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.
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
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
15 class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra
):
17 def __classcall_private__(cls
,
21 assume_associative
=False,
26 mult_table
= [b
.base_extend(field
) for b
in mult_table
]
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
)
33 cat
= MagmaticAlgebras(field
).FiniteDimensional().WithBasis()
34 cat
.or_subcategory(category
)
35 if assume_associative
:
36 cat
= cat
.Associative()
38 names
= normalize_names(n
, names
)
40 fda
= super(FiniteDimensionalEuclideanJordanAlgebra
, cls
)
41 return fda
.__classcall
__(cls
,
44 assume_associative
=assume_associative
,
48 natural_basis
=natural_basis
)
55 assume_associative
=False,
62 By definition, Jordan multiplication commutes::
64 sage: set_random_seed()
65 sage: J = random_eja()
66 sage: x = J.random_element()
67 sage: y = J.random_element()
73 self
._natural
_basis
= natural_basis
74 self
._multiplication
_table
= mult_table
75 fda
= super(FiniteDimensionalEuclideanJordanAlgebra
, self
)
84 Return a string representation of ``self``.
86 fmt
= "Euclidean Jordan algebra of degree {} over {}"
87 return fmt
.format(self
.degree(), self
.base_ring())
90 def _a_regular_element(self
):
92 Guess a regular element. Needed to compute the basis for our
93 characteristic polynomial coefficients.
96 z
= self
.sum( (i
+1)*gs
[i
] for i
in range(len(gs
)) )
97 if not z
.is_regular():
98 raise ValueError("don't know a regular element")
103 def _charpoly_basis_space(self
):
105 Return the vector space spanned by the basis used in our
106 characteristic polynomial coefficients. This is used not only to
107 compute those coefficients, but also any time we need to
108 evaluate the coefficients (like when we compute the trace or
111 z
= self
._a
_regular
_element
()
112 V
= z
.vector().parent().ambient_vector_space()
113 V1
= V
.span_of_basis( (z
**k
).vector() for k
in range(self
.rank()) )
114 b
= (V1
.basis() + V1
.complement().basis())
115 return V
.span_of_basis(b
)
119 def _charpoly_coeff(self
, i
):
121 Return the coefficient polynomial "a_{i}" of this algebra's
122 general characteristic polynomial.
124 Having this be a separate cached method lets us compute and
125 store the trace/determinant (a_{r-1} and a_{0} respectively)
126 separate from the entire characteristic polynomial.
128 (A_of_x
, x
, xr
, detA
) = self
._charpoly
_matrix
_system
()
129 R
= A_of_x
.base_ring()
131 # Guaranteed by theory
134 # Danger: the in-place modification is done for performance
135 # reasons (reconstructing a matrix with huge polynomial
136 # entries is slow), but I don't know how cached_method works,
137 # so it's highly possible that we're modifying some global
138 # list variable by reference, here. In other words, you
139 # probably shouldn't call this method twice on the same
140 # algebra, at the same time, in two threads
141 Ai_orig
= A_of_x
.column(i
)
142 A_of_x
.set_column(i
,xr
)
143 numerator
= A_of_x
.det()
144 A_of_x
.set_column(i
,Ai_orig
)
146 # We're relying on the theory here to ensure that each a_i is
147 # indeed back in R, and the added negative signs are to make
148 # the whole charpoly expression sum to zero.
149 return R(-numerator
/detA
)
153 def _charpoly_matrix_system(self
):
155 Compute the matrix whose entries A_ij are polynomials in
156 X1,...,XN, the vector ``x`` of variables X1,...,XN, the vector
157 corresponding to `x^r` and the determinent of the matrix A =
158 [A_ij]. In other words, all of the fixed (cachable) data needed
159 to compute the coefficients of the characteristic polynomial.
164 # Construct a new algebra over a multivariate polynomial ring...
165 names
= ['X' + str(i
) for i
in range(1,n
+1)]
166 R
= PolynomialRing(self
.base_ring(), names
)
167 J
= FiniteDimensionalEuclideanJordanAlgebra(R
,
168 self
._multiplication
_table
,
171 idmat
= identity_matrix(J
.base_ring(), n
)
173 W
= self
._charpoly
_basis
_space
()
174 W
= W
.change_ring(R
.fraction_field())
176 # Starting with the standard coordinates x = (X1,X2,...,Xn)
177 # and then converting the entries to W-coordinates allows us
178 # to pass in the standard coordinates to the charpoly and get
179 # back the right answer. Specifically, with x = (X1,X2,...,Xn),
182 # W.coordinates(x^2) eval'd at (standard z-coords)
186 # W-coords of (standard coords of x^2 eval'd at std-coords of z)
188 # We want the middle equivalent thing in our matrix, but use
189 # the first equivalent thing instead so that we can pass in
190 # standard coordinates.
191 x
= J(vector(R
, R
.gens()))
192 l1
= [column_matrix(W
.coordinates((x
**k
).vector())) for k
in range(r
)]
193 l2
= [idmat
.column(k
-1).column() for k
in range(r
+1, n
+1)]
194 A_of_x
= block_matrix(R
, 1, n
, (l1
+ l2
))
195 xr
= W
.coordinates((x
**r
).vector())
196 return (A_of_x
, x
, xr
, A_of_x
.det())
200 def characteristic_polynomial(self
):
205 This implementation doesn't guarantee that the polynomial
206 denominator in the coefficients is not identically zero, so
207 theoretically it could crash. The way that this is handled
208 in e.g. Faraut and Koranyi is to use a basis that guarantees
209 the denominator is non-zero. But, doing so requires knowledge
210 of at least one regular element, and we don't even know how
211 to do that. The trade-off is that, if we use the standard basis,
212 the resulting polynomial will accept the "usual" coordinates. In
213 other words, we don't have to do a change of basis before e.g.
214 computing the trace or determinant.
218 The characteristic polynomial in the spin algebra is given in
219 Alizadeh, Example 11.11::
221 sage: J = JordanSpinEJA(3)
222 sage: p = J.characteristic_polynomial(); p
223 X1^2 - X2^2 - X3^2 + (-2*t)*X1 + t^2
224 sage: xvec = J.one().vector()
232 # The list of coefficient polynomials a_1, a_2, ..., a_n.
233 a
= [ self
._charpoly
_coeff
(i
) for i
in range(n
) ]
235 # We go to a bit of trouble here to reorder the
236 # indeterminates, so that it's easier to evaluate the
237 # characteristic polynomial at x's coordinates and get back
238 # something in terms of t, which is what we want.
240 S
= PolynomialRing(self
.base_ring(),'t')
242 S
= PolynomialRing(S
, R
.variable_names())
245 # Note: all entries past the rth should be zero. The
246 # coefficient of the highest power (x^r) is 1, but it doesn't
247 # appear in the solution vector which contains coefficients
248 # for the other powers (to make them sum to x^r).
250 a
[r
] = 1 # corresponds to x^r
252 # When the rank is equal to the dimension, trying to
253 # assign a[r] goes out-of-bounds.
254 a
.append(1) # corresponds to x^r
256 return sum( a
[k
]*(t
**k
) for k
in range(len(a
)) )
259 def inner_product(self
, x
, y
):
261 The inner product associated with this Euclidean Jordan algebra.
263 Defaults to the trace inner product, but can be overridden by
264 subclasses if they are sure that the necessary properties are
269 The inner product must satisfy its axiom for this algebra to truly
270 be a Euclidean Jordan Algebra::
272 sage: set_random_seed()
273 sage: J = random_eja()
274 sage: x = J.random_element()
275 sage: y = J.random_element()
276 sage: z = J.random_element()
277 sage: (x*y).inner_product(z) == y.inner_product(x*z)
281 if (not x
in self
) or (not y
in self
):
282 raise TypeError("arguments must live in this algebra")
283 return x
.trace_inner_product(y
)
286 def natural_basis(self
):
288 Return a more-natural representation of this algebra's basis.
290 Every finite-dimensional Euclidean Jordan Algebra is a direct
291 sum of five simple algebras, four of which comprise Hermitian
292 matrices. This method returns the original "natural" basis
293 for our underlying vector space. (Typically, the natural basis
294 is used to construct the multiplication table in the first place.)
296 Note that this will always return a matrix. The standard basis
297 in `R^n` will be returned as `n`-by-`1` column matrices.
301 sage: J = RealSymmetricEJA(2)
304 sage: J.natural_basis()
312 sage: J = JordanSpinEJA(2)
315 sage: J.natural_basis()
322 if self
._natural
_basis
is None:
323 return tuple( b
.vector().column() for b
in self
.basis() )
325 return self
._natural
_basis
330 Return the rank of this EJA.
332 if self
._rank
is None:
333 raise ValueError("no rank specified at genesis")
338 class Element(FiniteDimensionalAlgebraElement
):
340 An element of a Euclidean Jordan algebra.
343 def __init__(self
, A
, elt
=None):
347 The identity in `S^n` is converted to the identity in the EJA::
349 sage: J = RealSymmetricEJA(3)
350 sage: I = identity_matrix(QQ,3)
351 sage: J(I) == J.one()
354 This skew-symmetric matrix can't be represented in the EJA::
356 sage: J = RealSymmetricEJA(3)
357 sage: A = matrix(QQ,3, lambda i,j: i-j)
359 Traceback (most recent call last):
361 ArithmeticError: vector is not in free module
364 # Goal: if we're given a matrix, and if it lives in our
365 # parent algebra's "natural ambient space," convert it
366 # into an algebra element.
368 # The catch is, we make a recursive call after converting
369 # the given matrix into a vector that lives in the algebra.
370 # This we need to try the parent class initializer first,
371 # to avoid recursing forever if we're given something that
372 # already fits into the algebra, but also happens to live
373 # in the parent's "natural ambient space" (this happens with
376 FiniteDimensionalAlgebraElement
.__init
__(self
, A
, elt
)
378 natural_basis
= A
.natural_basis()
379 if elt
in natural_basis
[0].matrix_space():
380 # Thanks for nothing! Matrix spaces aren't vector
381 # spaces in Sage, so we have to figure out its
382 # natural-basis coordinates ourselves.
383 V
= VectorSpace(elt
.base_ring(), elt
.nrows()**2)
384 W
= V
.span( _mat2vec(s
) for s
in natural_basis
)
385 coords
= W
.coordinates(_mat2vec(elt
))
386 FiniteDimensionalAlgebraElement
.__init
__(self
, A
, coords
)
388 def __pow__(self
, n
):
390 Return ``self`` raised to the power ``n``.
392 Jordan algebras are always power-associative; see for
393 example Faraut and Koranyi, Proposition II.1.2 (ii).
397 We have to override this because our superclass uses row vectors
398 instead of column vectors! We, on the other hand, assume column
403 sage: set_random_seed()
404 sage: x = random_eja().random_element()
405 sage: x.operator_matrix()*x.vector() == (x^2).vector()
408 A few examples of power-associativity::
410 sage: set_random_seed()
411 sage: x = random_eja().random_element()
412 sage: x*(x*x)*(x*x) == x^5
414 sage: (x*x)*(x*x*x) == x^5
417 We also know that powers operator-commute (Koecher, Chapter
420 sage: set_random_seed()
421 sage: x = random_eja().random_element()
422 sage: m = ZZ.random_element(0,10)
423 sage: n = ZZ.random_element(0,10)
424 sage: Lxm = (x^m).operator_matrix()
425 sage: Lxn = (x^n).operator_matrix()
426 sage: Lxm*Lxn == Lxn*Lxm
436 return A( (self
.operator_matrix()**(n
-1))*self
.vector() )
439 def apply_univariate_polynomial(self
, p
):
441 Apply the univariate polynomial ``p`` to this element.
443 A priori, SageMath won't allow us to apply a univariate
444 polynomial to an element of an EJA, because we don't know
445 that EJAs are rings (they are usually not associative). Of
446 course, we know that EJAs are power-associative, so the
447 operation is ultimately kosher. This function sidesteps
448 the CAS to get the answer we want and expect.
452 sage: R = PolynomialRing(QQ, 't')
454 sage: p = t^4 - t^3 + 5*t - 2
455 sage: J = RealCartesianProductEJA(5)
456 sage: J.one().apply_univariate_polynomial(p) == 3*J.one()
461 We should always get back an element of the algebra::
463 sage: set_random_seed()
464 sage: p = PolynomialRing(QQ, 't').random_element()
465 sage: J = random_eja()
466 sage: x = J.random_element()
467 sage: x.apply_univariate_polynomial(p) in J
471 if len(p
.variables()) > 1:
472 raise ValueError("not a univariate polynomial")
475 # Convert the coeficcients to the parent's base ring,
476 # because a priori they might live in an (unnecessarily)
477 # larger ring for which P.sum() would fail below.
478 cs
= [ R(c
) for c
in p
.coefficients(sparse
=False) ]
479 return P
.sum( cs
[k
]*(self
**k
) for k
in range(len(cs
)) )
482 def characteristic_polynomial(self
):
484 Return the characteristic polynomial of this element.
488 The rank of `R^3` is three, and the minimal polynomial of
489 the identity element is `(t-1)` from which it follows that
490 the characteristic polynomial should be `(t-1)^3`::
492 sage: J = RealCartesianProductEJA(3)
493 sage: J.one().characteristic_polynomial()
494 t^3 - 3*t^2 + 3*t - 1
496 Likewise, the characteristic of the zero element in the
497 rank-three algebra `R^{n}` should be `t^{3}`::
499 sage: J = RealCartesianProductEJA(3)
500 sage: J.zero().characteristic_polynomial()
503 The characteristic polynomial of an element should evaluate
504 to zero on that element::
506 sage: set_random_seed()
507 sage: x = RealCartesianProductEJA(3).random_element()
508 sage: p = x.characteristic_polynomial()
509 sage: x.apply_univariate_polynomial(p)
513 p
= self
.parent().characteristic_polynomial()
514 return p(*self
.vector())
517 def inner_product(self
, other
):
519 Return the parent algebra's inner product of myself and ``other``.
523 The inner product in the Jordan spin algebra is the usual
524 inner product on `R^n` (this example only works because the
525 basis for the Jordan algebra is the standard basis in `R^n`)::
527 sage: J = JordanSpinEJA(3)
528 sage: x = vector(QQ,[1,2,3])
529 sage: y = vector(QQ,[4,5,6])
530 sage: x.inner_product(y)
532 sage: J(x).inner_product(J(y))
535 The inner product on `S^n` is `<X,Y> = trace(X*Y)`, where
536 multiplication is the usual matrix multiplication in `S^n`,
537 so the inner product of the identity matrix with itself
540 sage: J = RealSymmetricEJA(3)
541 sage: J.one().inner_product(J.one())
544 Likewise, the inner product on `C^n` is `<X,Y> =
545 Re(trace(X*Y))`, where we must necessarily take the real
546 part because the product of Hermitian matrices may not be
549 sage: J = ComplexHermitianEJA(3)
550 sage: J.one().inner_product(J.one())
553 Ditto for the quaternions::
555 sage: J = QuaternionHermitianEJA(3)
556 sage: J.one().inner_product(J.one())
561 Ensure that we can always compute an inner product, and that
562 it gives us back a real number::
564 sage: set_random_seed()
565 sage: J = random_eja()
566 sage: x = J.random_element()
567 sage: y = J.random_element()
568 sage: x.inner_product(y) in RR
574 raise TypeError("'other' must live in the same algebra")
576 return P
.inner_product(self
, other
)
579 def operator_commutes_with(self
, other
):
581 Return whether or not this element operator-commutes
586 The definition of a Jordan algebra says that any element
587 operator-commutes with its square::
589 sage: set_random_seed()
590 sage: x = random_eja().random_element()
591 sage: x.operator_commutes_with(x^2)
596 Test Lemma 1 from Chapter III of Koecher::
598 sage: set_random_seed()
599 sage: J = random_eja()
600 sage: u = J.random_element()
601 sage: v = J.random_element()
602 sage: lhs = u.operator_commutes_with(u*v)
603 sage: rhs = v.operator_commutes_with(u^2)
608 if not other
in self
.parent():
609 raise TypeError("'other' must live in the same algebra")
611 A
= self
.operator_matrix()
612 B
= other
.operator_matrix()
618 Return my determinant, the product of my eigenvalues.
622 sage: J = JordanSpinEJA(2)
623 sage: e0,e1 = J.gens()
624 sage: x = sum( J.gens() )
630 sage: J = JordanSpinEJA(3)
631 sage: e0,e1,e2 = J.gens()
632 sage: x = sum( J.gens() )
638 An element is invertible if and only if its determinant is
641 sage: set_random_seed()
642 sage: x = random_eja().random_element()
643 sage: x.is_invertible() == (x.det() != 0)
649 p
= P
._charpoly
_coeff
(0)
650 # The _charpoly_coeff function already adds the factor of
651 # -1 to ensure that _charpoly_coeff(0) is really what
652 # appears in front of t^{0} in the charpoly. However,
653 # we want (-1)^r times THAT for the determinant.
654 return ((-1)**r
)*p(*self
.vector())
659 Return the Jordan-multiplicative inverse of this element.
661 We can't use the superclass method because it relies on the
662 algebra being associative.
666 The inverse in the spin factor algebra is given in Alizadeh's
669 sage: set_random_seed()
670 sage: n = ZZ.random_element(1,10)
671 sage: J = JordanSpinEJA(n)
672 sage: x = J.random_element()
673 sage: while x.is_zero():
674 ....: x = J.random_element()
675 sage: x_vec = x.vector()
677 sage: x_bar = x_vec[1:]
678 sage: coeff = 1/(x0^2 - x_bar.inner_product(x_bar))
679 sage: inv_vec = x_vec.parent()([x0] + (-x_bar).list())
680 sage: x_inverse = coeff*inv_vec
681 sage: x.inverse() == J(x_inverse)
686 The identity element is its own inverse::
688 sage: set_random_seed()
689 sage: J = random_eja()
690 sage: J.one().inverse() == J.one()
693 If an element has an inverse, it acts like one::
695 sage: set_random_seed()
696 sage: J = random_eja()
697 sage: x = J.random_element()
698 sage: (not x.is_invertible()) or (x.inverse()*x == J.one())
702 if self
.parent().is_associative():
703 elt
= FiniteDimensionalAlgebraElement(self
.parent(), self
)
706 # TODO: we can do better once the call to is_invertible()
707 # doesn't crash on irregular elements.
708 #if not self.is_invertible():
709 # raise ValueError('element is not invertible')
711 # We do this a little different than the usual recursive
712 # call to a finite-dimensional algebra element, because we
713 # wind up with an inverse that lives in the subalgebra and
714 # we need information about the parent to convert it back.
715 V
= self
.span_of_powers()
716 assoc_subalg
= self
.subalgebra_generated_by()
717 # Mis-design warning: the basis used for span_of_powers()
718 # and subalgebra_generated_by() must be the same, and in
720 elt
= assoc_subalg(V
.coordinates(self
.vector()))
722 # This will be in the subalgebra's coordinates...
723 fda_elt
= FiniteDimensionalAlgebraElement(assoc_subalg
, elt
)
724 subalg_inverse
= fda_elt
.inverse()
726 # So we have to convert back...
727 basis
= [ self
.parent(v
) for v
in V
.basis() ]
728 pairs
= zip(subalg_inverse
.vector(), basis
)
729 return self
.parent().linear_combination(pairs
)
732 def is_invertible(self
):
734 Return whether or not this element is invertible.
736 We can't use the superclass method because it relies on
737 the algebra being associative.
741 The usual way to do this is to check if the determinant is
742 zero, but we need the characteristic polynomial for the
743 determinant. The minimal polynomial is a lot easier to get,
744 so we use Corollary 2 in Chapter V of Koecher to check
745 whether or not the paren't algebra's zero element is a root
746 of this element's minimal polynomial.
750 The identity element is always invertible::
752 sage: set_random_seed()
753 sage: J = random_eja()
754 sage: J.one().is_invertible()
757 The zero element is never invertible::
759 sage: set_random_seed()
760 sage: J = random_eja()
761 sage: J.zero().is_invertible()
765 zero
= self
.parent().zero()
766 p
= self
.minimal_polynomial()
767 return not (p(zero
) == zero
)
770 def is_nilpotent(self
):
772 Return whether or not some power of this element is zero.
774 The superclass method won't work unless we're in an
775 associative algebra, and we aren't. However, we generate
776 an assocoative subalgebra and we're nilpotent there if and
777 only if we're nilpotent here (probably).
781 The identity element is never nilpotent::
783 sage: set_random_seed()
784 sage: random_eja().one().is_nilpotent()
787 The additive identity is always nilpotent::
789 sage: set_random_seed()
790 sage: random_eja().zero().is_nilpotent()
794 # The element we're going to call "is_nilpotent()" on.
795 # Either myself, interpreted as an element of a finite-
796 # dimensional algebra, or an element of an associative
800 if self
.parent().is_associative():
801 elt
= FiniteDimensionalAlgebraElement(self
.parent(), self
)
803 V
= self
.span_of_powers()
804 assoc_subalg
= self
.subalgebra_generated_by()
805 # Mis-design warning: the basis used for span_of_powers()
806 # and subalgebra_generated_by() must be the same, and in
808 elt
= assoc_subalg(V
.coordinates(self
.vector()))
810 # Recursive call, but should work since elt lives in an
811 # associative algebra.
812 return elt
.is_nilpotent()
815 def is_regular(self
):
817 Return whether or not this is a regular element.
821 The identity element always has degree one, but any element
822 linearly-independent from it is regular::
824 sage: J = JordanSpinEJA(5)
825 sage: J.one().is_regular()
827 sage: e0, e1, e2, e3, e4 = J.gens() # e0 is the identity
828 sage: for x in J.gens():
829 ....: (J.one() + x).is_regular()
837 return self
.degree() == self
.parent().rank()
842 Compute the degree of this element the straightforward way
843 according to the definition; by appending powers to a list
844 and figuring out its dimension (that is, whether or not
845 they're linearly dependent).
849 sage: J = JordanSpinEJA(4)
850 sage: J.one().degree()
852 sage: e0,e1,e2,e3 = J.gens()
853 sage: (e0 - e1).degree()
856 In the spin factor algebra (of rank two), all elements that
857 aren't multiples of the identity are regular::
859 sage: set_random_seed()
860 sage: n = ZZ.random_element(1,10)
861 sage: J = JordanSpinEJA(n)
862 sage: x = J.random_element()
863 sage: x == x.coefficient(0)*J.one() or x.degree() == 2
867 return self
.span_of_powers().dimension()
870 def minimal_polynomial(self
):
872 Return the minimal polynomial of this element,
873 as a function of the variable `t`.
877 We restrict ourselves to the associative subalgebra
878 generated by this element, and then return the minimal
879 polynomial of this element's operator matrix (in that
880 subalgebra). This works by Baes Proposition 2.3.16.
884 The minimal polynomial of the identity and zero elements are
887 sage: set_random_seed()
888 sage: J = random_eja()
889 sage: J.one().minimal_polynomial()
891 sage: J.zero().minimal_polynomial()
894 The degree of an element is (by one definition) the degree
895 of its minimal polynomial::
897 sage: set_random_seed()
898 sage: x = random_eja().random_element()
899 sage: x.degree() == x.minimal_polynomial().degree()
902 The minimal polynomial and the characteristic polynomial coincide
903 and are known (see Alizadeh, Example 11.11) for all elements of
904 the spin factor algebra that aren't scalar multiples of the
907 sage: set_random_seed()
908 sage: n = ZZ.random_element(2,10)
909 sage: J = JordanSpinEJA(n)
910 sage: y = J.random_element()
911 sage: while y == y.coefficient(0)*J.one():
912 ....: y = J.random_element()
913 sage: y0 = y.vector()[0]
914 sage: y_bar = y.vector()[1:]
915 sage: actual = y.minimal_polynomial()
916 sage: t = PolynomialRing(J.base_ring(),'t').gen(0)
917 sage: expected = t^2 - 2*y0*t + (y0^2 - norm(y_bar)^2)
918 sage: bool(actual == expected)
921 The minimal polynomial should always kill its element::
923 sage: set_random_seed()
924 sage: x = random_eja().random_element()
925 sage: p = x.minimal_polynomial()
926 sage: x.apply_univariate_polynomial(p)
930 V
= self
.span_of_powers()
931 assoc_subalg
= self
.subalgebra_generated_by()
932 # Mis-design warning: the basis used for span_of_powers()
933 # and subalgebra_generated_by() must be the same, and in
935 elt
= assoc_subalg(V
.coordinates(self
.vector()))
937 # We get back a symbolic polynomial in 'x' but want a real
939 p_of_x
= elt
.operator_matrix().minimal_polynomial()
940 return p_of_x
.change_variable_name('t')
943 def natural_representation(self
):
945 Return a more-natural representation of this element.
947 Every finite-dimensional Euclidean Jordan Algebra is a
948 direct sum of five simple algebras, four of which comprise
949 Hermitian matrices. This method returns the original
950 "natural" representation of this element as a Hermitian
951 matrix, if it has one. If not, you get the usual representation.
955 sage: J = ComplexHermitianEJA(3)
958 sage: J.one().natural_representation()
968 sage: J = QuaternionHermitianEJA(3)
971 sage: J.one().natural_representation()
972 [1 0 0 0 0 0 0 0 0 0 0 0]
973 [0 1 0 0 0 0 0 0 0 0 0 0]
974 [0 0 1 0 0 0 0 0 0 0 0 0]
975 [0 0 0 1 0 0 0 0 0 0 0 0]
976 [0 0 0 0 1 0 0 0 0 0 0 0]
977 [0 0 0 0 0 1 0 0 0 0 0 0]
978 [0 0 0 0 0 0 1 0 0 0 0 0]
979 [0 0 0 0 0 0 0 1 0 0 0 0]
980 [0 0 0 0 0 0 0 0 1 0 0 0]
981 [0 0 0 0 0 0 0 0 0 1 0 0]
982 [0 0 0 0 0 0 0 0 0 0 1 0]
983 [0 0 0 0 0 0 0 0 0 0 0 1]
986 B
= self
.parent().natural_basis()
987 W
= B
[0].matrix_space()
988 return W
.linear_combination(zip(self
.vector(), B
))
991 def operator_matrix(self
):
993 Return the matrix that represents left- (or right-)
994 multiplication by this element in the parent algebra.
996 We have to override this because the superclass method
997 returns a matrix that acts on row vectors (that is, on
1002 Test the first polarization identity from my notes, Koecher Chapter
1003 III, or from Baes (2.3)::
1005 sage: set_random_seed()
1006 sage: J = random_eja()
1007 sage: x = J.random_element()
1008 sage: y = J.random_element()
1009 sage: Lx = x.operator_matrix()
1010 sage: Ly = y.operator_matrix()
1011 sage: Lxx = (x*x).operator_matrix()
1012 sage: Lxy = (x*y).operator_matrix()
1013 sage: bool(2*Lx*Lxy + Ly*Lxx == 2*Lxy*Lx + Lxx*Ly)
1016 Test the second polarization identity from my notes or from
1019 sage: set_random_seed()
1020 sage: J = random_eja()
1021 sage: x = J.random_element()
1022 sage: y = J.random_element()
1023 sage: z = J.random_element()
1024 sage: Lx = x.operator_matrix()
1025 sage: Ly = y.operator_matrix()
1026 sage: Lz = z.operator_matrix()
1027 sage: Lzy = (z*y).operator_matrix()
1028 sage: Lxy = (x*y).operator_matrix()
1029 sage: Lxz = (x*z).operator_matrix()
1030 sage: bool(Lx*Lzy + Lz*Lxy + Ly*Lxz == Lzy*Lx + Lxy*Lz + Lxz*Ly)
1033 Test the third polarization identity from my notes or from
1036 sage: set_random_seed()
1037 sage: J = random_eja()
1038 sage: u = J.random_element()
1039 sage: y = J.random_element()
1040 sage: z = J.random_element()
1041 sage: Lu = u.operator_matrix()
1042 sage: Ly = y.operator_matrix()
1043 sage: Lz = z.operator_matrix()
1044 sage: Lzy = (z*y).operator_matrix()
1045 sage: Luy = (u*y).operator_matrix()
1046 sage: Luz = (u*z).operator_matrix()
1047 sage: Luyz = (u*(y*z)).operator_matrix()
1048 sage: lhs = Lu*Lzy + Lz*Luy + Ly*Luz
1049 sage: rhs = Luyz + Ly*Lu*Lz + Lz*Lu*Ly
1050 sage: bool(lhs == rhs)
1054 fda_elt
= FiniteDimensionalAlgebraElement(self
.parent(), self
)
1055 return fda_elt
.matrix().transpose()
1058 def quadratic_representation(self
, other
=None):
1060 Return the quadratic representation of this element.
1064 The explicit form in the spin factor algebra is given by
1065 Alizadeh's Example 11.12::
1067 sage: set_random_seed()
1068 sage: n = ZZ.random_element(1,10)
1069 sage: J = JordanSpinEJA(n)
1070 sage: x = J.random_element()
1071 sage: x_vec = x.vector()
1073 sage: x_bar = x_vec[1:]
1074 sage: A = matrix(QQ, 1, [x_vec.inner_product(x_vec)])
1075 sage: B = 2*x0*x_bar.row()
1076 sage: C = 2*x0*x_bar.column()
1077 sage: D = identity_matrix(QQ, n-1)
1078 sage: D = (x0^2 - x_bar.inner_product(x_bar))*D
1079 sage: D = D + 2*x_bar.tensor_product(x_bar)
1080 sage: Q = block_matrix(2,2,[A,B,C,D])
1081 sage: Q == x.quadratic_representation()
1084 Test all of the properties from Theorem 11.2 in Alizadeh::
1086 sage: set_random_seed()
1087 sage: J = random_eja()
1088 sage: x = J.random_element()
1089 sage: y = J.random_element()
1093 sage: actual = x.quadratic_representation(y)
1094 sage: expected = ( (x+y).quadratic_representation()
1095 ....: -x.quadratic_representation()
1096 ....: -y.quadratic_representation() ) / 2
1097 sage: actual == expected
1102 sage: alpha = QQ.random_element()
1103 sage: actual = (alpha*x).quadratic_representation()
1104 sage: expected = (alpha^2)*x.quadratic_representation()
1105 sage: actual == expected
1110 sage: Qy = y.quadratic_representation()
1111 sage: actual = J(Qy*x.vector()).quadratic_representation()
1112 sage: expected = Qy*x.quadratic_representation()*Qy
1113 sage: actual == expected
1118 sage: k = ZZ.random_element(1,10)
1119 sage: actual = (x^k).quadratic_representation()
1120 sage: expected = (x.quadratic_representation())^k
1121 sage: actual == expected
1127 elif not other
in self
.parent():
1128 raise TypeError("'other' must live in the same algebra")
1130 L
= self
.operator_matrix()
1131 M
= other
.operator_matrix()
1132 return ( L
*M
+ M
*L
- (self
*other
).operator_matrix() )
1135 def span_of_powers(self
):
1137 Return the vector space spanned by successive powers of
1140 # The dimension of the subalgebra can't be greater than
1141 # the big algebra, so just put everything into a list
1142 # and let span() get rid of the excess.
1144 # We do the extra ambient_vector_space() in case we're messing
1145 # with polynomials and the direct parent is a module.
1146 V
= self
.vector().parent().ambient_vector_space()
1147 return V
.span( (self
**d
).vector() for d
in xrange(V
.dimension()) )
1150 def subalgebra_generated_by(self
):
1152 Return the associative subalgebra of the parent EJA generated
1157 sage: set_random_seed()
1158 sage: x = random_eja().random_element()
1159 sage: x.subalgebra_generated_by().is_associative()
1162 Squaring in the subalgebra should be the same thing as
1163 squaring in the superalgebra::
1165 sage: set_random_seed()
1166 sage: x = random_eja().random_element()
1167 sage: u = x.subalgebra_generated_by().random_element()
1168 sage: u.operator_matrix()*u.vector() == (u**2).vector()
1172 # First get the subspace spanned by the powers of myself...
1173 V
= self
.span_of_powers()
1174 F
= self
.base_ring()
1176 # Now figure out the entries of the right-multiplication
1177 # matrix for the successive basis elements b0, b1,... of
1180 for b_right
in V
.basis():
1181 eja_b_right
= self
.parent()(b_right
)
1183 # The first row of the right-multiplication matrix by
1184 # b1 is what we get if we apply that matrix to b1. The
1185 # second row of the right multiplication matrix by b1
1186 # is what we get when we apply that matrix to b2...
1188 # IMPORTANT: this assumes that all vectors are COLUMN
1189 # vectors, unlike our superclass (which uses row vectors).
1190 for b_left
in V
.basis():
1191 eja_b_left
= self
.parent()(b_left
)
1192 # Multiply in the original EJA, but then get the
1193 # coordinates from the subalgebra in terms of its
1195 this_row
= V
.coordinates((eja_b_left
*eja_b_right
).vector())
1196 b_right_rows
.append(this_row
)
1197 b_right_matrix
= matrix(F
, b_right_rows
)
1198 mats
.append(b_right_matrix
)
1200 # It's an algebra of polynomials in one element, and EJAs
1201 # are power-associative.
1203 # TODO: choose generator names intelligently.
1204 return FiniteDimensionalEuclideanJordanAlgebra(F
, mats
, assume_associative
=True, names
='f')
1207 def subalgebra_idempotent(self
):
1209 Find an idempotent in the associative subalgebra I generate
1210 using Proposition 2.3.5 in Baes.
1214 sage: set_random_seed()
1215 sage: J = RealCartesianProductEJA(5)
1216 sage: c = J.random_element().subalgebra_idempotent()
1219 sage: J = JordanSpinEJA(5)
1220 sage: c = J.random_element().subalgebra_idempotent()
1225 if self
.is_nilpotent():
1226 raise ValueError("this only works with non-nilpotent elements!")
1228 V
= self
.span_of_powers()
1229 J
= self
.subalgebra_generated_by()
1230 # Mis-design warning: the basis used for span_of_powers()
1231 # and subalgebra_generated_by() must be the same, and in
1233 u
= J(V
.coordinates(self
.vector()))
1235 # The image of the matrix of left-u^m-multiplication
1236 # will be minimal for some natural number s...
1238 minimal_dim
= V
.dimension()
1239 for i
in xrange(1, V
.dimension()):
1240 this_dim
= (u
**i
).operator_matrix().image().dimension()
1241 if this_dim
< minimal_dim
:
1242 minimal_dim
= this_dim
1245 # Now minimal_matrix should correspond to the smallest
1246 # non-zero subspace in Baes's (or really, Koecher's)
1249 # However, we need to restrict the matrix to work on the
1250 # subspace... or do we? Can't we just solve, knowing that
1251 # A(c) = u^(s+1) should have a solution in the big space,
1254 # Beware, solve_right() means that we're using COLUMN vectors.
1255 # Our FiniteDimensionalAlgebraElement superclass uses rows.
1257 A
= u_next
.operator_matrix()
1258 c_coordinates
= A
.solve_right(u_next
.vector())
1260 # Now c_coordinates is the idempotent we want, but it's in
1261 # the coordinate system of the subalgebra.
1263 # We need the basis for J, but as elements of the parent algebra.
1265 basis
= [self
.parent(v
) for v
in V
.basis()]
1266 return self
.parent().linear_combination(zip(c_coordinates
, basis
))
1271 Return my trace, the sum of my eigenvalues.
1275 sage: J = JordanSpinEJA(3)
1276 sage: x = sum(J.gens())
1282 sage: J = RealCartesianProductEJA(5)
1283 sage: J.one().trace()
1288 The trace of an element is a real number::
1290 sage: set_random_seed()
1291 sage: J = random_eja()
1292 sage: J.random_element().trace() in J.base_ring()
1298 p
= P
._charpoly
_coeff
(r
-1)
1299 # The _charpoly_coeff function already adds the factor of
1300 # -1 to ensure that _charpoly_coeff(r-1) is really what
1301 # appears in front of t^{r-1} in the charpoly. However,
1302 # we want the negative of THAT for the trace.
1303 return -p(*self
.vector())
1306 def trace_inner_product(self
, other
):
1308 Return the trace inner product of myself and ``other``.
1312 The trace inner product is commutative::
1314 sage: set_random_seed()
1315 sage: J = random_eja()
1316 sage: x = J.random_element(); y = J.random_element()
1317 sage: x.trace_inner_product(y) == y.trace_inner_product(x)
1320 The trace inner product is bilinear::
1322 sage: set_random_seed()
1323 sage: J = random_eja()
1324 sage: x = J.random_element()
1325 sage: y = J.random_element()
1326 sage: z = J.random_element()
1327 sage: a = QQ.random_element();
1328 sage: actual = (a*(x+z)).trace_inner_product(y)
1329 sage: expected = ( a*x.trace_inner_product(y) +
1330 ....: a*z.trace_inner_product(y) )
1331 sage: actual == expected
1333 sage: actual = x.trace_inner_product(a*(y+z))
1334 sage: expected = ( a*x.trace_inner_product(y) +
1335 ....: a*x.trace_inner_product(z) )
1336 sage: actual == expected
1339 The trace inner product satisfies the compatibility
1340 condition in the definition of a Euclidean Jordan algebra::
1342 sage: set_random_seed()
1343 sage: J = random_eja()
1344 sage: x = J.random_element()
1345 sage: y = J.random_element()
1346 sage: z = J.random_element()
1347 sage: (x*y).trace_inner_product(z) == y.trace_inner_product(x*z)
1351 if not other
in self
.parent():
1352 raise TypeError("'other' must live in the same algebra")
1354 return (self
*other
).trace()
1357 class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1359 Return the Euclidean Jordan Algebra corresponding to the set
1360 `R^n` under the Hadamard product.
1362 Note: this is nothing more than the Cartesian product of ``n``
1363 copies of the spin algebra. Once Cartesian product algebras
1364 are implemented, this can go.
1368 This multiplication table can be verified by hand::
1370 sage: J = RealCartesianProductEJA(3)
1371 sage: e0,e1,e2 = J.gens()
1387 def __classcall_private__(cls
, n
, field
=QQ
):
1388 # The FiniteDimensionalAlgebra constructor takes a list of
1389 # matrices, the ith representing right multiplication by the ith
1390 # basis element in the vector space. So if e_1 = (1,0,0), then
1391 # right (Hadamard) multiplication of x by e_1 picks out the first
1392 # component of x; and likewise for the ith basis element e_i.
1393 Qs
= [ matrix(field
, n
, n
, lambda k
,j
: 1*(k
== j
== i
))
1394 for i
in xrange(n
) ]
1396 fdeja
= super(RealCartesianProductEJA
, cls
)
1397 return fdeja
.__classcall
_private
__(cls
, field
, Qs
, rank
=n
)
1399 def inner_product(self
, x
, y
):
1400 return _usual_ip(x
,y
)
1405 Return a "random" finite-dimensional Euclidean Jordan Algebra.
1409 For now, we choose a random natural number ``n`` (greater than zero)
1410 and then give you back one of the following:
1412 * The cartesian product of the rational numbers ``n`` times; this is
1413 ``QQ^n`` with the Hadamard product.
1415 * The Jordan spin algebra on ``QQ^n``.
1417 * The ``n``-by-``n`` rational symmetric matrices with the symmetric
1420 * The ``n``-by-``n`` complex-rational Hermitian matrices embedded
1421 in the space of ``2n``-by-``2n`` real symmetric matrices.
1423 * The ``n``-by-``n`` quaternion-rational Hermitian matrices embedded
1424 in the space of ``4n``-by-``4n`` real symmetric matrices.
1426 Later this might be extended to return Cartesian products of the
1432 Euclidean Jordan algebra of degree...
1436 # The max_n component lets us choose different upper bounds on the
1437 # value "n" that gets passed to the constructor. This is needed
1438 # because e.g. R^{10} is reasonable to test, while the Hermitian
1439 # 10-by-10 quaternion matrices are not.
1440 (constructor
, max_n
) = choice([(RealCartesianProductEJA
, 6),
1442 (RealSymmetricEJA
, 5),
1443 (ComplexHermitianEJA
, 4),
1444 (QuaternionHermitianEJA
, 3)])
1445 n
= ZZ
.random_element(1, max_n
)
1446 return constructor(n
, field
=QQ
)
1450 def _real_symmetric_basis(n
, field
=QQ
):
1452 Return a basis for the space of real symmetric n-by-n matrices.
1454 # The basis of symmetric matrices, as matrices, in their R^(n-by-n)
1458 for j
in xrange(i
+1):
1459 Eij
= matrix(field
, n
, lambda k
,l
: k
==i
and l
==j
)
1463 # Beware, orthogonal but not normalized!
1464 Sij
= Eij
+ Eij
.transpose()
1469 def _complex_hermitian_basis(n
, field
=QQ
):
1471 Returns a basis for the space of complex Hermitian n-by-n matrices.
1475 sage: set_random_seed()
1476 sage: n = ZZ.random_element(1,5)
1477 sage: all( M.is_symmetric() for M in _complex_hermitian_basis(n) )
1481 F
= QuadraticField(-1, 'I')
1484 # This is like the symmetric case, but we need to be careful:
1486 # * We want conjugate-symmetry, not just symmetry.
1487 # * The diagonal will (as a result) be real.
1491 for j
in xrange(i
+1):
1492 Eij
= matrix(field
, n
, lambda k
,l
: k
==i
and l
==j
)
1494 Sij
= _embed_complex_matrix(Eij
)
1497 # Beware, orthogonal but not normalized! The second one
1498 # has a minus because it's conjugated.
1499 Sij_real
= _embed_complex_matrix(Eij
+ Eij
.transpose())
1501 Sij_imag
= _embed_complex_matrix(I
*Eij
- I
*Eij
.transpose())
1506 def _quaternion_hermitian_basis(n
, field
=QQ
):
1508 Returns a basis for the space of quaternion Hermitian n-by-n matrices.
1512 sage: set_random_seed()
1513 sage: n = ZZ.random_element(1,5)
1514 sage: all( M.is_symmetric() for M in _quaternion_hermitian_basis(n) )
1518 Q
= QuaternionAlgebra(QQ
,-1,-1)
1521 # This is like the symmetric case, but we need to be careful:
1523 # * We want conjugate-symmetry, not just symmetry.
1524 # * The diagonal will (as a result) be real.
1528 for j
in xrange(i
+1):
1529 Eij
= matrix(Q
, n
, lambda k
,l
: k
==i
and l
==j
)
1531 Sij
= _embed_quaternion_matrix(Eij
)
1534 # Beware, orthogonal but not normalized! The second,
1535 # third, and fourth ones have a minus because they're
1537 Sij_real
= _embed_quaternion_matrix(Eij
+ Eij
.transpose())
1539 Sij_I
= _embed_quaternion_matrix(I
*Eij
- I
*Eij
.transpose())
1541 Sij_J
= _embed_quaternion_matrix(J
*Eij
- J
*Eij
.transpose())
1543 Sij_K
= _embed_quaternion_matrix(K
*Eij
- K
*Eij
.transpose())
1549 return vector(m
.base_ring(), m
.list())
1552 return matrix(v
.base_ring(), sqrt(v
.degree()), v
.list())
1554 def _multiplication_table_from_matrix_basis(basis
):
1556 At least three of the five simple Euclidean Jordan algebras have the
1557 symmetric multiplication (A,B) |-> (AB + BA)/2, where the
1558 multiplication on the right is matrix multiplication. Given a basis
1559 for the underlying matrix space, this function returns a
1560 multiplication table (obtained by looping through the basis
1561 elements) for an algebra of those matrices. A reordered copy
1562 of the basis is also returned to work around the fact that
1563 the ``span()`` in this function will change the order of the basis
1564 from what we think it is, to... something else.
1566 # In S^2, for example, we nominally have four coordinates even
1567 # though the space is of dimension three only. The vector space V
1568 # is supposed to hold the entire long vector, and the subspace W
1569 # of V will be spanned by the vectors that arise from symmetric
1570 # matrices. Thus for S^2, dim(V) == 4 and dim(W) == 3.
1571 field
= basis
[0].base_ring()
1572 dimension
= basis
[0].nrows()
1574 V
= VectorSpace(field
, dimension
**2)
1575 W
= V
.span( _mat2vec(s
) for s
in basis
)
1577 # Taking the span above reorders our basis (thanks, jerk!) so we
1578 # need to put our "matrix basis" in the same order as the
1579 # (reordered) vector basis.
1580 S
= tuple( _vec2mat(b
) for b
in W
.basis() )
1584 # Brute force the multiplication-by-s matrix by looping
1585 # through all elements of the basis and doing the computation
1586 # to find out what the corresponding row should be. BEWARE:
1587 # these multiplication tables won't be symmetric! It therefore
1588 # becomes REALLY IMPORTANT that the underlying algebra
1589 # constructor uses ROW vectors and not COLUMN vectors. That's
1590 # why we're computing rows here and not columns.
1593 this_row
= _mat2vec((s
*t
+ t
*s
)/2)
1594 Q_rows
.append(W
.coordinates(this_row
))
1595 Q
= matrix(field
, W
.dimension(), Q_rows
)
1601 def _embed_complex_matrix(M
):
1603 Embed the n-by-n complex matrix ``M`` into the space of real
1604 matrices of size 2n-by-2n via the map the sends each entry `z = a +
1605 bi` to the block matrix ``[[a,b],[-b,a]]``.
1609 sage: F = QuadraticField(-1,'i')
1610 sage: x1 = F(4 - 2*i)
1611 sage: x2 = F(1 + 2*i)
1614 sage: M = matrix(F,2,[[x1,x2],[x3,x4]])
1615 sage: _embed_complex_matrix(M)
1624 Embedding is a homomorphism (isomorphism, in fact)::
1626 sage: set_random_seed()
1627 sage: n = ZZ.random_element(5)
1628 sage: F = QuadraticField(-1, 'i')
1629 sage: X = random_matrix(F, n)
1630 sage: Y = random_matrix(F, n)
1631 sage: actual = _embed_complex_matrix(X) * _embed_complex_matrix(Y)
1632 sage: expected = _embed_complex_matrix(X*Y)
1633 sage: actual == expected
1639 raise ValueError("the matrix 'M' must be square")
1640 field
= M
.base_ring()
1645 blocks
.append(matrix(field
, 2, [[a
,b
],[-b
,a
]]))
1647 # We can drop the imaginaries here.
1648 return block_matrix(field
.base_ring(), n
, blocks
)
1651 def _unembed_complex_matrix(M
):
1653 The inverse of _embed_complex_matrix().
1657 sage: A = matrix(QQ,[ [ 1, 2, 3, 4],
1658 ....: [-2, 1, -4, 3],
1659 ....: [ 9, 10, 11, 12],
1660 ....: [-10, 9, -12, 11] ])
1661 sage: _unembed_complex_matrix(A)
1663 [ 10*i + 9 12*i + 11]
1667 Unembedding is the inverse of embedding::
1669 sage: set_random_seed()
1670 sage: F = QuadraticField(-1, 'i')
1671 sage: M = random_matrix(F, 3)
1672 sage: _unembed_complex_matrix(_embed_complex_matrix(M)) == M
1678 raise ValueError("the matrix 'M' must be square")
1679 if not n
.mod(2).is_zero():
1680 raise ValueError("the matrix 'M' must be a complex embedding")
1682 F
= QuadraticField(-1, 'i')
1685 # Go top-left to bottom-right (reading order), converting every
1686 # 2-by-2 block we see to a single complex element.
1688 for k
in xrange(n
/2):
1689 for j
in xrange(n
/2):
1690 submat
= M
[2*k
:2*k
+2,2*j
:2*j
+2]
1691 if submat
[0,0] != submat
[1,1]:
1692 raise ValueError('bad on-diagonal submatrix')
1693 if submat
[0,1] != -submat
[1,0]:
1694 raise ValueError('bad off-diagonal submatrix')
1695 z
= submat
[0,0] + submat
[0,1]*i
1698 return matrix(F
, n
/2, elements
)
1701 def _embed_quaternion_matrix(M
):
1703 Embed the n-by-n quaternion matrix ``M`` into the space of real
1704 matrices of size 4n-by-4n by first sending each quaternion entry
1705 `z = a + bi + cj + dk` to the block-complex matrix
1706 ``[[a + bi, c+di],[-c + di, a-bi]]`, and then embedding those into
1711 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1712 sage: i,j,k = Q.gens()
1713 sage: x = 1 + 2*i + 3*j + 4*k
1714 sage: M = matrix(Q, 1, [[x]])
1715 sage: _embed_quaternion_matrix(M)
1721 Embedding is a homomorphism (isomorphism, in fact)::
1723 sage: set_random_seed()
1724 sage: n = ZZ.random_element(5)
1725 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1726 sage: X = random_matrix(Q, n)
1727 sage: Y = random_matrix(Q, n)
1728 sage: actual = _embed_quaternion_matrix(X)*_embed_quaternion_matrix(Y)
1729 sage: expected = _embed_quaternion_matrix(X*Y)
1730 sage: actual == expected
1734 quaternions
= M
.base_ring()
1737 raise ValueError("the matrix 'M' must be square")
1739 F
= QuadraticField(-1, 'i')
1744 t
= z
.coefficient_tuple()
1749 cplx_matrix
= matrix(F
, 2, [[ a
+ b
*i
, c
+ d
*i
],
1750 [-c
+ d
*i
, a
- b
*i
]])
1751 blocks
.append(_embed_complex_matrix(cplx_matrix
))
1753 # We should have real entries by now, so use the realest field
1754 # we've got for the return value.
1755 return block_matrix(quaternions
.base_ring(), n
, blocks
)
1758 def _unembed_quaternion_matrix(M
):
1760 The inverse of _embed_quaternion_matrix().
1764 sage: M = matrix(QQ, [[ 1, 2, 3, 4],
1765 ....: [-2, 1, -4, 3],
1766 ....: [-3, 4, 1, -2],
1767 ....: [-4, -3, 2, 1]])
1768 sage: _unembed_quaternion_matrix(M)
1769 [1 + 2*i + 3*j + 4*k]
1773 Unembedding is the inverse of embedding::
1775 sage: set_random_seed()
1776 sage: Q = QuaternionAlgebra(QQ, -1, -1)
1777 sage: M = random_matrix(Q, 3)
1778 sage: _unembed_quaternion_matrix(_embed_quaternion_matrix(M)) == M
1784 raise ValueError("the matrix 'M' must be square")
1785 if not n
.mod(4).is_zero():
1786 raise ValueError("the matrix 'M' must be a complex embedding")
1788 Q
= QuaternionAlgebra(QQ
,-1,-1)
1791 # Go top-left to bottom-right (reading order), converting every
1792 # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1
1795 for l
in xrange(n
/4):
1796 for m
in xrange(n
/4):
1797 submat
= _unembed_complex_matrix(M
[4*l
:4*l
+4,4*m
:4*m
+4])
1798 if submat
[0,0] != submat
[1,1].conjugate():
1799 raise ValueError('bad on-diagonal submatrix')
1800 if submat
[0,1] != -submat
[1,0].conjugate():
1801 raise ValueError('bad off-diagonal submatrix')
1802 z
= submat
[0,0].real() + submat
[0,0].imag()*i
1803 z
+= submat
[0,1].real()*j
+ submat
[0,1].imag()*k
1806 return matrix(Q
, n
/4, elements
)
1809 # The usual inner product on R^n.
1811 return x
.vector().inner_product(y
.vector())
1813 # The inner product used for the real symmetric simple EJA.
1814 # We keep it as a separate function because e.g. the complex
1815 # algebra uses the same inner product, except divided by 2.
1816 def _matrix_ip(X
,Y
):
1817 X_mat
= X
.natural_representation()
1818 Y_mat
= Y
.natural_representation()
1819 return (X_mat
*Y_mat
).trace()
1822 class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1824 The rank-n simple EJA consisting of real symmetric n-by-n
1825 matrices, the usual symmetric Jordan product, and the trace inner
1826 product. It has dimension `(n^2 + n)/2` over the reals.
1830 sage: J = RealSymmetricEJA(2)
1831 sage: e0, e1, e2 = J.gens()
1841 The degree of this algebra is `(n^2 + n) / 2`::
1843 sage: set_random_seed()
1844 sage: n = ZZ.random_element(1,5)
1845 sage: J = RealSymmetricEJA(n)
1846 sage: J.degree() == (n^2 + n)/2
1849 The Jordan multiplication is what we think it is::
1851 sage: set_random_seed()
1852 sage: n = ZZ.random_element(1,5)
1853 sage: J = RealSymmetricEJA(n)
1854 sage: x = J.random_element()
1855 sage: y = J.random_element()
1856 sage: actual = (x*y).natural_representation()
1857 sage: X = x.natural_representation()
1858 sage: Y = y.natural_representation()
1859 sage: expected = (X*Y + Y*X)/2
1860 sage: actual == expected
1862 sage: J(expected) == x*y
1867 def __classcall_private__(cls
, n
, field
=QQ
):
1868 S
= _real_symmetric_basis(n
, field
=field
)
1869 (Qs
, T
) = _multiplication_table_from_matrix_basis(S
)
1871 fdeja
= super(RealSymmetricEJA
, cls
)
1872 return fdeja
.__classcall
_private
__(cls
,
1878 def inner_product(self
, x
, y
):
1879 return _matrix_ip(x
,y
)
1882 class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1884 The rank-n simple EJA consisting of complex Hermitian n-by-n
1885 matrices over the real numbers, the usual symmetric Jordan product,
1886 and the real-part-of-trace inner product. It has dimension `n^2` over
1891 The degree of this algebra is `n^2`::
1893 sage: set_random_seed()
1894 sage: n = ZZ.random_element(1,5)
1895 sage: J = ComplexHermitianEJA(n)
1896 sage: J.degree() == n^2
1899 The Jordan multiplication is what we think it is::
1901 sage: set_random_seed()
1902 sage: n = ZZ.random_element(1,5)
1903 sage: J = ComplexHermitianEJA(n)
1904 sage: x = J.random_element()
1905 sage: y = J.random_element()
1906 sage: actual = (x*y).natural_representation()
1907 sage: X = x.natural_representation()
1908 sage: Y = y.natural_representation()
1909 sage: expected = (X*Y + Y*X)/2
1910 sage: actual == expected
1912 sage: J(expected) == x*y
1917 def __classcall_private__(cls
, n
, field
=QQ
):
1918 S
= _complex_hermitian_basis(n
)
1919 (Qs
, T
) = _multiplication_table_from_matrix_basis(S
)
1921 fdeja
= super(ComplexHermitianEJA
, cls
)
1922 return fdeja
.__classcall
_private
__(cls
,
1928 def inner_product(self
, x
, y
):
1929 # Since a+bi on the diagonal is represented as
1934 # we'll double-count the "a" entries if we take the trace of
1936 return _matrix_ip(x
,y
)/2
1939 class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1941 The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
1942 matrices, the usual symmetric Jordan product, and the
1943 real-part-of-trace inner product. It has dimension `2n^2 - n` over
1948 The degree of this algebra is `n^2`::
1950 sage: set_random_seed()
1951 sage: n = ZZ.random_element(1,5)
1952 sage: J = QuaternionHermitianEJA(n)
1953 sage: J.degree() == 2*(n^2) - n
1956 The Jordan multiplication is what we think it is::
1958 sage: set_random_seed()
1959 sage: n = ZZ.random_element(1,5)
1960 sage: J = QuaternionHermitianEJA(n)
1961 sage: x = J.random_element()
1962 sage: y = J.random_element()
1963 sage: actual = (x*y).natural_representation()
1964 sage: X = x.natural_representation()
1965 sage: Y = y.natural_representation()
1966 sage: expected = (X*Y + Y*X)/2
1967 sage: actual == expected
1969 sage: J(expected) == x*y
1974 def __classcall_private__(cls
, n
, field
=QQ
):
1975 S
= _quaternion_hermitian_basis(n
)
1976 (Qs
, T
) = _multiplication_table_from_matrix_basis(S
)
1978 fdeja
= super(QuaternionHermitianEJA
, cls
)
1979 return fdeja
.__classcall
_private
__(cls
,
1985 def inner_product(self
, x
, y
):
1986 # Since a+bi+cj+dk on the diagonal is represented as
1988 # a + bi +cj + dk = [ a b c d]
1993 # we'll quadruple-count the "a" entries if we take the trace of
1995 return _matrix_ip(x
,y
)/4
1998 class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra
):
2000 The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
2001 with the usual inner product and jordan product ``x*y =
2002 (<x_bar,y_bar>, x0*y_bar + y0*x_bar)``. It has dimension `n` over
2007 This multiplication table can be verified by hand::
2009 sage: J = JordanSpinEJA(4)
2010 sage: e0,e1,e2,e3 = J.gens()
2028 def __classcall_private__(cls
, n
, field
=QQ
):
2030 id_matrix
= identity_matrix(field
, n
)
2032 ei
= id_matrix
.column(i
)
2033 Qi
= zero_matrix(field
, n
)
2035 Qi
.set_column(0, ei
)
2036 Qi
+= diagonal_matrix(n
, [ei
[0]]*n
)
2037 # The addition of the diagonal matrix adds an extra ei[0] in the
2038 # upper-left corner of the matrix.
2039 Qi
[0,0] = Qi
[0,0] * ~
field(2)
2042 # The rank of the spin algebra is two, unless we're in a
2043 # one-dimensional ambient space (because the rank is bounded by
2044 # the ambient dimension).
2045 fdeja
= super(JordanSpinEJA
, cls
)
2046 return fdeja
.__classcall
_private
__(cls
, field
, Qs
, rank
=min(n
,2))
2048 def inner_product(self
, x
, y
):
2049 return _usual_ip(x
,y
)