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
14 from sage
.algebras
.finite_dimensional_algebras
.finite_dimensional_algebra_morphism
import FiniteDimensionalAlgebraMorphism
17 class FiniteDimensionalEuclideanJordanAlgebraMorphism(FiniteDimensionalAlgebraMorphism
):
19 A very thin wrapper around FiniteDimensionalAlgebraMorphism that
22 1. Avoids the ``unitary`` and ``check`` arguments to the constructor
23 that will always be ``False``. This is necessary because these
24 are homomorphisms with respect to ADDITION, but the SageMath
25 machinery wants to check that they're homomorphisms with respect
26 to (Jordan) MULTIPLICATION. That obviously doesn't work.
28 2. Inputs and outputs the underlying matrix with respect to COLUMN
29 vectors, unlike the parent class.
31 If this seems a bit heavyweight, it is. I would have been happy to
32 use a the ring morphism that underlies the finite-dimensional
33 algebra morphism, but they don't seem to be callable on elements of
36 def __init__(self
, parent
, f
):
37 FiniteDimensionalAlgebraMorphism
.__init
__(self
,
46 We override only the representation that is shown to the user,
47 because we want the matrix to be with respect to COLUMN vectors.
51 Ensure that we see the transpose of the underlying matrix object:
53 sage: J = RealSymmetricEJA(3)
54 sage: x = J.linear_combination(zip(range(len(J.gens())), J.gens()))
55 sage: L = x.operator()
57 Morphism from Euclidean Jordan algebra of degree 6 over Rational
58 Field to Euclidean Jordan algebra of degree 6 over Rational Field
75 return "Morphism from {} to {} given by matrix\n{}".format(
76 self
.domain(), self
.codomain(), self
.matrix())
80 Return the matrix of this morphism with respect to a left-action
83 return FiniteDimensionalAlgebraMorphism
.matrix(self
).transpose()
86 class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra
):
88 def __classcall_private__(cls
,
92 assume_associative
=False,
97 mult_table
= [b
.base_extend(field
) for b
in mult_table
]
100 if not (is_Matrix(b
) and b
.dimensions() == (n
, n
)):
101 raise ValueError("input is not a multiplication table")
102 mult_table
= tuple(mult_table
)
104 cat
= MagmaticAlgebras(field
).FiniteDimensional().WithBasis()
105 cat
.or_subcategory(category
)
106 if assume_associative
:
107 cat
= cat
.Associative()
109 names
= normalize_names(n
, names
)
111 fda
= super(FiniteDimensionalEuclideanJordanAlgebra
, cls
)
112 return fda
.__classcall
__(cls
,
115 assume_associative
=assume_associative
,
119 natural_basis
=natural_basis
)
126 assume_associative
=False,
133 By definition, Jordan multiplication commutes::
135 sage: set_random_seed()
136 sage: J = random_eja()
137 sage: x = J.random_element()
138 sage: y = J.random_element()
144 self
._natural
_basis
= natural_basis
145 self
._multiplication
_table
= mult_table
146 fda
= super(FiniteDimensionalEuclideanJordanAlgebra
, self
)
155 Return a string representation of ``self``.
157 fmt
= "Euclidean Jordan algebra of degree {} over {}"
158 return fmt
.format(self
.degree(), self
.base_ring())
161 def _a_regular_element(self
):
163 Guess a regular element. Needed to compute the basis for our
164 characteristic polynomial coefficients.
167 z
= self
.sum( (i
+1)*gs
[i
] for i
in range(len(gs
)) )
168 if not z
.is_regular():
169 raise ValueError("don't know a regular element")
174 def _charpoly_basis_space(self
):
176 Return the vector space spanned by the basis used in our
177 characteristic polynomial coefficients. This is used not only to
178 compute those coefficients, but also any time we need to
179 evaluate the coefficients (like when we compute the trace or
182 z
= self
._a
_regular
_element
()
183 V
= z
.vector().parent().ambient_vector_space()
184 V1
= V
.span_of_basis( (z
**k
).vector() for k
in range(self
.rank()) )
185 b
= (V1
.basis() + V1
.complement().basis())
186 return V
.span_of_basis(b
)
190 def _charpoly_coeff(self
, i
):
192 Return the coefficient polynomial "a_{i}" of this algebra's
193 general characteristic polynomial.
195 Having this be a separate cached method lets us compute and
196 store the trace/determinant (a_{r-1} and a_{0} respectively)
197 separate from the entire characteristic polynomial.
199 (A_of_x
, x
, xr
, detA
) = self
._charpoly
_matrix
_system
()
200 R
= A_of_x
.base_ring()
202 # Guaranteed by theory
205 # Danger: the in-place modification is done for performance
206 # reasons (reconstructing a matrix with huge polynomial
207 # entries is slow), but I don't know how cached_method works,
208 # so it's highly possible that we're modifying some global
209 # list variable by reference, here. In other words, you
210 # probably shouldn't call this method twice on the same
211 # algebra, at the same time, in two threads
212 Ai_orig
= A_of_x
.column(i
)
213 A_of_x
.set_column(i
,xr
)
214 numerator
= A_of_x
.det()
215 A_of_x
.set_column(i
,Ai_orig
)
217 # We're relying on the theory here to ensure that each a_i is
218 # indeed back in R, and the added negative signs are to make
219 # the whole charpoly expression sum to zero.
220 return R(-numerator
/detA
)
224 def _charpoly_matrix_system(self
):
226 Compute the matrix whose entries A_ij are polynomials in
227 X1,...,XN, the vector ``x`` of variables X1,...,XN, the vector
228 corresponding to `x^r` and the determinent of the matrix A =
229 [A_ij]. In other words, all of the fixed (cachable) data needed
230 to compute the coefficients of the characteristic polynomial.
235 # Construct a new algebra over a multivariate polynomial ring...
236 names
= ['X' + str(i
) for i
in range(1,n
+1)]
237 R
= PolynomialRing(self
.base_ring(), names
)
238 J
= FiniteDimensionalEuclideanJordanAlgebra(R
,
239 self
._multiplication
_table
,
242 idmat
= identity_matrix(J
.base_ring(), n
)
244 W
= self
._charpoly
_basis
_space
()
245 W
= W
.change_ring(R
.fraction_field())
247 # Starting with the standard coordinates x = (X1,X2,...,Xn)
248 # and then converting the entries to W-coordinates allows us
249 # to pass in the standard coordinates to the charpoly and get
250 # back the right answer. Specifically, with x = (X1,X2,...,Xn),
253 # W.coordinates(x^2) eval'd at (standard z-coords)
257 # W-coords of (standard coords of x^2 eval'd at std-coords of z)
259 # We want the middle equivalent thing in our matrix, but use
260 # the first equivalent thing instead so that we can pass in
261 # standard coordinates.
262 x
= J(vector(R
, R
.gens()))
263 l1
= [column_matrix(W
.coordinates((x
**k
).vector())) for k
in range(r
)]
264 l2
= [idmat
.column(k
-1).column() for k
in range(r
+1, n
+1)]
265 A_of_x
= block_matrix(R
, 1, n
, (l1
+ l2
))
266 xr
= W
.coordinates((x
**r
).vector())
267 return (A_of_x
, x
, xr
, A_of_x
.det())
271 def characteristic_polynomial(self
):
276 This implementation doesn't guarantee that the polynomial
277 denominator in the coefficients is not identically zero, so
278 theoretically it could crash. The way that this is handled
279 in e.g. Faraut and Koranyi is to use a basis that guarantees
280 the denominator is non-zero. But, doing so requires knowledge
281 of at least one regular element, and we don't even know how
282 to do that. The trade-off is that, if we use the standard basis,
283 the resulting polynomial will accept the "usual" coordinates. In
284 other words, we don't have to do a change of basis before e.g.
285 computing the trace or determinant.
289 The characteristic polynomial in the spin algebra is given in
290 Alizadeh, Example 11.11::
292 sage: J = JordanSpinEJA(3)
293 sage: p = J.characteristic_polynomial(); p
294 X1^2 - X2^2 - X3^2 + (-2*t)*X1 + t^2
295 sage: xvec = J.one().vector()
303 # The list of coefficient polynomials a_1, a_2, ..., a_n.
304 a
= [ self
._charpoly
_coeff
(i
) for i
in range(n
) ]
306 # We go to a bit of trouble here to reorder the
307 # indeterminates, so that it's easier to evaluate the
308 # characteristic polynomial at x's coordinates and get back
309 # something in terms of t, which is what we want.
311 S
= PolynomialRing(self
.base_ring(),'t')
313 S
= PolynomialRing(S
, R
.variable_names())
316 # Note: all entries past the rth should be zero. The
317 # coefficient of the highest power (x^r) is 1, but it doesn't
318 # appear in the solution vector which contains coefficients
319 # for the other powers (to make them sum to x^r).
321 a
[r
] = 1 # corresponds to x^r
323 # When the rank is equal to the dimension, trying to
324 # assign a[r] goes out-of-bounds.
325 a
.append(1) # corresponds to x^r
327 return sum( a
[k
]*(t
**k
) for k
in range(len(a
)) )
330 def inner_product(self
, x
, y
):
332 The inner product associated with this Euclidean Jordan algebra.
334 Defaults to the trace inner product, but can be overridden by
335 subclasses if they are sure that the necessary properties are
340 The inner product must satisfy its axiom for this algebra to truly
341 be a Euclidean Jordan Algebra::
343 sage: set_random_seed()
344 sage: J = random_eja()
345 sage: x = J.random_element()
346 sage: y = J.random_element()
347 sage: z = J.random_element()
348 sage: (x*y).inner_product(z) == y.inner_product(x*z)
352 if (not x
in self
) or (not y
in self
):
353 raise TypeError("arguments must live in this algebra")
354 return x
.trace_inner_product(y
)
357 def natural_basis(self
):
359 Return a more-natural representation of this algebra's basis.
361 Every finite-dimensional Euclidean Jordan Algebra is a direct
362 sum of five simple algebras, four of which comprise Hermitian
363 matrices. This method returns the original "natural" basis
364 for our underlying vector space. (Typically, the natural basis
365 is used to construct the multiplication table in the first place.)
367 Note that this will always return a matrix. The standard basis
368 in `R^n` will be returned as `n`-by-`1` column matrices.
372 sage: J = RealSymmetricEJA(2)
375 sage: J.natural_basis()
383 sage: J = JordanSpinEJA(2)
386 sage: J.natural_basis()
393 if self
._natural
_basis
is None:
394 return tuple( b
.vector().column() for b
in self
.basis() )
396 return self
._natural
_basis
401 Return the rank of this EJA.
403 if self
._rank
is None:
404 raise ValueError("no rank specified at genesis")
409 class Element(FiniteDimensionalAlgebraElement
):
411 An element of a Euclidean Jordan algebra.
416 Oh man, I should not be doing this. This hides the "disabled"
417 methods ``left_matrix`` and ``matrix`` from introspection;
418 in particular it removes them from tab-completion.
420 return filter(lambda s
: s
not in ['left_matrix', 'matrix'],
421 dir(self
.__class
__) )
424 def __init__(self
, A
, elt
=None):
428 The identity in `S^n` is converted to the identity in the EJA::
430 sage: J = RealSymmetricEJA(3)
431 sage: I = identity_matrix(QQ,3)
432 sage: J(I) == J.one()
435 This skew-symmetric matrix can't be represented in the EJA::
437 sage: J = RealSymmetricEJA(3)
438 sage: A = matrix(QQ,3, lambda i,j: i-j)
440 Traceback (most recent call last):
442 ArithmeticError: vector is not in free module
445 # Goal: if we're given a matrix, and if it lives in our
446 # parent algebra's "natural ambient space," convert it
447 # into an algebra element.
449 # The catch is, we make a recursive call after converting
450 # the given matrix into a vector that lives in the algebra.
451 # This we need to try the parent class initializer first,
452 # to avoid recursing forever if we're given something that
453 # already fits into the algebra, but also happens to live
454 # in the parent's "natural ambient space" (this happens with
457 FiniteDimensionalAlgebraElement
.__init
__(self
, A
, elt
)
459 natural_basis
= A
.natural_basis()
460 if elt
in natural_basis
[0].matrix_space():
461 # Thanks for nothing! Matrix spaces aren't vector
462 # spaces in Sage, so we have to figure out its
463 # natural-basis coordinates ourselves.
464 V
= VectorSpace(elt
.base_ring(), elt
.nrows()**2)
465 W
= V
.span( _mat2vec(s
) for s
in natural_basis
)
466 coords
= W
.coordinates(_mat2vec(elt
))
467 FiniteDimensionalAlgebraElement
.__init
__(self
, A
, coords
)
469 def __pow__(self
, n
):
471 Return ``self`` raised to the power ``n``.
473 Jordan algebras are always power-associative; see for
474 example Faraut and Koranyi, Proposition II.1.2 (ii).
478 We have to override this because our superclass uses row vectors
479 instead of column vectors! We, on the other hand, assume column
484 sage: set_random_seed()
485 sage: x = random_eja().random_element()
486 sage: x.operator_matrix()*x.vector() == (x^2).vector()
489 A few examples of power-associativity::
491 sage: set_random_seed()
492 sage: x = random_eja().random_element()
493 sage: x*(x*x)*(x*x) == x^5
495 sage: (x*x)*(x*x*x) == x^5
498 We also know that powers operator-commute (Koecher, Chapter
501 sage: set_random_seed()
502 sage: x = random_eja().random_element()
503 sage: m = ZZ.random_element(0,10)
504 sage: n = ZZ.random_element(0,10)
505 sage: Lxm = (x^m).operator_matrix()
506 sage: Lxn = (x^n).operator_matrix()
507 sage: Lxm*Lxn == Lxn*Lxm
517 return A( (self
.operator_matrix()**(n
-1))*self
.vector() )
520 def apply_univariate_polynomial(self
, p
):
522 Apply the univariate polynomial ``p`` to this element.
524 A priori, SageMath won't allow us to apply a univariate
525 polynomial to an element of an EJA, because we don't know
526 that EJAs are rings (they are usually not associative). Of
527 course, we know that EJAs are power-associative, so the
528 operation is ultimately kosher. This function sidesteps
529 the CAS to get the answer we want and expect.
533 sage: R = PolynomialRing(QQ, 't')
535 sage: p = t^4 - t^3 + 5*t - 2
536 sage: J = RealCartesianProductEJA(5)
537 sage: J.one().apply_univariate_polynomial(p) == 3*J.one()
542 We should always get back an element of the algebra::
544 sage: set_random_seed()
545 sage: p = PolynomialRing(QQ, 't').random_element()
546 sage: J = random_eja()
547 sage: x = J.random_element()
548 sage: x.apply_univariate_polynomial(p) in J
552 if len(p
.variables()) > 1:
553 raise ValueError("not a univariate polynomial")
556 # Convert the coeficcients to the parent's base ring,
557 # because a priori they might live in an (unnecessarily)
558 # larger ring for which P.sum() would fail below.
559 cs
= [ R(c
) for c
in p
.coefficients(sparse
=False) ]
560 return P
.sum( cs
[k
]*(self
**k
) for k
in range(len(cs
)) )
563 def characteristic_polynomial(self
):
565 Return the characteristic polynomial of this element.
569 The rank of `R^3` is three, and the minimal polynomial of
570 the identity element is `(t-1)` from which it follows that
571 the characteristic polynomial should be `(t-1)^3`::
573 sage: J = RealCartesianProductEJA(3)
574 sage: J.one().characteristic_polynomial()
575 t^3 - 3*t^2 + 3*t - 1
577 Likewise, the characteristic of the zero element in the
578 rank-three algebra `R^{n}` should be `t^{3}`::
580 sage: J = RealCartesianProductEJA(3)
581 sage: J.zero().characteristic_polynomial()
584 The characteristic polynomial of an element should evaluate
585 to zero on that element::
587 sage: set_random_seed()
588 sage: x = RealCartesianProductEJA(3).random_element()
589 sage: p = x.characteristic_polynomial()
590 sage: x.apply_univariate_polynomial(p)
594 p
= self
.parent().characteristic_polynomial()
595 return p(*self
.vector())
598 def inner_product(self
, other
):
600 Return the parent algebra's inner product of myself and ``other``.
604 The inner product in the Jordan spin algebra is the usual
605 inner product on `R^n` (this example only works because the
606 basis for the Jordan algebra is the standard basis in `R^n`)::
608 sage: J = JordanSpinEJA(3)
609 sage: x = vector(QQ,[1,2,3])
610 sage: y = vector(QQ,[4,5,6])
611 sage: x.inner_product(y)
613 sage: J(x).inner_product(J(y))
616 The inner product on `S^n` is `<X,Y> = trace(X*Y)`, where
617 multiplication is the usual matrix multiplication in `S^n`,
618 so the inner product of the identity matrix with itself
621 sage: J = RealSymmetricEJA(3)
622 sage: J.one().inner_product(J.one())
625 Likewise, the inner product on `C^n` is `<X,Y> =
626 Re(trace(X*Y))`, where we must necessarily take the real
627 part because the product of Hermitian matrices may not be
630 sage: J = ComplexHermitianEJA(3)
631 sage: J.one().inner_product(J.one())
634 Ditto for the quaternions::
636 sage: J = QuaternionHermitianEJA(3)
637 sage: J.one().inner_product(J.one())
642 Ensure that we can always compute an inner product, and that
643 it gives us back a real number::
645 sage: set_random_seed()
646 sage: J = random_eja()
647 sage: x = J.random_element()
648 sage: y = J.random_element()
649 sage: x.inner_product(y) in RR
655 raise TypeError("'other' must live in the same algebra")
657 return P
.inner_product(self
, other
)
660 def operator_commutes_with(self
, other
):
662 Return whether or not this element operator-commutes
667 The definition of a Jordan algebra says that any element
668 operator-commutes with its square::
670 sage: set_random_seed()
671 sage: x = random_eja().random_element()
672 sage: x.operator_commutes_with(x^2)
677 Test Lemma 1 from Chapter III of Koecher::
679 sage: set_random_seed()
680 sage: J = random_eja()
681 sage: u = J.random_element()
682 sage: v = J.random_element()
683 sage: lhs = u.operator_commutes_with(u*v)
684 sage: rhs = v.operator_commutes_with(u^2)
689 if not other
in self
.parent():
690 raise TypeError("'other' must live in the same algebra")
692 A
= self
.operator_matrix()
693 B
= other
.operator_matrix()
699 Return my determinant, the product of my eigenvalues.
703 sage: J = JordanSpinEJA(2)
704 sage: e0,e1 = J.gens()
705 sage: x = sum( J.gens() )
711 sage: J = JordanSpinEJA(3)
712 sage: e0,e1,e2 = J.gens()
713 sage: x = sum( J.gens() )
719 An element is invertible if and only if its determinant is
722 sage: set_random_seed()
723 sage: x = random_eja().random_element()
724 sage: x.is_invertible() == (x.det() != 0)
730 p
= P
._charpoly
_coeff
(0)
731 # The _charpoly_coeff function already adds the factor of
732 # -1 to ensure that _charpoly_coeff(0) is really what
733 # appears in front of t^{0} in the charpoly. However,
734 # we want (-1)^r times THAT for the determinant.
735 return ((-1)**r
)*p(*self
.vector())
740 Return the Jordan-multiplicative inverse of this element.
744 We appeal to the quadratic representation as in Koecher's
745 Theorem 12 in Chapter III, Section 5.
749 The inverse in the spin factor algebra is given in Alizadeh's
752 sage: set_random_seed()
753 sage: n = ZZ.random_element(1,10)
754 sage: J = JordanSpinEJA(n)
755 sage: x = J.random_element()
756 sage: while x.is_zero():
757 ....: x = J.random_element()
758 sage: x_vec = x.vector()
760 sage: x_bar = x_vec[1:]
761 sage: coeff = 1/(x0^2 - x_bar.inner_product(x_bar))
762 sage: inv_vec = x_vec.parent()([x0] + (-x_bar).list())
763 sage: x_inverse = coeff*inv_vec
764 sage: x.inverse() == J(x_inverse)
769 The identity element is its own inverse::
771 sage: set_random_seed()
772 sage: J = random_eja()
773 sage: J.one().inverse() == J.one()
776 If an element has an inverse, it acts like one::
778 sage: set_random_seed()
779 sage: J = random_eja()
780 sage: x = J.random_element()
781 sage: (not x.is_invertible()) or (x.inverse()*x == J.one())
784 The inverse of the inverse is what we started with::
786 sage: set_random_seed()
787 sage: J = random_eja()
788 sage: x = J.random_element()
789 sage: (not x.is_invertible()) or (x.inverse().inverse() == x)
792 The zero element is never invertible::
794 sage: set_random_seed()
795 sage: J = random_eja().zero().inverse()
796 Traceback (most recent call last):
798 ValueError: element is not invertible
801 if not self
.is_invertible():
802 raise ValueError("element is not invertible")
805 return P(self
.quadratic_representation().inverse()*self
.vector())
808 def is_invertible(self
):
810 Return whether or not this element is invertible.
812 We can't use the superclass method because it relies on
813 the algebra being associative.
817 The usual way to do this is to check if the determinant is
818 zero, but we need the characteristic polynomial for the
819 determinant. The minimal polynomial is a lot easier to get,
820 so we use Corollary 2 in Chapter V of Koecher to check
821 whether or not the paren't algebra's zero element is a root
822 of this element's minimal polynomial.
826 The identity element is always invertible::
828 sage: set_random_seed()
829 sage: J = random_eja()
830 sage: J.one().is_invertible()
833 The zero element is never invertible::
835 sage: set_random_seed()
836 sage: J = random_eja()
837 sage: J.zero().is_invertible()
841 zero
= self
.parent().zero()
842 p
= self
.minimal_polynomial()
843 return not (p(zero
) == zero
)
846 def is_nilpotent(self
):
848 Return whether or not some power of this element is zero.
850 The superclass method won't work unless we're in an
851 associative algebra, and we aren't. However, we generate
852 an assocoative subalgebra and we're nilpotent there if and
853 only if we're nilpotent here (probably).
857 The identity element is never nilpotent::
859 sage: set_random_seed()
860 sage: random_eja().one().is_nilpotent()
863 The additive identity is always nilpotent::
865 sage: set_random_seed()
866 sage: random_eja().zero().is_nilpotent()
870 # The element we're going to call "is_nilpotent()" on.
871 # Either myself, interpreted as an element of a finite-
872 # dimensional algebra, or an element of an associative
876 if self
.parent().is_associative():
877 elt
= FiniteDimensionalAlgebraElement(self
.parent(), self
)
879 V
= self
.span_of_powers()
880 assoc_subalg
= self
.subalgebra_generated_by()
881 # Mis-design warning: the basis used for span_of_powers()
882 # and subalgebra_generated_by() must be the same, and in
884 elt
= assoc_subalg(V
.coordinates(self
.vector()))
886 # Recursive call, but should work since elt lives in an
887 # associative algebra.
888 return elt
.is_nilpotent()
891 def is_regular(self
):
893 Return whether or not this is a regular element.
897 The identity element always has degree one, but any element
898 linearly-independent from it is regular::
900 sage: J = JordanSpinEJA(5)
901 sage: J.one().is_regular()
903 sage: e0, e1, e2, e3, e4 = J.gens() # e0 is the identity
904 sage: for x in J.gens():
905 ....: (J.one() + x).is_regular()
913 return self
.degree() == self
.parent().rank()
918 Compute the degree of this element the straightforward way
919 according to the definition; by appending powers to a list
920 and figuring out its dimension (that is, whether or not
921 they're linearly dependent).
925 sage: J = JordanSpinEJA(4)
926 sage: J.one().degree()
928 sage: e0,e1,e2,e3 = J.gens()
929 sage: (e0 - e1).degree()
932 In the spin factor algebra (of rank two), all elements that
933 aren't multiples of the identity are regular::
935 sage: set_random_seed()
936 sage: n = ZZ.random_element(1,10)
937 sage: J = JordanSpinEJA(n)
938 sage: x = J.random_element()
939 sage: x == x.coefficient(0)*J.one() or x.degree() == 2
943 return self
.span_of_powers().dimension()
946 def left_matrix(self
):
948 Our parent class defines ``left_matrix`` and ``matrix``
949 methods whose names are misleading. We don't want them.
951 raise NotImplementedError("use operator_matrix() instead")
956 def minimal_polynomial(self
):
958 Return the minimal polynomial of this element,
959 as a function of the variable `t`.
963 We restrict ourselves to the associative subalgebra
964 generated by this element, and then return the minimal
965 polynomial of this element's operator matrix (in that
966 subalgebra). This works by Baes Proposition 2.3.16.
970 The minimal polynomial of the identity and zero elements are
973 sage: set_random_seed()
974 sage: J = random_eja()
975 sage: J.one().minimal_polynomial()
977 sage: J.zero().minimal_polynomial()
980 The degree of an element is (by one definition) the degree
981 of its minimal polynomial::
983 sage: set_random_seed()
984 sage: x = random_eja().random_element()
985 sage: x.degree() == x.minimal_polynomial().degree()
988 The minimal polynomial and the characteristic polynomial coincide
989 and are known (see Alizadeh, Example 11.11) for all elements of
990 the spin factor algebra that aren't scalar multiples of the
993 sage: set_random_seed()
994 sage: n = ZZ.random_element(2,10)
995 sage: J = JordanSpinEJA(n)
996 sage: y = J.random_element()
997 sage: while y == y.coefficient(0)*J.one():
998 ....: y = J.random_element()
999 sage: y0 = y.vector()[0]
1000 sage: y_bar = y.vector()[1:]
1001 sage: actual = y.minimal_polynomial()
1002 sage: t = PolynomialRing(J.base_ring(),'t').gen(0)
1003 sage: expected = t^2 - 2*y0*t + (y0^2 - norm(y_bar)^2)
1004 sage: bool(actual == expected)
1007 The minimal polynomial should always kill its element::
1009 sage: set_random_seed()
1010 sage: x = random_eja().random_element()
1011 sage: p = x.minimal_polynomial()
1012 sage: x.apply_univariate_polynomial(p)
1016 V
= self
.span_of_powers()
1017 assoc_subalg
= self
.subalgebra_generated_by()
1018 # Mis-design warning: the basis used for span_of_powers()
1019 # and subalgebra_generated_by() must be the same, and in
1021 elt
= assoc_subalg(V
.coordinates(self
.vector()))
1023 # We get back a symbolic polynomial in 'x' but want a real
1024 # polynomial in 't'.
1025 p_of_x
= elt
.operator_matrix().minimal_polynomial()
1026 return p_of_x
.change_variable_name('t')
1029 def natural_representation(self
):
1031 Return a more-natural representation of this element.
1033 Every finite-dimensional Euclidean Jordan Algebra is a
1034 direct sum of five simple algebras, four of which comprise
1035 Hermitian matrices. This method returns the original
1036 "natural" representation of this element as a Hermitian
1037 matrix, if it has one. If not, you get the usual representation.
1041 sage: J = ComplexHermitianEJA(3)
1044 sage: J.one().natural_representation()
1054 sage: J = QuaternionHermitianEJA(3)
1057 sage: J.one().natural_representation()
1058 [1 0 0 0 0 0 0 0 0 0 0 0]
1059 [0 1 0 0 0 0 0 0 0 0 0 0]
1060 [0 0 1 0 0 0 0 0 0 0 0 0]
1061 [0 0 0 1 0 0 0 0 0 0 0 0]
1062 [0 0 0 0 1 0 0 0 0 0 0 0]
1063 [0 0 0 0 0 1 0 0 0 0 0 0]
1064 [0 0 0 0 0 0 1 0 0 0 0 0]
1065 [0 0 0 0 0 0 0 1 0 0 0 0]
1066 [0 0 0 0 0 0 0 0 1 0 0 0]
1067 [0 0 0 0 0 0 0 0 0 1 0 0]
1068 [0 0 0 0 0 0 0 0 0 0 1 0]
1069 [0 0 0 0 0 0 0 0 0 0 0 1]
1072 B
= self
.parent().natural_basis()
1073 W
= B
[0].matrix_space()
1074 return W
.linear_combination(zip(self
.vector(), B
))
1077 def operator_matrix(self
):
1079 Return the matrix that represents left- (or right-)
1080 multiplication by this element in the parent algebra.
1082 We implement this ourselves to work around the fact that
1083 our parent class represents everything with row vectors.
1087 Test the first polarization identity from my notes, Koecher Chapter
1088 III, or from Baes (2.3)::
1090 sage: set_random_seed()
1091 sage: J = random_eja()
1092 sage: x = J.random_element()
1093 sage: y = J.random_element()
1094 sage: Lx = x.operator_matrix()
1095 sage: Ly = y.operator_matrix()
1096 sage: Lxx = (x*x).operator_matrix()
1097 sage: Lxy = (x*y).operator_matrix()
1098 sage: bool(2*Lx*Lxy + Ly*Lxx == 2*Lxy*Lx + Lxx*Ly)
1101 Test the second polarization identity from my notes or from
1104 sage: set_random_seed()
1105 sage: J = random_eja()
1106 sage: x = J.random_element()
1107 sage: y = J.random_element()
1108 sage: z = J.random_element()
1109 sage: Lx = x.operator_matrix()
1110 sage: Ly = y.operator_matrix()
1111 sage: Lz = z.operator_matrix()
1112 sage: Lzy = (z*y).operator_matrix()
1113 sage: Lxy = (x*y).operator_matrix()
1114 sage: Lxz = (x*z).operator_matrix()
1115 sage: bool(Lx*Lzy + Lz*Lxy + Ly*Lxz == Lzy*Lx + Lxy*Lz + Lxz*Ly)
1118 Test the third polarization identity from my notes or from
1121 sage: set_random_seed()
1122 sage: J = random_eja()
1123 sage: u = J.random_element()
1124 sage: y = J.random_element()
1125 sage: z = J.random_element()
1126 sage: Lu = u.operator_matrix()
1127 sage: Ly = y.operator_matrix()
1128 sage: Lz = z.operator_matrix()
1129 sage: Lzy = (z*y).operator_matrix()
1130 sage: Luy = (u*y).operator_matrix()
1131 sage: Luz = (u*z).operator_matrix()
1132 sage: Luyz = (u*(y*z)).operator_matrix()
1133 sage: lhs = Lu*Lzy + Lz*Luy + Ly*Luz
1134 sage: rhs = Luyz + Ly*Lu*Lz + Lz*Lu*Ly
1135 sage: bool(lhs == rhs)
1139 fda_elt
= FiniteDimensionalAlgebraElement(self
.parent(), self
)
1140 return fda_elt
.matrix().transpose()
1143 def quadratic_representation(self
, other
=None):
1145 Return the quadratic representation of this element.
1149 The explicit form in the spin factor algebra is given by
1150 Alizadeh's Example 11.12::
1152 sage: set_random_seed()
1153 sage: n = ZZ.random_element(1,10)
1154 sage: J = JordanSpinEJA(n)
1155 sage: x = J.random_element()
1156 sage: x_vec = x.vector()
1158 sage: x_bar = x_vec[1:]
1159 sage: A = matrix(QQ, 1, [x_vec.inner_product(x_vec)])
1160 sage: B = 2*x0*x_bar.row()
1161 sage: C = 2*x0*x_bar.column()
1162 sage: D = identity_matrix(QQ, n-1)
1163 sage: D = (x0^2 - x_bar.inner_product(x_bar))*D
1164 sage: D = D + 2*x_bar.tensor_product(x_bar)
1165 sage: Q = block_matrix(2,2,[A,B,C,D])
1166 sage: Q == x.quadratic_representation()
1169 Test all of the properties from Theorem 11.2 in Alizadeh::
1171 sage: set_random_seed()
1172 sage: J = random_eja()
1173 sage: x = J.random_element()
1174 sage: y = J.random_element()
1175 sage: Lx = x.operator_matrix()
1176 sage: Lxx = (x*x).operator_matrix()
1177 sage: Qx = x.quadratic_representation()
1178 sage: Qy = y.quadratic_representation()
1179 sage: Qxy = x.quadratic_representation(y)
1180 sage: Qex = J.one().quadratic_representation(x)
1181 sage: n = ZZ.random_element(10)
1182 sage: Qxn = (x^n).quadratic_representation()
1186 sage: 2*Qxy == (x+y).quadratic_representation() - Qx - Qy
1191 sage: alpha = QQ.random_element()
1192 sage: (alpha*x).quadratic_representation() == (alpha^2)*Qx
1197 sage: not x.is_invertible() or (
1198 ....: Qx*x.inverse().vector() == x.vector() )
1201 sage: not x.is_invertible() or (
1204 ....: x.inverse().quadratic_representation() )
1207 sage: Qxy*(J.one().vector()) == (x*y).vector()
1212 sage: not x.is_invertible() or (
1213 ....: x.quadratic_representation(x.inverse())*Qx
1214 ....: == Qx*x.quadratic_representation(x.inverse()) )
1217 sage: not x.is_invertible() or (
1218 ....: x.quadratic_representation(x.inverse())*Qx
1220 ....: 2*x.operator_matrix()*Qex - Qx )
1223 sage: 2*x.operator_matrix()*Qex - Qx == Lxx
1228 sage: J(Qy*x.vector()).quadratic_representation() == Qy*Qx*Qy
1238 sage: not x.is_invertible() or (
1239 ....: Qx*x.inverse().operator_matrix() == Lx )
1244 sage: not x.operator_commutes_with(y) or (
1245 ....: J(Qx*y.vector())^n == J(Qxn*(y^n).vector()) )
1251 elif not other
in self
.parent():
1252 raise TypeError("'other' must live in the same algebra")
1254 L
= self
.operator_matrix()
1255 M
= other
.operator_matrix()
1256 return ( L
*M
+ M
*L
- (self
*other
).operator_matrix() )
1259 def span_of_powers(self
):
1261 Return the vector space spanned by successive powers of
1264 # The dimension of the subalgebra can't be greater than
1265 # the big algebra, so just put everything into a list
1266 # and let span() get rid of the excess.
1268 # We do the extra ambient_vector_space() in case we're messing
1269 # with polynomials and the direct parent is a module.
1270 V
= self
.vector().parent().ambient_vector_space()
1271 return V
.span( (self
**d
).vector() for d
in xrange(V
.dimension()) )
1274 def subalgebra_generated_by(self
):
1276 Return the associative subalgebra of the parent EJA generated
1281 sage: set_random_seed()
1282 sage: x = random_eja().random_element()
1283 sage: x.subalgebra_generated_by().is_associative()
1286 Squaring in the subalgebra should be the same thing as
1287 squaring in the superalgebra::
1289 sage: set_random_seed()
1290 sage: x = random_eja().random_element()
1291 sage: u = x.subalgebra_generated_by().random_element()
1292 sage: u.operator_matrix()*u.vector() == (u**2).vector()
1296 # First get the subspace spanned by the powers of myself...
1297 V
= self
.span_of_powers()
1298 F
= self
.base_ring()
1300 # Now figure out the entries of the right-multiplication
1301 # matrix for the successive basis elements b0, b1,... of
1304 for b_right
in V
.basis():
1305 eja_b_right
= self
.parent()(b_right
)
1307 # The first row of the right-multiplication matrix by
1308 # b1 is what we get if we apply that matrix to b1. The
1309 # second row of the right multiplication matrix by b1
1310 # is what we get when we apply that matrix to b2...
1312 # IMPORTANT: this assumes that all vectors are COLUMN
1313 # vectors, unlike our superclass (which uses row vectors).
1314 for b_left
in V
.basis():
1315 eja_b_left
= self
.parent()(b_left
)
1316 # Multiply in the original EJA, but then get the
1317 # coordinates from the subalgebra in terms of its
1319 this_row
= V
.coordinates((eja_b_left
*eja_b_right
).vector())
1320 b_right_rows
.append(this_row
)
1321 b_right_matrix
= matrix(F
, b_right_rows
)
1322 mats
.append(b_right_matrix
)
1324 # It's an algebra of polynomials in one element, and EJAs
1325 # are power-associative.
1327 # TODO: choose generator names intelligently.
1328 return FiniteDimensionalEuclideanJordanAlgebra(F
, mats
, assume_associative
=True, names
='f')
1331 def subalgebra_idempotent(self
):
1333 Find an idempotent in the associative subalgebra I generate
1334 using Proposition 2.3.5 in Baes.
1338 sage: set_random_seed()
1339 sage: J = random_eja()
1340 sage: x = J.random_element()
1341 sage: while x.is_nilpotent():
1342 ....: x = J.random_element()
1343 sage: c = x.subalgebra_idempotent()
1348 if self
.is_nilpotent():
1349 raise ValueError("this only works with non-nilpotent elements!")
1351 V
= self
.span_of_powers()
1352 J
= self
.subalgebra_generated_by()
1353 # Mis-design warning: the basis used for span_of_powers()
1354 # and subalgebra_generated_by() must be the same, and in
1356 u
= J(V
.coordinates(self
.vector()))
1358 # The image of the matrix of left-u^m-multiplication
1359 # will be minimal for some natural number s...
1361 minimal_dim
= V
.dimension()
1362 for i
in xrange(1, V
.dimension()):
1363 this_dim
= (u
**i
).operator_matrix().image().dimension()
1364 if this_dim
< minimal_dim
:
1365 minimal_dim
= this_dim
1368 # Now minimal_matrix should correspond to the smallest
1369 # non-zero subspace in Baes's (or really, Koecher's)
1372 # However, we need to restrict the matrix to work on the
1373 # subspace... or do we? Can't we just solve, knowing that
1374 # A(c) = u^(s+1) should have a solution in the big space,
1377 # Beware, solve_right() means that we're using COLUMN vectors.
1378 # Our FiniteDimensionalAlgebraElement superclass uses rows.
1380 A
= u_next
.operator_matrix()
1381 c_coordinates
= A
.solve_right(u_next
.vector())
1383 # Now c_coordinates is the idempotent we want, but it's in
1384 # the coordinate system of the subalgebra.
1386 # We need the basis for J, but as elements of the parent algebra.
1388 basis
= [self
.parent(v
) for v
in V
.basis()]
1389 return self
.parent().linear_combination(zip(c_coordinates
, basis
))
1394 Return my trace, the sum of my eigenvalues.
1398 sage: J = JordanSpinEJA(3)
1399 sage: x = sum(J.gens())
1405 sage: J = RealCartesianProductEJA(5)
1406 sage: J.one().trace()
1411 The trace of an element is a real number::
1413 sage: set_random_seed()
1414 sage: J = random_eja()
1415 sage: J.random_element().trace() in J.base_ring()
1421 p
= P
._charpoly
_coeff
(r
-1)
1422 # The _charpoly_coeff function already adds the factor of
1423 # -1 to ensure that _charpoly_coeff(r-1) is really what
1424 # appears in front of t^{r-1} in the charpoly. However,
1425 # we want the negative of THAT for the trace.
1426 return -p(*self
.vector())
1429 def trace_inner_product(self
, other
):
1431 Return the trace inner product of myself and ``other``.
1435 The trace inner product is commutative::
1437 sage: set_random_seed()
1438 sage: J = random_eja()
1439 sage: x = J.random_element(); y = J.random_element()
1440 sage: x.trace_inner_product(y) == y.trace_inner_product(x)
1443 The trace inner product is bilinear::
1445 sage: set_random_seed()
1446 sage: J = random_eja()
1447 sage: x = J.random_element()
1448 sage: y = J.random_element()
1449 sage: z = J.random_element()
1450 sage: a = QQ.random_element();
1451 sage: actual = (a*(x+z)).trace_inner_product(y)
1452 sage: expected = ( a*x.trace_inner_product(y) +
1453 ....: a*z.trace_inner_product(y) )
1454 sage: actual == expected
1456 sage: actual = x.trace_inner_product(a*(y+z))
1457 sage: expected = ( a*x.trace_inner_product(y) +
1458 ....: a*x.trace_inner_product(z) )
1459 sage: actual == expected
1462 The trace inner product satisfies the compatibility
1463 condition in the definition of a Euclidean Jordan algebra::
1465 sage: set_random_seed()
1466 sage: J = random_eja()
1467 sage: x = J.random_element()
1468 sage: y = J.random_element()
1469 sage: z = J.random_element()
1470 sage: (x*y).trace_inner_product(z) == y.trace_inner_product(x*z)
1474 if not other
in self
.parent():
1475 raise TypeError("'other' must live in the same algebra")
1477 return (self
*other
).trace()
1480 class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1482 Return the Euclidean Jordan Algebra corresponding to the set
1483 `R^n` under the Hadamard product.
1485 Note: this is nothing more than the Cartesian product of ``n``
1486 copies of the spin algebra. Once Cartesian product algebras
1487 are implemented, this can go.
1491 This multiplication table can be verified by hand::
1493 sage: J = RealCartesianProductEJA(3)
1494 sage: e0,e1,e2 = J.gens()
1510 def __classcall_private__(cls
, n
, field
=QQ
):
1511 # The FiniteDimensionalAlgebra constructor takes a list of
1512 # matrices, the ith representing right multiplication by the ith
1513 # basis element in the vector space. So if e_1 = (1,0,0), then
1514 # right (Hadamard) multiplication of x by e_1 picks out the first
1515 # component of x; and likewise for the ith basis element e_i.
1516 Qs
= [ matrix(field
, n
, n
, lambda k
,j
: 1*(k
== j
== i
))
1517 for i
in xrange(n
) ]
1519 fdeja
= super(RealCartesianProductEJA
, cls
)
1520 return fdeja
.__classcall
_private
__(cls
, field
, Qs
, rank
=n
)
1522 def inner_product(self
, x
, y
):
1523 return _usual_ip(x
,y
)
1528 Return a "random" finite-dimensional Euclidean Jordan Algebra.
1532 For now, we choose a random natural number ``n`` (greater than zero)
1533 and then give you back one of the following:
1535 * The cartesian product of the rational numbers ``n`` times; this is
1536 ``QQ^n`` with the Hadamard product.
1538 * The Jordan spin algebra on ``QQ^n``.
1540 * The ``n``-by-``n`` rational symmetric matrices with the symmetric
1543 * The ``n``-by-``n`` complex-rational Hermitian matrices embedded
1544 in the space of ``2n``-by-``2n`` real symmetric matrices.
1546 * The ``n``-by-``n`` quaternion-rational Hermitian matrices embedded
1547 in the space of ``4n``-by-``4n`` real symmetric matrices.
1549 Later this might be extended to return Cartesian products of the
1555 Euclidean Jordan algebra of degree...
1559 # The max_n component lets us choose different upper bounds on the
1560 # value "n" that gets passed to the constructor. This is needed
1561 # because e.g. R^{10} is reasonable to test, while the Hermitian
1562 # 10-by-10 quaternion matrices are not.
1563 (constructor
, max_n
) = choice([(RealCartesianProductEJA
, 6),
1565 (RealSymmetricEJA
, 5),
1566 (ComplexHermitianEJA
, 4),
1567 (QuaternionHermitianEJA
, 3)])
1568 n
= ZZ
.random_element(1, max_n
)
1569 return constructor(n
, field
=QQ
)
1573 def _real_symmetric_basis(n
, field
=QQ
):
1575 Return a basis for the space of real symmetric n-by-n matrices.
1577 # The basis of symmetric matrices, as matrices, in their R^(n-by-n)
1581 for j
in xrange(i
+1):
1582 Eij
= matrix(field
, n
, lambda k
,l
: k
==i
and l
==j
)
1586 # Beware, orthogonal but not normalized!
1587 Sij
= Eij
+ Eij
.transpose()
1592 def _complex_hermitian_basis(n
, field
=QQ
):
1594 Returns a basis for the space of complex Hermitian n-by-n matrices.
1598 sage: set_random_seed()
1599 sage: n = ZZ.random_element(1,5)
1600 sage: all( M.is_symmetric() for M in _complex_hermitian_basis(n) )
1604 F
= QuadraticField(-1, 'I')
1607 # This is like the symmetric case, but we need to be careful:
1609 # * We want conjugate-symmetry, not just symmetry.
1610 # * The diagonal will (as a result) be real.
1614 for j
in xrange(i
+1):
1615 Eij
= matrix(field
, n
, lambda k
,l
: k
==i
and l
==j
)
1617 Sij
= _embed_complex_matrix(Eij
)
1620 # Beware, orthogonal but not normalized! The second one
1621 # has a minus because it's conjugated.
1622 Sij_real
= _embed_complex_matrix(Eij
+ Eij
.transpose())
1624 Sij_imag
= _embed_complex_matrix(I
*Eij
- I
*Eij
.transpose())
1629 def _quaternion_hermitian_basis(n
, field
=QQ
):
1631 Returns a basis for the space of quaternion Hermitian n-by-n matrices.
1635 sage: set_random_seed()
1636 sage: n = ZZ.random_element(1,5)
1637 sage: all( M.is_symmetric() for M in _quaternion_hermitian_basis(n) )
1641 Q
= QuaternionAlgebra(QQ
,-1,-1)
1644 # This is like the symmetric case, but we need to be careful:
1646 # * We want conjugate-symmetry, not just symmetry.
1647 # * The diagonal will (as a result) be real.
1651 for j
in xrange(i
+1):
1652 Eij
= matrix(Q
, n
, lambda k
,l
: k
==i
and l
==j
)
1654 Sij
= _embed_quaternion_matrix(Eij
)
1657 # Beware, orthogonal but not normalized! The second,
1658 # third, and fourth ones have a minus because they're
1660 Sij_real
= _embed_quaternion_matrix(Eij
+ Eij
.transpose())
1662 Sij_I
= _embed_quaternion_matrix(I
*Eij
- I
*Eij
.transpose())
1664 Sij_J
= _embed_quaternion_matrix(J
*Eij
- J
*Eij
.transpose())
1666 Sij_K
= _embed_quaternion_matrix(K
*Eij
- K
*Eij
.transpose())
1672 return vector(m
.base_ring(), m
.list())
1675 return matrix(v
.base_ring(), sqrt(v
.degree()), v
.list())
1677 def _multiplication_table_from_matrix_basis(basis
):
1679 At least three of the five simple Euclidean Jordan algebras have the
1680 symmetric multiplication (A,B) |-> (AB + BA)/2, where the
1681 multiplication on the right is matrix multiplication. Given a basis
1682 for the underlying matrix space, this function returns a
1683 multiplication table (obtained by looping through the basis
1684 elements) for an algebra of those matrices. A reordered copy
1685 of the basis is also returned to work around the fact that
1686 the ``span()`` in this function will change the order of the basis
1687 from what we think it is, to... something else.
1689 # In S^2, for example, we nominally have four coordinates even
1690 # though the space is of dimension three only. The vector space V
1691 # is supposed to hold the entire long vector, and the subspace W
1692 # of V will be spanned by the vectors that arise from symmetric
1693 # matrices. Thus for S^2, dim(V) == 4 and dim(W) == 3.
1694 field
= basis
[0].base_ring()
1695 dimension
= basis
[0].nrows()
1697 V
= VectorSpace(field
, dimension
**2)
1698 W
= V
.span( _mat2vec(s
) for s
in basis
)
1700 # Taking the span above reorders our basis (thanks, jerk!) so we
1701 # need to put our "matrix basis" in the same order as the
1702 # (reordered) vector basis.
1703 S
= tuple( _vec2mat(b
) for b
in W
.basis() )
1707 # Brute force the multiplication-by-s matrix by looping
1708 # through all elements of the basis and doing the computation
1709 # to find out what the corresponding row should be. BEWARE:
1710 # these multiplication tables won't be symmetric! It therefore
1711 # becomes REALLY IMPORTANT that the underlying algebra
1712 # constructor uses ROW vectors and not COLUMN vectors. That's
1713 # why we're computing rows here and not columns.
1716 this_row
= _mat2vec((s
*t
+ t
*s
)/2)
1717 Q_rows
.append(W
.coordinates(this_row
))
1718 Q
= matrix(field
, W
.dimension(), Q_rows
)
1724 def _embed_complex_matrix(M
):
1726 Embed the n-by-n complex matrix ``M`` into the space of real
1727 matrices of size 2n-by-2n via the map the sends each entry `z = a +
1728 bi` to the block matrix ``[[a,b],[-b,a]]``.
1732 sage: F = QuadraticField(-1,'i')
1733 sage: x1 = F(4 - 2*i)
1734 sage: x2 = F(1 + 2*i)
1737 sage: M = matrix(F,2,[[x1,x2],[x3,x4]])
1738 sage: _embed_complex_matrix(M)
1747 Embedding is a homomorphism (isomorphism, in fact)::
1749 sage: set_random_seed()
1750 sage: n = ZZ.random_element(5)
1751 sage: F = QuadraticField(-1, 'i')
1752 sage: X = random_matrix(F, n)
1753 sage: Y = random_matrix(F, n)
1754 sage: actual = _embed_complex_matrix(X) * _embed_complex_matrix(Y)
1755 sage: expected = _embed_complex_matrix(X*Y)
1756 sage: actual == expected
1762 raise ValueError("the matrix 'M' must be square")
1763 field
= M
.base_ring()
1768 blocks
.append(matrix(field
, 2, [[a
,b
],[-b
,a
]]))
1770 # We can drop the imaginaries here.
1771 return block_matrix(field
.base_ring(), n
, blocks
)
1774 def _unembed_complex_matrix(M
):
1776 The inverse of _embed_complex_matrix().
1780 sage: A = matrix(QQ,[ [ 1, 2, 3, 4],
1781 ....: [-2, 1, -4, 3],
1782 ....: [ 9, 10, 11, 12],
1783 ....: [-10, 9, -12, 11] ])
1784 sage: _unembed_complex_matrix(A)
1786 [ 10*i + 9 12*i + 11]
1790 Unembedding is the inverse of embedding::
1792 sage: set_random_seed()
1793 sage: F = QuadraticField(-1, 'i')
1794 sage: M = random_matrix(F, 3)
1795 sage: _unembed_complex_matrix(_embed_complex_matrix(M)) == M
1801 raise ValueError("the matrix 'M' must be square")
1802 if not n
.mod(2).is_zero():
1803 raise ValueError("the matrix 'M' must be a complex embedding")
1805 F
= QuadraticField(-1, 'i')
1808 # Go top-left to bottom-right (reading order), converting every
1809 # 2-by-2 block we see to a single complex element.
1811 for k
in xrange(n
/2):
1812 for j
in xrange(n
/2):
1813 submat
= M
[2*k
:2*k
+2,2*j
:2*j
+2]
1814 if submat
[0,0] != submat
[1,1]:
1815 raise ValueError('bad on-diagonal submatrix')
1816 if submat
[0,1] != -submat
[1,0]:
1817 raise ValueError('bad off-diagonal submatrix')
1818 z
= submat
[0,0] + submat
[0,1]*i
1821 return matrix(F
, n
/2, elements
)
1824 def _embed_quaternion_matrix(M
):
1826 Embed the n-by-n quaternion matrix ``M`` into the space of real
1827 matrices of size 4n-by-4n by first sending each quaternion entry
1828 `z = a + bi + cj + dk` to the block-complex matrix
1829 ``[[a + bi, c+di],[-c + di, a-bi]]`, and then embedding those into
1834 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1835 sage: i,j,k = Q.gens()
1836 sage: x = 1 + 2*i + 3*j + 4*k
1837 sage: M = matrix(Q, 1, [[x]])
1838 sage: _embed_quaternion_matrix(M)
1844 Embedding is a homomorphism (isomorphism, in fact)::
1846 sage: set_random_seed()
1847 sage: n = ZZ.random_element(5)
1848 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1849 sage: X = random_matrix(Q, n)
1850 sage: Y = random_matrix(Q, n)
1851 sage: actual = _embed_quaternion_matrix(X)*_embed_quaternion_matrix(Y)
1852 sage: expected = _embed_quaternion_matrix(X*Y)
1853 sage: actual == expected
1857 quaternions
= M
.base_ring()
1860 raise ValueError("the matrix 'M' must be square")
1862 F
= QuadraticField(-1, 'i')
1867 t
= z
.coefficient_tuple()
1872 cplx_matrix
= matrix(F
, 2, [[ a
+ b
*i
, c
+ d
*i
],
1873 [-c
+ d
*i
, a
- b
*i
]])
1874 blocks
.append(_embed_complex_matrix(cplx_matrix
))
1876 # We should have real entries by now, so use the realest field
1877 # we've got for the return value.
1878 return block_matrix(quaternions
.base_ring(), n
, blocks
)
1881 def _unembed_quaternion_matrix(M
):
1883 The inverse of _embed_quaternion_matrix().
1887 sage: M = matrix(QQ, [[ 1, 2, 3, 4],
1888 ....: [-2, 1, -4, 3],
1889 ....: [-3, 4, 1, -2],
1890 ....: [-4, -3, 2, 1]])
1891 sage: _unembed_quaternion_matrix(M)
1892 [1 + 2*i + 3*j + 4*k]
1896 Unembedding is the inverse of embedding::
1898 sage: set_random_seed()
1899 sage: Q = QuaternionAlgebra(QQ, -1, -1)
1900 sage: M = random_matrix(Q, 3)
1901 sage: _unembed_quaternion_matrix(_embed_quaternion_matrix(M)) == M
1907 raise ValueError("the matrix 'M' must be square")
1908 if not n
.mod(4).is_zero():
1909 raise ValueError("the matrix 'M' must be a complex embedding")
1911 Q
= QuaternionAlgebra(QQ
,-1,-1)
1914 # Go top-left to bottom-right (reading order), converting every
1915 # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1
1918 for l
in xrange(n
/4):
1919 for m
in xrange(n
/4):
1920 submat
= _unembed_complex_matrix(M
[4*l
:4*l
+4,4*m
:4*m
+4])
1921 if submat
[0,0] != submat
[1,1].conjugate():
1922 raise ValueError('bad on-diagonal submatrix')
1923 if submat
[0,1] != -submat
[1,0].conjugate():
1924 raise ValueError('bad off-diagonal submatrix')
1925 z
= submat
[0,0].real() + submat
[0,0].imag()*i
1926 z
+= submat
[0,1].real()*j
+ submat
[0,1].imag()*k
1929 return matrix(Q
, n
/4, elements
)
1932 # The usual inner product on R^n.
1934 return x
.vector().inner_product(y
.vector())
1936 # The inner product used for the real symmetric simple EJA.
1937 # We keep it as a separate function because e.g. the complex
1938 # algebra uses the same inner product, except divided by 2.
1939 def _matrix_ip(X
,Y
):
1940 X_mat
= X
.natural_representation()
1941 Y_mat
= Y
.natural_representation()
1942 return (X_mat
*Y_mat
).trace()
1945 class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1947 The rank-n simple EJA consisting of real symmetric n-by-n
1948 matrices, the usual symmetric Jordan product, and the trace inner
1949 product. It has dimension `(n^2 + n)/2` over the reals.
1953 sage: J = RealSymmetricEJA(2)
1954 sage: e0, e1, e2 = J.gens()
1964 The degree of this algebra is `(n^2 + n) / 2`::
1966 sage: set_random_seed()
1967 sage: n = ZZ.random_element(1,5)
1968 sage: J = RealSymmetricEJA(n)
1969 sage: J.degree() == (n^2 + n)/2
1972 The Jordan multiplication is what we think it is::
1974 sage: set_random_seed()
1975 sage: n = ZZ.random_element(1,5)
1976 sage: J = RealSymmetricEJA(n)
1977 sage: x = J.random_element()
1978 sage: y = J.random_element()
1979 sage: actual = (x*y).natural_representation()
1980 sage: X = x.natural_representation()
1981 sage: Y = y.natural_representation()
1982 sage: expected = (X*Y + Y*X)/2
1983 sage: actual == expected
1985 sage: J(expected) == x*y
1990 def __classcall_private__(cls
, n
, field
=QQ
):
1991 S
= _real_symmetric_basis(n
, field
=field
)
1992 (Qs
, T
) = _multiplication_table_from_matrix_basis(S
)
1994 fdeja
= super(RealSymmetricEJA
, cls
)
1995 return fdeja
.__classcall
_private
__(cls
,
2001 def inner_product(self
, x
, y
):
2002 return _matrix_ip(x
,y
)
2005 class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra
):
2007 The rank-n simple EJA consisting of complex Hermitian n-by-n
2008 matrices over the real numbers, the usual symmetric Jordan product,
2009 and the real-part-of-trace inner product. It has dimension `n^2` over
2014 The degree of this algebra is `n^2`::
2016 sage: set_random_seed()
2017 sage: n = ZZ.random_element(1,5)
2018 sage: J = ComplexHermitianEJA(n)
2019 sage: J.degree() == n^2
2022 The Jordan multiplication is what we think it is::
2024 sage: set_random_seed()
2025 sage: n = ZZ.random_element(1,5)
2026 sage: J = ComplexHermitianEJA(n)
2027 sage: x = J.random_element()
2028 sage: y = J.random_element()
2029 sage: actual = (x*y).natural_representation()
2030 sage: X = x.natural_representation()
2031 sage: Y = y.natural_representation()
2032 sage: expected = (X*Y + Y*X)/2
2033 sage: actual == expected
2035 sage: J(expected) == x*y
2040 def __classcall_private__(cls
, n
, field
=QQ
):
2041 S
= _complex_hermitian_basis(n
)
2042 (Qs
, T
) = _multiplication_table_from_matrix_basis(S
)
2044 fdeja
= super(ComplexHermitianEJA
, cls
)
2045 return fdeja
.__classcall
_private
__(cls
,
2051 def inner_product(self
, x
, y
):
2052 # Since a+bi on the diagonal is represented as
2057 # we'll double-count the "a" entries if we take the trace of
2059 return _matrix_ip(x
,y
)/2
2062 class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra
):
2064 The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
2065 matrices, the usual symmetric Jordan product, and the
2066 real-part-of-trace inner product. It has dimension `2n^2 - n` over
2071 The degree of this algebra is `n^2`::
2073 sage: set_random_seed()
2074 sage: n = ZZ.random_element(1,5)
2075 sage: J = QuaternionHermitianEJA(n)
2076 sage: J.degree() == 2*(n^2) - n
2079 The Jordan multiplication is what we think it is::
2081 sage: set_random_seed()
2082 sage: n = ZZ.random_element(1,5)
2083 sage: J = QuaternionHermitianEJA(n)
2084 sage: x = J.random_element()
2085 sage: y = J.random_element()
2086 sage: actual = (x*y).natural_representation()
2087 sage: X = x.natural_representation()
2088 sage: Y = y.natural_representation()
2089 sage: expected = (X*Y + Y*X)/2
2090 sage: actual == expected
2092 sage: J(expected) == x*y
2097 def __classcall_private__(cls
, n
, field
=QQ
):
2098 S
= _quaternion_hermitian_basis(n
)
2099 (Qs
, T
) = _multiplication_table_from_matrix_basis(S
)
2101 fdeja
= super(QuaternionHermitianEJA
, cls
)
2102 return fdeja
.__classcall
_private
__(cls
,
2108 def inner_product(self
, x
, y
):
2109 # Since a+bi+cj+dk on the diagonal is represented as
2111 # a + bi +cj + dk = [ a b c d]
2116 # we'll quadruple-count the "a" entries if we take the trace of
2118 return _matrix_ip(x
,y
)/4
2121 class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra
):
2123 The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
2124 with the usual inner product and jordan product ``x*y =
2125 (<x_bar,y_bar>, x0*y_bar + y0*x_bar)``. It has dimension `n` over
2130 This multiplication table can be verified by hand::
2132 sage: J = JordanSpinEJA(4)
2133 sage: e0,e1,e2,e3 = J.gens()
2151 def __classcall_private__(cls
, n
, field
=QQ
):
2153 id_matrix
= identity_matrix(field
, n
)
2155 ei
= id_matrix
.column(i
)
2156 Qi
= zero_matrix(field
, n
)
2158 Qi
.set_column(0, ei
)
2159 Qi
+= diagonal_matrix(n
, [ei
[0]]*n
)
2160 # The addition of the diagonal matrix adds an extra ei[0] in the
2161 # upper-left corner of the matrix.
2162 Qi
[0,0] = Qi
[0,0] * ~
field(2)
2165 # The rank of the spin algebra is two, unless we're in a
2166 # one-dimensional ambient space (because the rank is bounded by
2167 # the ambient dimension).
2168 fdeja
= super(JordanSpinEJA
, cls
)
2169 return fdeja
.__classcall
_private
__(cls
, field
, Qs
, rank
=min(n
,2))
2171 def inner_product(self
, x
, y
):
2172 return _usual_ip(x
,y
)