1908 Qs,
1909 rank=n,
1910 natural_basis=T)
1911
1912 def inner_product(self, x, y):
1913 # Since a+bi+cj+dk on the diagonal is represented as
1914 #
1915 # a + bi +cj + dk = [ a b c d]
1916 # [ -b a -d c]
1917 # [ -c d a -b]
1918 # [ -d -c b a],
1919 #
1920 # we'll quadruple-count the "a" entries if we take the trace of
1921 # the embedding.
1922 return _matrix_ip(x,y)/4
1923
1924
1925 class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra):
1926 """
1927 The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
1928 with the usual inner product and jordan product ``x*y =
1929 (<x_bar,y_bar>, x0*y_bar + y0*x_bar)``. It has dimension `n` over
1930 the reals.
1931
1932 EXAMPLES:
1933
1934 This multiplication table can be verified by hand::
1935
1936 sage: J = JordanSpinEJA(4)
1937 sage: e0,e1,e2,e3 = J.gens()
1938 sage: e0*e0
1939 e0
1940 sage: e0*e1
1941 e1
1942 sage: e0*e2
1943 e2
1944 sage: e0*e3
1945 e3
1946 sage: e1*e2
1947 0
1948 sage: e1*e3
1949 0
1950 sage: e2*e3
1951 0
1952
1953 """
1954 @staticmethod
1955 def __classcall_private__(cls, n, field=QQ):
1956 Qs = []
1957 id_matrix = identity_matrix(field, n)
1958 for i in xrange(n):
1959 ei = id_matrix.column(i)
1960 Qi = zero_matrix(field, n)
1961 Qi.set_row(0, ei)
1962 Qi.set_column(0, ei)
1963 Qi += diagonal_matrix(n, [ei[0]]*n)
1964 # The addition of the diagonal matrix adds an extra ei[0] in the
1965 # upper-left corner of the matrix.
1966 Qi[0,0] = Qi[0,0] * ~field(2)
1967 Qs.append(Qi)
1968
1969 # The rank of the spin algebra is two, unless we're in a
1970 # one-dimensional ambient space (because the rank is bounded by
1971 # the ambient dimension).
1972 fdeja = super(JordanSpinEJA, cls)
1973 return fdeja.__classcall_private__(cls, field, Qs, rank=min(n,2))
1974
1975 def inner_product(self, x, y):
1976 return _usual_ip(x,y)
My personal library of Sage code.