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
20 does only three things:
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 3. Allows us to add morphisms in the obvious way.
33 If this seems a bit heavyweight, it is. I would have been happy to
34 use a the ring morphism that underlies the finite-dimensional
35 algebra morphism, but they don't seem to be callable on elements of
36 our EJA, and you can't add them.
39 def __add__(self
, other
):
41 Add two EJA morphisms in the obvious way.
45 sage: J = RealSymmetricEJA(3)
48 sage: x.operator() + y.operator()
49 Morphism from Euclidean Jordan algebra of degree 6 over Rational
50 Field to Euclidean Jordan algebra of degree 6 over Rational Field
62 raise ValueError("summands must live in the same space")
64 return FiniteDimensionalEuclideanJordanAlgebraMorphism(
66 self
.matrix() + other
.matrix() )
69 def __init__(self
, parent
, f
):
70 FiniteDimensionalAlgebraMorphism
.__init
__(self
,
79 We override only the representation that is shown to the user,
80 because we want the matrix to be with respect to COLUMN vectors.
84 Ensure that we see the transpose of the underlying matrix object:
86 sage: J = RealSymmetricEJA(3)
87 sage: x = J.linear_combination(zip(range(len(J.gens())), J.gens()))
88 sage: L = x.operator()
90 Morphism from Euclidean Jordan algebra of degree 6 over Rational
91 Field to Euclidean Jordan algebra of degree 6 over Rational Field
108 return "Morphism from {} to {} given by matrix\n{}".format(
109 self
.domain(), self
.codomain(), self
.matrix())
113 Return the matrix of this morphism with respect to a left-action
116 return FiniteDimensionalAlgebraMorphism
.matrix(self
).transpose()
119 class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra
):
121 def __classcall_private__(cls
,
125 assume_associative
=False,
130 mult_table
= [b
.base_extend(field
) for b
in mult_table
]
133 if not (is_Matrix(b
) and b
.dimensions() == (n
, n
)):
134 raise ValueError("input is not a multiplication table")
135 mult_table
= tuple(mult_table
)
137 cat
= MagmaticAlgebras(field
).FiniteDimensional().WithBasis()
138 cat
.or_subcategory(category
)
139 if assume_associative
:
140 cat
= cat
.Associative()
142 names
= normalize_names(n
, names
)
144 fda
= super(FiniteDimensionalEuclideanJordanAlgebra
, cls
)
145 return fda
.__classcall
__(cls
,
148 assume_associative
=assume_associative
,
152 natural_basis
=natural_basis
)
159 assume_associative
=False,
166 By definition, Jordan multiplication commutes::
168 sage: set_random_seed()
169 sage: J = random_eja()
170 sage: x = J.random_element()
171 sage: y = J.random_element()
177 self
._natural
_basis
= natural_basis
178 self
._multiplication
_table
= mult_table
179 fda
= super(FiniteDimensionalEuclideanJordanAlgebra
, self
)
188 Return a string representation of ``self``.
190 fmt
= "Euclidean Jordan algebra of degree {} over {}"
191 return fmt
.format(self
.degree(), self
.base_ring())
194 def _a_regular_element(self
):
196 Guess a regular element. Needed to compute the basis for our
197 characteristic polynomial coefficients.
200 z
= self
.sum( (i
+1)*gs
[i
] for i
in range(len(gs
)) )
201 if not z
.is_regular():
202 raise ValueError("don't know a regular element")
207 def _charpoly_basis_space(self
):
209 Return the vector space spanned by the basis used in our
210 characteristic polynomial coefficients. This is used not only to
211 compute those coefficients, but also any time we need to
212 evaluate the coefficients (like when we compute the trace or
215 z
= self
._a
_regular
_element
()
216 V
= z
.vector().parent().ambient_vector_space()
217 V1
= V
.span_of_basis( (z
**k
).vector() for k
in range(self
.rank()) )
218 b
= (V1
.basis() + V1
.complement().basis())
219 return V
.span_of_basis(b
)
223 def _charpoly_coeff(self
, i
):
225 Return the coefficient polynomial "a_{i}" of this algebra's
226 general characteristic polynomial.
228 Having this be a separate cached method lets us compute and
229 store the trace/determinant (a_{r-1} and a_{0} respectively)
230 separate from the entire characteristic polynomial.
232 (A_of_x
, x
, xr
, detA
) = self
._charpoly
_matrix
_system
()
233 R
= A_of_x
.base_ring()
235 # Guaranteed by theory
238 # Danger: the in-place modification is done for performance
239 # reasons (reconstructing a matrix with huge polynomial
240 # entries is slow), but I don't know how cached_method works,
241 # so it's highly possible that we're modifying some global
242 # list variable by reference, here. In other words, you
243 # probably shouldn't call this method twice on the same
244 # algebra, at the same time, in two threads
245 Ai_orig
= A_of_x
.column(i
)
246 A_of_x
.set_column(i
,xr
)
247 numerator
= A_of_x
.det()
248 A_of_x
.set_column(i
,Ai_orig
)
250 # We're relying on the theory here to ensure that each a_i is
251 # indeed back in R, and the added negative signs are to make
252 # the whole charpoly expression sum to zero.
253 return R(-numerator
/detA
)
257 def _charpoly_matrix_system(self
):
259 Compute the matrix whose entries A_ij are polynomials in
260 X1,...,XN, the vector ``x`` of variables X1,...,XN, the vector
261 corresponding to `x^r` and the determinent of the matrix A =
262 [A_ij]. In other words, all of the fixed (cachable) data needed
263 to compute the coefficients of the characteristic polynomial.
268 # Construct a new algebra over a multivariate polynomial ring...
269 names
= ['X' + str(i
) for i
in range(1,n
+1)]
270 R
= PolynomialRing(self
.base_ring(), names
)
271 J
= FiniteDimensionalEuclideanJordanAlgebra(R
,
272 self
._multiplication
_table
,
275 idmat
= identity_matrix(J
.base_ring(), n
)
277 W
= self
._charpoly
_basis
_space
()
278 W
= W
.change_ring(R
.fraction_field())
280 # Starting with the standard coordinates x = (X1,X2,...,Xn)
281 # and then converting the entries to W-coordinates allows us
282 # to pass in the standard coordinates to the charpoly and get
283 # back the right answer. Specifically, with x = (X1,X2,...,Xn),
286 # W.coordinates(x^2) eval'd at (standard z-coords)
290 # W-coords of (standard coords of x^2 eval'd at std-coords of z)
292 # We want the middle equivalent thing in our matrix, but use
293 # the first equivalent thing instead so that we can pass in
294 # standard coordinates.
295 x
= J(vector(R
, R
.gens()))
296 l1
= [column_matrix(W
.coordinates((x
**k
).vector())) for k
in range(r
)]
297 l2
= [idmat
.column(k
-1).column() for k
in range(r
+1, n
+1)]
298 A_of_x
= block_matrix(R
, 1, n
, (l1
+ l2
))
299 xr
= W
.coordinates((x
**r
).vector())
300 return (A_of_x
, x
, xr
, A_of_x
.det())
304 def characteristic_polynomial(self
):
309 This implementation doesn't guarantee that the polynomial
310 denominator in the coefficients is not identically zero, so
311 theoretically it could crash. The way that this is handled
312 in e.g. Faraut and Koranyi is to use a basis that guarantees
313 the denominator is non-zero. But, doing so requires knowledge
314 of at least one regular element, and we don't even know how
315 to do that. The trade-off is that, if we use the standard basis,
316 the resulting polynomial will accept the "usual" coordinates. In
317 other words, we don't have to do a change of basis before e.g.
318 computing the trace or determinant.
322 The characteristic polynomial in the spin algebra is given in
323 Alizadeh, Example 11.11::
325 sage: J = JordanSpinEJA(3)
326 sage: p = J.characteristic_polynomial(); p
327 X1^2 - X2^2 - X3^2 + (-2*t)*X1 + t^2
328 sage: xvec = J.one().vector()
336 # The list of coefficient polynomials a_1, a_2, ..., a_n.
337 a
= [ self
._charpoly
_coeff
(i
) for i
in range(n
) ]
339 # We go to a bit of trouble here to reorder the
340 # indeterminates, so that it's easier to evaluate the
341 # characteristic polynomial at x's coordinates and get back
342 # something in terms of t, which is what we want.
344 S
= PolynomialRing(self
.base_ring(),'t')
346 S
= PolynomialRing(S
, R
.variable_names())
349 # Note: all entries past the rth should be zero. The
350 # coefficient of the highest power (x^r) is 1, but it doesn't
351 # appear in the solution vector which contains coefficients
352 # for the other powers (to make them sum to x^r).
354 a
[r
] = 1 # corresponds to x^r
356 # When the rank is equal to the dimension, trying to
357 # assign a[r] goes out-of-bounds.
358 a
.append(1) # corresponds to x^r
360 return sum( a
[k
]*(t
**k
) for k
in range(len(a
)) )
363 def inner_product(self
, x
, y
):
365 The inner product associated with this Euclidean Jordan algebra.
367 Defaults to the trace inner product, but can be overridden by
368 subclasses if they are sure that the necessary properties are
373 The inner product must satisfy its axiom for this algebra to truly
374 be a Euclidean Jordan Algebra::
376 sage: set_random_seed()
377 sage: J = random_eja()
378 sage: x = J.random_element()
379 sage: y = J.random_element()
380 sage: z = J.random_element()
381 sage: (x*y).inner_product(z) == y.inner_product(x*z)
385 if (not x
in self
) or (not y
in self
):
386 raise TypeError("arguments must live in this algebra")
387 return x
.trace_inner_product(y
)
390 def natural_basis(self
):
392 Return a more-natural representation of this algebra's basis.
394 Every finite-dimensional Euclidean Jordan Algebra is a direct
395 sum of five simple algebras, four of which comprise Hermitian
396 matrices. This method returns the original "natural" basis
397 for our underlying vector space. (Typically, the natural basis
398 is used to construct the multiplication table in the first place.)
400 Note that this will always return a matrix. The standard basis
401 in `R^n` will be returned as `n`-by-`1` column matrices.
405 sage: J = RealSymmetricEJA(2)
408 sage: J.natural_basis()
416 sage: J = JordanSpinEJA(2)
419 sage: J.natural_basis()
426 if self
._natural
_basis
is None:
427 return tuple( b
.vector().column() for b
in self
.basis() )
429 return self
._natural
_basis
434 Return the rank of this EJA.
436 if self
._rank
is None:
437 raise ValueError("no rank specified at genesis")
442 class Element(FiniteDimensionalAlgebraElement
):
444 An element of a Euclidean Jordan algebra.
449 Oh man, I should not be doing this. This hides the "disabled"
450 methods ``left_matrix`` and ``matrix`` from introspection;
451 in particular it removes them from tab-completion.
453 return filter(lambda s
: s
not in ['left_matrix', 'matrix'],
454 dir(self
.__class
__) )
457 def __init__(self
, A
, elt
=None):
461 The identity in `S^n` is converted to the identity in the EJA::
463 sage: J = RealSymmetricEJA(3)
464 sage: I = identity_matrix(QQ,3)
465 sage: J(I) == J.one()
468 This skew-symmetric matrix can't be represented in the EJA::
470 sage: J = RealSymmetricEJA(3)
471 sage: A = matrix(QQ,3, lambda i,j: i-j)
473 Traceback (most recent call last):
475 ArithmeticError: vector is not in free module
478 # Goal: if we're given a matrix, and if it lives in our
479 # parent algebra's "natural ambient space," convert it
480 # into an algebra element.
482 # The catch is, we make a recursive call after converting
483 # the given matrix into a vector that lives in the algebra.
484 # This we need to try the parent class initializer first,
485 # to avoid recursing forever if we're given something that
486 # already fits into the algebra, but also happens to live
487 # in the parent's "natural ambient space" (this happens with
490 FiniteDimensionalAlgebraElement
.__init
__(self
, A
, elt
)
492 natural_basis
= A
.natural_basis()
493 if elt
in natural_basis
[0].matrix_space():
494 # Thanks for nothing! Matrix spaces aren't vector
495 # spaces in Sage, so we have to figure out its
496 # natural-basis coordinates ourselves.
497 V
= VectorSpace(elt
.base_ring(), elt
.nrows()**2)
498 W
= V
.span( _mat2vec(s
) for s
in natural_basis
)
499 coords
= W
.coordinates(_mat2vec(elt
))
500 FiniteDimensionalAlgebraElement
.__init
__(self
, A
, coords
)
502 def __pow__(self
, n
):
504 Return ``self`` raised to the power ``n``.
506 Jordan algebras are always power-associative; see for
507 example Faraut and Koranyi, Proposition II.1.2 (ii).
511 We have to override this because our superclass uses row vectors
512 instead of column vectors! We, on the other hand, assume column
517 sage: set_random_seed()
518 sage: x = random_eja().random_element()
519 sage: x.operator_matrix()*x.vector() == (x^2).vector()
522 A few examples of power-associativity::
524 sage: set_random_seed()
525 sage: x = random_eja().random_element()
526 sage: x*(x*x)*(x*x) == x^5
528 sage: (x*x)*(x*x*x) == x^5
531 We also know that powers operator-commute (Koecher, Chapter
534 sage: set_random_seed()
535 sage: x = random_eja().random_element()
536 sage: m = ZZ.random_element(0,10)
537 sage: n = ZZ.random_element(0,10)
538 sage: Lxm = (x^m).operator_matrix()
539 sage: Lxn = (x^n).operator_matrix()
540 sage: Lxm*Lxn == Lxn*Lxm
550 return A( (self
.operator_matrix()**(n
-1))*self
.vector() )
553 def apply_univariate_polynomial(self
, p
):
555 Apply the univariate polynomial ``p`` to this element.
557 A priori, SageMath won't allow us to apply a univariate
558 polynomial to an element of an EJA, because we don't know
559 that EJAs are rings (they are usually not associative). Of
560 course, we know that EJAs are power-associative, so the
561 operation is ultimately kosher. This function sidesteps
562 the CAS to get the answer we want and expect.
566 sage: R = PolynomialRing(QQ, 't')
568 sage: p = t^4 - t^3 + 5*t - 2
569 sage: J = RealCartesianProductEJA(5)
570 sage: J.one().apply_univariate_polynomial(p) == 3*J.one()
575 We should always get back an element of the algebra::
577 sage: set_random_seed()
578 sage: p = PolynomialRing(QQ, 't').random_element()
579 sage: J = random_eja()
580 sage: x = J.random_element()
581 sage: x.apply_univariate_polynomial(p) in J
585 if len(p
.variables()) > 1:
586 raise ValueError("not a univariate polynomial")
589 # Convert the coeficcients to the parent's base ring,
590 # because a priori they might live in an (unnecessarily)
591 # larger ring for which P.sum() would fail below.
592 cs
= [ R(c
) for c
in p
.coefficients(sparse
=False) ]
593 return P
.sum( cs
[k
]*(self
**k
) for k
in range(len(cs
)) )
596 def characteristic_polynomial(self
):
598 Return the characteristic polynomial of this element.
602 The rank of `R^3` is three, and the minimal polynomial of
603 the identity element is `(t-1)` from which it follows that
604 the characteristic polynomial should be `(t-1)^3`::
606 sage: J = RealCartesianProductEJA(3)
607 sage: J.one().characteristic_polynomial()
608 t^3 - 3*t^2 + 3*t - 1
610 Likewise, the characteristic of the zero element in the
611 rank-three algebra `R^{n}` should be `t^{3}`::
613 sage: J = RealCartesianProductEJA(3)
614 sage: J.zero().characteristic_polynomial()
617 The characteristic polynomial of an element should evaluate
618 to zero on that element::
620 sage: set_random_seed()
621 sage: x = RealCartesianProductEJA(3).random_element()
622 sage: p = x.characteristic_polynomial()
623 sage: x.apply_univariate_polynomial(p)
627 p
= self
.parent().characteristic_polynomial()
628 return p(*self
.vector())
631 def inner_product(self
, other
):
633 Return the parent algebra's inner product of myself and ``other``.
637 The inner product in the Jordan spin algebra is the usual
638 inner product on `R^n` (this example only works because the
639 basis for the Jordan algebra is the standard basis in `R^n`)::
641 sage: J = JordanSpinEJA(3)
642 sage: x = vector(QQ,[1,2,3])
643 sage: y = vector(QQ,[4,5,6])
644 sage: x.inner_product(y)
646 sage: J(x).inner_product(J(y))
649 The inner product on `S^n` is `<X,Y> = trace(X*Y)`, where
650 multiplication is the usual matrix multiplication in `S^n`,
651 so the inner product of the identity matrix with itself
654 sage: J = RealSymmetricEJA(3)
655 sage: J.one().inner_product(J.one())
658 Likewise, the inner product on `C^n` is `<X,Y> =
659 Re(trace(X*Y))`, where we must necessarily take the real
660 part because the product of Hermitian matrices may not be
663 sage: J = ComplexHermitianEJA(3)
664 sage: J.one().inner_product(J.one())
667 Ditto for the quaternions::
669 sage: J = QuaternionHermitianEJA(3)
670 sage: J.one().inner_product(J.one())
675 Ensure that we can always compute an inner product, and that
676 it gives us back a real number::
678 sage: set_random_seed()
679 sage: J = random_eja()
680 sage: x = J.random_element()
681 sage: y = J.random_element()
682 sage: x.inner_product(y) in RR
688 raise TypeError("'other' must live in the same algebra")
690 return P
.inner_product(self
, other
)
693 def operator_commutes_with(self
, other
):
695 Return whether or not this element operator-commutes
700 The definition of a Jordan algebra says that any element
701 operator-commutes with its square::
703 sage: set_random_seed()
704 sage: x = random_eja().random_element()
705 sage: x.operator_commutes_with(x^2)
710 Test Lemma 1 from Chapter III of Koecher::
712 sage: set_random_seed()
713 sage: J = random_eja()
714 sage: u = J.random_element()
715 sage: v = J.random_element()
716 sage: lhs = u.operator_commutes_with(u*v)
717 sage: rhs = v.operator_commutes_with(u^2)
722 if not other
in self
.parent():
723 raise TypeError("'other' must live in the same algebra")
725 A
= self
.operator_matrix()
726 B
= other
.operator_matrix()
732 Return my determinant, the product of my eigenvalues.
736 sage: J = JordanSpinEJA(2)
737 sage: e0,e1 = J.gens()
738 sage: x = sum( J.gens() )
744 sage: J = JordanSpinEJA(3)
745 sage: e0,e1,e2 = J.gens()
746 sage: x = sum( J.gens() )
752 An element is invertible if and only if its determinant is
755 sage: set_random_seed()
756 sage: x = random_eja().random_element()
757 sage: x.is_invertible() == (x.det() != 0)
763 p
= P
._charpoly
_coeff
(0)
764 # The _charpoly_coeff function already adds the factor of
765 # -1 to ensure that _charpoly_coeff(0) is really what
766 # appears in front of t^{0} in the charpoly. However,
767 # we want (-1)^r times THAT for the determinant.
768 return ((-1)**r
)*p(*self
.vector())
773 Return the Jordan-multiplicative inverse of this element.
777 We appeal to the quadratic representation as in Koecher's
778 Theorem 12 in Chapter III, Section 5.
782 The inverse in the spin factor algebra is given in Alizadeh's
785 sage: set_random_seed()
786 sage: n = ZZ.random_element(1,10)
787 sage: J = JordanSpinEJA(n)
788 sage: x = J.random_element()
789 sage: while x.is_zero():
790 ....: x = J.random_element()
791 sage: x_vec = x.vector()
793 sage: x_bar = x_vec[1:]
794 sage: coeff = 1/(x0^2 - x_bar.inner_product(x_bar))
795 sage: inv_vec = x_vec.parent()([x0] + (-x_bar).list())
796 sage: x_inverse = coeff*inv_vec
797 sage: x.inverse() == J(x_inverse)
802 The identity element is its own inverse::
804 sage: set_random_seed()
805 sage: J = random_eja()
806 sage: J.one().inverse() == J.one()
809 If an element has an inverse, it acts like one::
811 sage: set_random_seed()
812 sage: J = random_eja()
813 sage: x = J.random_element()
814 sage: (not x.is_invertible()) or (x.inverse()*x == J.one())
817 The inverse of the inverse is what we started with::
819 sage: set_random_seed()
820 sage: J = random_eja()
821 sage: x = J.random_element()
822 sage: (not x.is_invertible()) or (x.inverse().inverse() == x)
825 The zero element is never invertible::
827 sage: set_random_seed()
828 sage: J = random_eja().zero().inverse()
829 Traceback (most recent call last):
831 ValueError: element is not invertible
834 if not self
.is_invertible():
835 raise ValueError("element is not invertible")
838 return P(self
.quadratic_representation().inverse()*self
.vector())
841 def is_invertible(self
):
843 Return whether or not this element is invertible.
845 We can't use the superclass method because it relies on
846 the algebra being associative.
850 The usual way to do this is to check if the determinant is
851 zero, but we need the characteristic polynomial for the
852 determinant. The minimal polynomial is a lot easier to get,
853 so we use Corollary 2 in Chapter V of Koecher to check
854 whether or not the paren't algebra's zero element is a root
855 of this element's minimal polynomial.
859 The identity element is always invertible::
861 sage: set_random_seed()
862 sage: J = random_eja()
863 sage: J.one().is_invertible()
866 The zero element is never invertible::
868 sage: set_random_seed()
869 sage: J = random_eja()
870 sage: J.zero().is_invertible()
874 zero
= self
.parent().zero()
875 p
= self
.minimal_polynomial()
876 return not (p(zero
) == zero
)
879 def is_nilpotent(self
):
881 Return whether or not some power of this element is zero.
883 The superclass method won't work unless we're in an
884 associative algebra, and we aren't. However, we generate
885 an assocoative subalgebra and we're nilpotent there if and
886 only if we're nilpotent here (probably).
890 The identity element is never nilpotent::
892 sage: set_random_seed()
893 sage: random_eja().one().is_nilpotent()
896 The additive identity is always nilpotent::
898 sage: set_random_seed()
899 sage: random_eja().zero().is_nilpotent()
903 # The element we're going to call "is_nilpotent()" on.
904 # Either myself, interpreted as an element of a finite-
905 # dimensional algebra, or an element of an associative
909 if self
.parent().is_associative():
910 elt
= FiniteDimensionalAlgebraElement(self
.parent(), self
)
912 V
= self
.span_of_powers()
913 assoc_subalg
= self
.subalgebra_generated_by()
914 # Mis-design warning: the basis used for span_of_powers()
915 # and subalgebra_generated_by() must be the same, and in
917 elt
= assoc_subalg(V
.coordinates(self
.vector()))
919 # Recursive call, but should work since elt lives in an
920 # associative algebra.
921 return elt
.is_nilpotent()
924 def is_regular(self
):
926 Return whether or not this is a regular element.
930 The identity element always has degree one, but any element
931 linearly-independent from it is regular::
933 sage: J = JordanSpinEJA(5)
934 sage: J.one().is_regular()
936 sage: e0, e1, e2, e3, e4 = J.gens() # e0 is the identity
937 sage: for x in J.gens():
938 ....: (J.one() + x).is_regular()
946 return self
.degree() == self
.parent().rank()
951 Compute the degree of this element the straightforward way
952 according to the definition; by appending powers to a list
953 and figuring out its dimension (that is, whether or not
954 they're linearly dependent).
958 sage: J = JordanSpinEJA(4)
959 sage: J.one().degree()
961 sage: e0,e1,e2,e3 = J.gens()
962 sage: (e0 - e1).degree()
965 In the spin factor algebra (of rank two), all elements that
966 aren't multiples of the identity are regular::
968 sage: set_random_seed()
969 sage: n = ZZ.random_element(1,10)
970 sage: J = JordanSpinEJA(n)
971 sage: x = J.random_element()
972 sage: x == x.coefficient(0)*J.one() or x.degree() == 2
976 return self
.span_of_powers().dimension()
979 def left_matrix(self
):
981 Our parent class defines ``left_matrix`` and ``matrix``
982 methods whose names are misleading. We don't want them.
984 raise NotImplementedError("use operator_matrix() instead")
989 def minimal_polynomial(self
):
991 Return the minimal polynomial of this element,
992 as a function of the variable `t`.
996 We restrict ourselves to the associative subalgebra
997 generated by this element, and then return the minimal
998 polynomial of this element's operator matrix (in that
999 subalgebra). This works by Baes Proposition 2.3.16.
1003 The minimal polynomial of the identity and zero elements are
1006 sage: set_random_seed()
1007 sage: J = random_eja()
1008 sage: J.one().minimal_polynomial()
1010 sage: J.zero().minimal_polynomial()
1013 The degree of an element is (by one definition) the degree
1014 of its minimal polynomial::
1016 sage: set_random_seed()
1017 sage: x = random_eja().random_element()
1018 sage: x.degree() == x.minimal_polynomial().degree()
1021 The minimal polynomial and the characteristic polynomial coincide
1022 and are known (see Alizadeh, Example 11.11) for all elements of
1023 the spin factor algebra that aren't scalar multiples of the
1026 sage: set_random_seed()
1027 sage: n = ZZ.random_element(2,10)
1028 sage: J = JordanSpinEJA(n)
1029 sage: y = J.random_element()
1030 sage: while y == y.coefficient(0)*J.one():
1031 ....: y = J.random_element()
1032 sage: y0 = y.vector()[0]
1033 sage: y_bar = y.vector()[1:]
1034 sage: actual = y.minimal_polynomial()
1035 sage: t = PolynomialRing(J.base_ring(),'t').gen(0)
1036 sage: expected = t^2 - 2*y0*t + (y0^2 - norm(y_bar)^2)
1037 sage: bool(actual == expected)
1040 The minimal polynomial should always kill its element::
1042 sage: set_random_seed()
1043 sage: x = random_eja().random_element()
1044 sage: p = x.minimal_polynomial()
1045 sage: x.apply_univariate_polynomial(p)
1049 V
= self
.span_of_powers()
1050 assoc_subalg
= self
.subalgebra_generated_by()
1051 # Mis-design warning: the basis used for span_of_powers()
1052 # and subalgebra_generated_by() must be the same, and in
1054 elt
= assoc_subalg(V
.coordinates(self
.vector()))
1056 # We get back a symbolic polynomial in 'x' but want a real
1057 # polynomial in 't'.
1058 p_of_x
= elt
.operator_matrix().minimal_polynomial()
1059 return p_of_x
.change_variable_name('t')
1062 def natural_representation(self
):
1064 Return a more-natural representation of this element.
1066 Every finite-dimensional Euclidean Jordan Algebra is a
1067 direct sum of five simple algebras, four of which comprise
1068 Hermitian matrices. This method returns the original
1069 "natural" representation of this element as a Hermitian
1070 matrix, if it has one. If not, you get the usual representation.
1074 sage: J = ComplexHermitianEJA(3)
1077 sage: J.one().natural_representation()
1087 sage: J = QuaternionHermitianEJA(3)
1090 sage: J.one().natural_representation()
1091 [1 0 0 0 0 0 0 0 0 0 0 0]
1092 [0 1 0 0 0 0 0 0 0 0 0 0]
1093 [0 0 1 0 0 0 0 0 0 0 0 0]
1094 [0 0 0 1 0 0 0 0 0 0 0 0]
1095 [0 0 0 0 1 0 0 0 0 0 0 0]
1096 [0 0 0 0 0 1 0 0 0 0 0 0]
1097 [0 0 0 0 0 0 1 0 0 0 0 0]
1098 [0 0 0 0 0 0 0 1 0 0 0 0]
1099 [0 0 0 0 0 0 0 0 1 0 0 0]
1100 [0 0 0 0 0 0 0 0 0 1 0 0]
1101 [0 0 0 0 0 0 0 0 0 0 1 0]
1102 [0 0 0 0 0 0 0 0 0 0 0 1]
1105 B
= self
.parent().natural_basis()
1106 W
= B
[0].matrix_space()
1107 return W
.linear_combination(zip(self
.vector(), B
))
1112 Return the left-multiplication-by-this-element
1113 operator on the ambient algebra.
1117 sage: set_random_seed()
1118 sage: J = random_eja()
1119 sage: x = J.random_element()
1120 sage: y = J.random_element()
1121 sage: x.operator()(y) == x*y
1123 sage: y.operator()(x) == x*y
1128 return FiniteDimensionalEuclideanJordanAlgebraMorphism(
1130 self
.operator_matrix() )
1134 def operator_matrix(self
):
1136 Return the matrix that represents left- (or right-)
1137 multiplication by this element in the parent algebra.
1139 We implement this ourselves to work around the fact that
1140 our parent class represents everything with row vectors.
1144 Test the first polarization identity from my notes, Koecher Chapter
1145 III, or from Baes (2.3)::
1147 sage: set_random_seed()
1148 sage: J = random_eja()
1149 sage: x = J.random_element()
1150 sage: y = J.random_element()
1151 sage: Lx = x.operator_matrix()
1152 sage: Ly = y.operator_matrix()
1153 sage: Lxx = (x*x).operator_matrix()
1154 sage: Lxy = (x*y).operator_matrix()
1155 sage: bool(2*Lx*Lxy + Ly*Lxx == 2*Lxy*Lx + Lxx*Ly)
1158 Test the second polarization identity from my notes or from
1161 sage: set_random_seed()
1162 sage: J = random_eja()
1163 sage: x = J.random_element()
1164 sage: y = J.random_element()
1165 sage: z = J.random_element()
1166 sage: Lx = x.operator_matrix()
1167 sage: Ly = y.operator_matrix()
1168 sage: Lz = z.operator_matrix()
1169 sage: Lzy = (z*y).operator_matrix()
1170 sage: Lxy = (x*y).operator_matrix()
1171 sage: Lxz = (x*z).operator_matrix()
1172 sage: bool(Lx*Lzy + Lz*Lxy + Ly*Lxz == Lzy*Lx + Lxy*Lz + Lxz*Ly)
1175 Test the third polarization identity from my notes or from
1178 sage: set_random_seed()
1179 sage: J = random_eja()
1180 sage: u = J.random_element()
1181 sage: y = J.random_element()
1182 sage: z = J.random_element()
1183 sage: Lu = u.operator_matrix()
1184 sage: Ly = y.operator_matrix()
1185 sage: Lz = z.operator_matrix()
1186 sage: Lzy = (z*y).operator_matrix()
1187 sage: Luy = (u*y).operator_matrix()
1188 sage: Luz = (u*z).operator_matrix()
1189 sage: Luyz = (u*(y*z)).operator_matrix()
1190 sage: lhs = Lu*Lzy + Lz*Luy + Ly*Luz
1191 sage: rhs = Luyz + Ly*Lu*Lz + Lz*Lu*Ly
1192 sage: bool(lhs == rhs)
1196 fda_elt
= FiniteDimensionalAlgebraElement(self
.parent(), self
)
1197 return fda_elt
.matrix().transpose()
1200 def quadratic_representation(self
, other
=None):
1202 Return the quadratic representation of this element.
1206 The explicit form in the spin factor algebra is given by
1207 Alizadeh's Example 11.12::
1209 sage: set_random_seed()
1210 sage: n = ZZ.random_element(1,10)
1211 sage: J = JordanSpinEJA(n)
1212 sage: x = J.random_element()
1213 sage: x_vec = x.vector()
1215 sage: x_bar = x_vec[1:]
1216 sage: A = matrix(QQ, 1, [x_vec.inner_product(x_vec)])
1217 sage: B = 2*x0*x_bar.row()
1218 sage: C = 2*x0*x_bar.column()
1219 sage: D = identity_matrix(QQ, n-1)
1220 sage: D = (x0^2 - x_bar.inner_product(x_bar))*D
1221 sage: D = D + 2*x_bar.tensor_product(x_bar)
1222 sage: Q = block_matrix(2,2,[A,B,C,D])
1223 sage: Q == x.quadratic_representation()
1226 Test all of the properties from Theorem 11.2 in Alizadeh::
1228 sage: set_random_seed()
1229 sage: J = random_eja()
1230 sage: x = J.random_element()
1231 sage: y = J.random_element()
1232 sage: Lx = x.operator_matrix()
1233 sage: Lxx = (x*x).operator_matrix()
1234 sage: Qx = x.quadratic_representation()
1235 sage: Qy = y.quadratic_representation()
1236 sage: Qxy = x.quadratic_representation(y)
1237 sage: Qex = J.one().quadratic_representation(x)
1238 sage: n = ZZ.random_element(10)
1239 sage: Qxn = (x^n).quadratic_representation()
1243 sage: 2*Qxy == (x+y).quadratic_representation() - Qx - Qy
1248 sage: alpha = QQ.random_element()
1249 sage: (alpha*x).quadratic_representation() == (alpha^2)*Qx
1254 sage: not x.is_invertible() or (
1255 ....: Qx*x.inverse().vector() == x.vector() )
1258 sage: not x.is_invertible() or (
1261 ....: x.inverse().quadratic_representation() )
1264 sage: Qxy*(J.one().vector()) == (x*y).vector()
1269 sage: not x.is_invertible() or (
1270 ....: x.quadratic_representation(x.inverse())*Qx
1271 ....: == Qx*x.quadratic_representation(x.inverse()) )
1274 sage: not x.is_invertible() or (
1275 ....: x.quadratic_representation(x.inverse())*Qx
1277 ....: 2*x.operator_matrix()*Qex - Qx )
1280 sage: 2*x.operator_matrix()*Qex - Qx == Lxx
1285 sage: J(Qy*x.vector()).quadratic_representation() == Qy*Qx*Qy
1295 sage: not x.is_invertible() or (
1296 ....: Qx*x.inverse().operator_matrix() == Lx )
1301 sage: not x.operator_commutes_with(y) or (
1302 ....: J(Qx*y.vector())^n == J(Qxn*(y^n).vector()) )
1308 elif not other
in self
.parent():
1309 raise TypeError("'other' must live in the same algebra")
1311 L
= self
.operator_matrix()
1312 M
= other
.operator_matrix()
1313 return ( L
*M
+ M
*L
- (self
*other
).operator_matrix() )
1316 def span_of_powers(self
):
1318 Return the vector space spanned by successive powers of
1321 # The dimension of the subalgebra can't be greater than
1322 # the big algebra, so just put everything into a list
1323 # and let span() get rid of the excess.
1325 # We do the extra ambient_vector_space() in case we're messing
1326 # with polynomials and the direct parent is a module.
1327 V
= self
.vector().parent().ambient_vector_space()
1328 return V
.span( (self
**d
).vector() for d
in xrange(V
.dimension()) )
1331 def subalgebra_generated_by(self
):
1333 Return the associative subalgebra of the parent EJA generated
1338 sage: set_random_seed()
1339 sage: x = random_eja().random_element()
1340 sage: x.subalgebra_generated_by().is_associative()
1343 Squaring in the subalgebra should be the same thing as
1344 squaring in the superalgebra::
1346 sage: set_random_seed()
1347 sage: x = random_eja().random_element()
1348 sage: u = x.subalgebra_generated_by().random_element()
1349 sage: u.operator_matrix()*u.vector() == (u**2).vector()
1353 # First get the subspace spanned by the powers of myself...
1354 V
= self
.span_of_powers()
1355 F
= self
.base_ring()
1357 # Now figure out the entries of the right-multiplication
1358 # matrix for the successive basis elements b0, b1,... of
1361 for b_right
in V
.basis():
1362 eja_b_right
= self
.parent()(b_right
)
1364 # The first row of the right-multiplication matrix by
1365 # b1 is what we get if we apply that matrix to b1. The
1366 # second row of the right multiplication matrix by b1
1367 # is what we get when we apply that matrix to b2...
1369 # IMPORTANT: this assumes that all vectors are COLUMN
1370 # vectors, unlike our superclass (which uses row vectors).
1371 for b_left
in V
.basis():
1372 eja_b_left
= self
.parent()(b_left
)
1373 # Multiply in the original EJA, but then get the
1374 # coordinates from the subalgebra in terms of its
1376 this_row
= V
.coordinates((eja_b_left
*eja_b_right
).vector())
1377 b_right_rows
.append(this_row
)
1378 b_right_matrix
= matrix(F
, b_right_rows
)
1379 mats
.append(b_right_matrix
)
1381 # It's an algebra of polynomials in one element, and EJAs
1382 # are power-associative.
1384 # TODO: choose generator names intelligently.
1385 return FiniteDimensionalEuclideanJordanAlgebra(F
, mats
, assume_associative
=True, names
='f')
1388 def subalgebra_idempotent(self
):
1390 Find an idempotent in the associative subalgebra I generate
1391 using Proposition 2.3.5 in Baes.
1395 sage: set_random_seed()
1396 sage: J = random_eja()
1397 sage: x = J.random_element()
1398 sage: while x.is_nilpotent():
1399 ....: x = J.random_element()
1400 sage: c = x.subalgebra_idempotent()
1405 if self
.is_nilpotent():
1406 raise ValueError("this only works with non-nilpotent elements!")
1408 V
= self
.span_of_powers()
1409 J
= self
.subalgebra_generated_by()
1410 # Mis-design warning: the basis used for span_of_powers()
1411 # and subalgebra_generated_by() must be the same, and in
1413 u
= J(V
.coordinates(self
.vector()))
1415 # The image of the matrix of left-u^m-multiplication
1416 # will be minimal for some natural number s...
1418 minimal_dim
= V
.dimension()
1419 for i
in xrange(1, V
.dimension()):
1420 this_dim
= (u
**i
).operator_matrix().image().dimension()
1421 if this_dim
< minimal_dim
:
1422 minimal_dim
= this_dim
1425 # Now minimal_matrix should correspond to the smallest
1426 # non-zero subspace in Baes's (or really, Koecher's)
1429 # However, we need to restrict the matrix to work on the
1430 # subspace... or do we? Can't we just solve, knowing that
1431 # A(c) = u^(s+1) should have a solution in the big space,
1434 # Beware, solve_right() means that we're using COLUMN vectors.
1435 # Our FiniteDimensionalAlgebraElement superclass uses rows.
1437 A
= u_next
.operator_matrix()
1438 c_coordinates
= A
.solve_right(u_next
.vector())
1440 # Now c_coordinates is the idempotent we want, but it's in
1441 # the coordinate system of the subalgebra.
1443 # We need the basis for J, but as elements of the parent algebra.
1445 basis
= [self
.parent(v
) for v
in V
.basis()]
1446 return self
.parent().linear_combination(zip(c_coordinates
, basis
))
1451 Return my trace, the sum of my eigenvalues.
1455 sage: J = JordanSpinEJA(3)
1456 sage: x = sum(J.gens())
1462 sage: J = RealCartesianProductEJA(5)
1463 sage: J.one().trace()
1468 The trace of an element is a real number::
1470 sage: set_random_seed()
1471 sage: J = random_eja()
1472 sage: J.random_element().trace() in J.base_ring()
1478 p
= P
._charpoly
_coeff
(r
-1)
1479 # The _charpoly_coeff function already adds the factor of
1480 # -1 to ensure that _charpoly_coeff(r-1) is really what
1481 # appears in front of t^{r-1} in the charpoly. However,
1482 # we want the negative of THAT for the trace.
1483 return -p(*self
.vector())
1486 def trace_inner_product(self
, other
):
1488 Return the trace inner product of myself and ``other``.
1492 The trace inner product is commutative::
1494 sage: set_random_seed()
1495 sage: J = random_eja()
1496 sage: x = J.random_element(); y = J.random_element()
1497 sage: x.trace_inner_product(y) == y.trace_inner_product(x)
1500 The trace inner product is bilinear::
1502 sage: set_random_seed()
1503 sage: J = random_eja()
1504 sage: x = J.random_element()
1505 sage: y = J.random_element()
1506 sage: z = J.random_element()
1507 sage: a = QQ.random_element();
1508 sage: actual = (a*(x+z)).trace_inner_product(y)
1509 sage: expected = ( a*x.trace_inner_product(y) +
1510 ....: a*z.trace_inner_product(y) )
1511 sage: actual == expected
1513 sage: actual = x.trace_inner_product(a*(y+z))
1514 sage: expected = ( a*x.trace_inner_product(y) +
1515 ....: a*x.trace_inner_product(z) )
1516 sage: actual == expected
1519 The trace inner product satisfies the compatibility
1520 condition in the definition of a Euclidean Jordan algebra::
1522 sage: set_random_seed()
1523 sage: J = random_eja()
1524 sage: x = J.random_element()
1525 sage: y = J.random_element()
1526 sage: z = J.random_element()
1527 sage: (x*y).trace_inner_product(z) == y.trace_inner_product(x*z)
1531 if not other
in self
.parent():
1532 raise TypeError("'other' must live in the same algebra")
1534 return (self
*other
).trace()
1537 class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1539 Return the Euclidean Jordan Algebra corresponding to the set
1540 `R^n` under the Hadamard product.
1542 Note: this is nothing more than the Cartesian product of ``n``
1543 copies of the spin algebra. Once Cartesian product algebras
1544 are implemented, this can go.
1548 This multiplication table can be verified by hand::
1550 sage: J = RealCartesianProductEJA(3)
1551 sage: e0,e1,e2 = J.gens()
1567 def __classcall_private__(cls
, n
, field
=QQ
):
1568 # The FiniteDimensionalAlgebra constructor takes a list of
1569 # matrices, the ith representing right multiplication by the ith
1570 # basis element in the vector space. So if e_1 = (1,0,0), then
1571 # right (Hadamard) multiplication of x by e_1 picks out the first
1572 # component of x; and likewise for the ith basis element e_i.
1573 Qs
= [ matrix(field
, n
, n
, lambda k
,j
: 1*(k
== j
== i
))
1574 for i
in xrange(n
) ]
1576 fdeja
= super(RealCartesianProductEJA
, cls
)
1577 return fdeja
.__classcall
_private
__(cls
, field
, Qs
, rank
=n
)
1579 def inner_product(self
, x
, y
):
1580 return _usual_ip(x
,y
)
1585 Return a "random" finite-dimensional Euclidean Jordan Algebra.
1589 For now, we choose a random natural number ``n`` (greater than zero)
1590 and then give you back one of the following:
1592 * The cartesian product of the rational numbers ``n`` times; this is
1593 ``QQ^n`` with the Hadamard product.
1595 * The Jordan spin algebra on ``QQ^n``.
1597 * The ``n``-by-``n`` rational symmetric matrices with the symmetric
1600 * The ``n``-by-``n`` complex-rational Hermitian matrices embedded
1601 in the space of ``2n``-by-``2n`` real symmetric matrices.
1603 * The ``n``-by-``n`` quaternion-rational Hermitian matrices embedded
1604 in the space of ``4n``-by-``4n`` real symmetric matrices.
1606 Later this might be extended to return Cartesian products of the
1612 Euclidean Jordan algebra of degree...
1616 # The max_n component lets us choose different upper bounds on the
1617 # value "n" that gets passed to the constructor. This is needed
1618 # because e.g. R^{10} is reasonable to test, while the Hermitian
1619 # 10-by-10 quaternion matrices are not.
1620 (constructor
, max_n
) = choice([(RealCartesianProductEJA
, 6),
1622 (RealSymmetricEJA
, 5),
1623 (ComplexHermitianEJA
, 4),
1624 (QuaternionHermitianEJA
, 3)])
1625 n
= ZZ
.random_element(1, max_n
)
1626 return constructor(n
, field
=QQ
)
1630 def _real_symmetric_basis(n
, field
=QQ
):
1632 Return a basis for the space of real symmetric n-by-n matrices.
1634 # The basis of symmetric matrices, as matrices, in their R^(n-by-n)
1638 for j
in xrange(i
+1):
1639 Eij
= matrix(field
, n
, lambda k
,l
: k
==i
and l
==j
)
1643 # Beware, orthogonal but not normalized!
1644 Sij
= Eij
+ Eij
.transpose()
1649 def _complex_hermitian_basis(n
, field
=QQ
):
1651 Returns a basis for the space of complex Hermitian n-by-n matrices.
1655 sage: set_random_seed()
1656 sage: n = ZZ.random_element(1,5)
1657 sage: all( M.is_symmetric() for M in _complex_hermitian_basis(n) )
1661 F
= QuadraticField(-1, 'I')
1664 # This is like the symmetric case, but we need to be careful:
1666 # * We want conjugate-symmetry, not just symmetry.
1667 # * The diagonal will (as a result) be real.
1671 for j
in xrange(i
+1):
1672 Eij
= matrix(field
, n
, lambda k
,l
: k
==i
and l
==j
)
1674 Sij
= _embed_complex_matrix(Eij
)
1677 # Beware, orthogonal but not normalized! The second one
1678 # has a minus because it's conjugated.
1679 Sij_real
= _embed_complex_matrix(Eij
+ Eij
.transpose())
1681 Sij_imag
= _embed_complex_matrix(I
*Eij
- I
*Eij
.transpose())
1686 def _quaternion_hermitian_basis(n
, field
=QQ
):
1688 Returns a basis for the space of quaternion Hermitian n-by-n matrices.
1692 sage: set_random_seed()
1693 sage: n = ZZ.random_element(1,5)
1694 sage: all( M.is_symmetric() for M in _quaternion_hermitian_basis(n) )
1698 Q
= QuaternionAlgebra(QQ
,-1,-1)
1701 # This is like the symmetric case, but we need to be careful:
1703 # * We want conjugate-symmetry, not just symmetry.
1704 # * The diagonal will (as a result) be real.
1708 for j
in xrange(i
+1):
1709 Eij
= matrix(Q
, n
, lambda k
,l
: k
==i
and l
==j
)
1711 Sij
= _embed_quaternion_matrix(Eij
)
1714 # Beware, orthogonal but not normalized! The second,
1715 # third, and fourth ones have a minus because they're
1717 Sij_real
= _embed_quaternion_matrix(Eij
+ Eij
.transpose())
1719 Sij_I
= _embed_quaternion_matrix(I
*Eij
- I
*Eij
.transpose())
1721 Sij_J
= _embed_quaternion_matrix(J
*Eij
- J
*Eij
.transpose())
1723 Sij_K
= _embed_quaternion_matrix(K
*Eij
- K
*Eij
.transpose())
1729 return vector(m
.base_ring(), m
.list())
1732 return matrix(v
.base_ring(), sqrt(v
.degree()), v
.list())
1734 def _multiplication_table_from_matrix_basis(basis
):
1736 At least three of the five simple Euclidean Jordan algebras have the
1737 symmetric multiplication (A,B) |-> (AB + BA)/2, where the
1738 multiplication on the right is matrix multiplication. Given a basis
1739 for the underlying matrix space, this function returns a
1740 multiplication table (obtained by looping through the basis
1741 elements) for an algebra of those matrices. A reordered copy
1742 of the basis is also returned to work around the fact that
1743 the ``span()`` in this function will change the order of the basis
1744 from what we think it is, to... something else.
1746 # In S^2, for example, we nominally have four coordinates even
1747 # though the space is of dimension three only. The vector space V
1748 # is supposed to hold the entire long vector, and the subspace W
1749 # of V will be spanned by the vectors that arise from symmetric
1750 # matrices. Thus for S^2, dim(V) == 4 and dim(W) == 3.
1751 field
= basis
[0].base_ring()
1752 dimension
= basis
[0].nrows()
1754 V
= VectorSpace(field
, dimension
**2)
1755 W
= V
.span( _mat2vec(s
) for s
in basis
)
1757 # Taking the span above reorders our basis (thanks, jerk!) so we
1758 # need to put our "matrix basis" in the same order as the
1759 # (reordered) vector basis.
1760 S
= tuple( _vec2mat(b
) for b
in W
.basis() )
1764 # Brute force the multiplication-by-s matrix by looping
1765 # through all elements of the basis and doing the computation
1766 # to find out what the corresponding row should be. BEWARE:
1767 # these multiplication tables won't be symmetric! It therefore
1768 # becomes REALLY IMPORTANT that the underlying algebra
1769 # constructor uses ROW vectors and not COLUMN vectors. That's
1770 # why we're computing rows here and not columns.
1773 this_row
= _mat2vec((s
*t
+ t
*s
)/2)
1774 Q_rows
.append(W
.coordinates(this_row
))
1775 Q
= matrix(field
, W
.dimension(), Q_rows
)
1781 def _embed_complex_matrix(M
):
1783 Embed the n-by-n complex matrix ``M`` into the space of real
1784 matrices of size 2n-by-2n via the map the sends each entry `z = a +
1785 bi` to the block matrix ``[[a,b],[-b,a]]``.
1789 sage: F = QuadraticField(-1,'i')
1790 sage: x1 = F(4 - 2*i)
1791 sage: x2 = F(1 + 2*i)
1794 sage: M = matrix(F,2,[[x1,x2],[x3,x4]])
1795 sage: _embed_complex_matrix(M)
1804 Embedding is a homomorphism (isomorphism, in fact)::
1806 sage: set_random_seed()
1807 sage: n = ZZ.random_element(5)
1808 sage: F = QuadraticField(-1, 'i')
1809 sage: X = random_matrix(F, n)
1810 sage: Y = random_matrix(F, n)
1811 sage: actual = _embed_complex_matrix(X) * _embed_complex_matrix(Y)
1812 sage: expected = _embed_complex_matrix(X*Y)
1813 sage: actual == expected
1819 raise ValueError("the matrix 'M' must be square")
1820 field
= M
.base_ring()
1825 blocks
.append(matrix(field
, 2, [[a
,b
],[-b
,a
]]))
1827 # We can drop the imaginaries here.
1828 return block_matrix(field
.base_ring(), n
, blocks
)
1831 def _unembed_complex_matrix(M
):
1833 The inverse of _embed_complex_matrix().
1837 sage: A = matrix(QQ,[ [ 1, 2, 3, 4],
1838 ....: [-2, 1, -4, 3],
1839 ....: [ 9, 10, 11, 12],
1840 ....: [-10, 9, -12, 11] ])
1841 sage: _unembed_complex_matrix(A)
1843 [ 10*i + 9 12*i + 11]
1847 Unembedding is the inverse of embedding::
1849 sage: set_random_seed()
1850 sage: F = QuadraticField(-1, 'i')
1851 sage: M = random_matrix(F, 3)
1852 sage: _unembed_complex_matrix(_embed_complex_matrix(M)) == M
1858 raise ValueError("the matrix 'M' must be square")
1859 if not n
.mod(2).is_zero():
1860 raise ValueError("the matrix 'M' must be a complex embedding")
1862 F
= QuadraticField(-1, 'i')
1865 # Go top-left to bottom-right (reading order), converting every
1866 # 2-by-2 block we see to a single complex element.
1868 for k
in xrange(n
/2):
1869 for j
in xrange(n
/2):
1870 submat
= M
[2*k
:2*k
+2,2*j
:2*j
+2]
1871 if submat
[0,0] != submat
[1,1]:
1872 raise ValueError('bad on-diagonal submatrix')
1873 if submat
[0,1] != -submat
[1,0]:
1874 raise ValueError('bad off-diagonal submatrix')
1875 z
= submat
[0,0] + submat
[0,1]*i
1878 return matrix(F
, n
/2, elements
)
1881 def _embed_quaternion_matrix(M
):
1883 Embed the n-by-n quaternion matrix ``M`` into the space of real
1884 matrices of size 4n-by-4n by first sending each quaternion entry
1885 `z = a + bi + cj + dk` to the block-complex matrix
1886 ``[[a + bi, c+di],[-c + di, a-bi]]`, and then embedding those into
1891 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1892 sage: i,j,k = Q.gens()
1893 sage: x = 1 + 2*i + 3*j + 4*k
1894 sage: M = matrix(Q, 1, [[x]])
1895 sage: _embed_quaternion_matrix(M)
1901 Embedding is a homomorphism (isomorphism, in fact)::
1903 sage: set_random_seed()
1904 sage: n = ZZ.random_element(5)
1905 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1906 sage: X = random_matrix(Q, n)
1907 sage: Y = random_matrix(Q, n)
1908 sage: actual = _embed_quaternion_matrix(X)*_embed_quaternion_matrix(Y)
1909 sage: expected = _embed_quaternion_matrix(X*Y)
1910 sage: actual == expected
1914 quaternions
= M
.base_ring()
1917 raise ValueError("the matrix 'M' must be square")
1919 F
= QuadraticField(-1, 'i')
1924 t
= z
.coefficient_tuple()
1929 cplx_matrix
= matrix(F
, 2, [[ a
+ b
*i
, c
+ d
*i
],
1930 [-c
+ d
*i
, a
- b
*i
]])
1931 blocks
.append(_embed_complex_matrix(cplx_matrix
))
1933 # We should have real entries by now, so use the realest field
1934 # we've got for the return value.
1935 return block_matrix(quaternions
.base_ring(), n
, blocks
)
1938 def _unembed_quaternion_matrix(M
):
1940 The inverse of _embed_quaternion_matrix().
1944 sage: M = matrix(QQ, [[ 1, 2, 3, 4],
1945 ....: [-2, 1, -4, 3],
1946 ....: [-3, 4, 1, -2],
1947 ....: [-4, -3, 2, 1]])
1948 sage: _unembed_quaternion_matrix(M)
1949 [1 + 2*i + 3*j + 4*k]
1953 Unembedding is the inverse of embedding::
1955 sage: set_random_seed()
1956 sage: Q = QuaternionAlgebra(QQ, -1, -1)
1957 sage: M = random_matrix(Q, 3)
1958 sage: _unembed_quaternion_matrix(_embed_quaternion_matrix(M)) == M
1964 raise ValueError("the matrix 'M' must be square")
1965 if not n
.mod(4).is_zero():
1966 raise ValueError("the matrix 'M' must be a complex embedding")
1968 Q
= QuaternionAlgebra(QQ
,-1,-1)
1971 # Go top-left to bottom-right (reading order), converting every
1972 # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1
1975 for l
in xrange(n
/4):
1976 for m
in xrange(n
/4):
1977 submat
= _unembed_complex_matrix(M
[4*l
:4*l
+4,4*m
:4*m
+4])
1978 if submat
[0,0] != submat
[1,1].conjugate():
1979 raise ValueError('bad on-diagonal submatrix')
1980 if submat
[0,1] != -submat
[1,0].conjugate():
1981 raise ValueError('bad off-diagonal submatrix')
1982 z
= submat
[0,0].real() + submat
[0,0].imag()*i
1983 z
+= submat
[0,1].real()*j
+ submat
[0,1].imag()*k
1986 return matrix(Q
, n
/4, elements
)
1989 # The usual inner product on R^n.
1991 return x
.vector().inner_product(y
.vector())
1993 # The inner product used for the real symmetric simple EJA.
1994 # We keep it as a separate function because e.g. the complex
1995 # algebra uses the same inner product, except divided by 2.
1996 def _matrix_ip(X
,Y
):
1997 X_mat
= X
.natural_representation()
1998 Y_mat
= Y
.natural_representation()
1999 return (X_mat
*Y_mat
).trace()
2002 class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra
):
2004 The rank-n simple EJA consisting of real symmetric n-by-n
2005 matrices, the usual symmetric Jordan product, and the trace inner
2006 product. It has dimension `(n^2 + n)/2` over the reals.
2010 sage: J = RealSymmetricEJA(2)
2011 sage: e0, e1, e2 = J.gens()
2021 The degree of this algebra is `(n^2 + n) / 2`::
2023 sage: set_random_seed()
2024 sage: n = ZZ.random_element(1,5)
2025 sage: J = RealSymmetricEJA(n)
2026 sage: J.degree() == (n^2 + n)/2
2029 The Jordan multiplication is what we think it is::
2031 sage: set_random_seed()
2032 sage: n = ZZ.random_element(1,5)
2033 sage: J = RealSymmetricEJA(n)
2034 sage: x = J.random_element()
2035 sage: y = J.random_element()
2036 sage: actual = (x*y).natural_representation()
2037 sage: X = x.natural_representation()
2038 sage: Y = y.natural_representation()
2039 sage: expected = (X*Y + Y*X)/2
2040 sage: actual == expected
2042 sage: J(expected) == x*y
2047 def __classcall_private__(cls
, n
, field
=QQ
):
2048 S
= _real_symmetric_basis(n
, field
=field
)
2049 (Qs
, T
) = _multiplication_table_from_matrix_basis(S
)
2051 fdeja
= super(RealSymmetricEJA
, cls
)
2052 return fdeja
.__classcall
_private
__(cls
,
2058 def inner_product(self
, x
, y
):
2059 return _matrix_ip(x
,y
)
2062 class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra
):
2064 The rank-n simple EJA consisting of complex Hermitian n-by-n
2065 matrices over the real numbers, the usual symmetric Jordan product,
2066 and the real-part-of-trace inner product. It has dimension `n^2` 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 = ComplexHermitianEJA(n)
2076 sage: J.degree() == n^2
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 = ComplexHermitianEJA(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
= _complex_hermitian_basis(n
)
2099 (Qs
, T
) = _multiplication_table_from_matrix_basis(S
)
2101 fdeja
= super(ComplexHermitianEJA
, cls
)
2102 return fdeja
.__classcall
_private
__(cls
,
2108 def inner_product(self
, x
, y
):
2109 # Since a+bi on the diagonal is represented as
2114 # we'll double-count the "a" entries if we take the trace of
2116 return _matrix_ip(x
,y
)/2
2119 class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra
):
2121 The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
2122 matrices, the usual symmetric Jordan product, and the
2123 real-part-of-trace inner product. It has dimension `2n^2 - n` over
2128 The degree of this algebra is `n^2`::
2130 sage: set_random_seed()
2131 sage: n = ZZ.random_element(1,5)
2132 sage: J = QuaternionHermitianEJA(n)
2133 sage: J.degree() == 2*(n^2) - n
2136 The Jordan multiplication is what we think it is::
2138 sage: set_random_seed()
2139 sage: n = ZZ.random_element(1,5)
2140 sage: J = QuaternionHermitianEJA(n)
2141 sage: x = J.random_element()
2142 sage: y = J.random_element()
2143 sage: actual = (x*y).natural_representation()
2144 sage: X = x.natural_representation()
2145 sage: Y = y.natural_representation()
2146 sage: expected = (X*Y + Y*X)/2
2147 sage: actual == expected
2149 sage: J(expected) == x*y
2154 def __classcall_private__(cls
, n
, field
=QQ
):
2155 S
= _quaternion_hermitian_basis(n
)
2156 (Qs
, T
) = _multiplication_table_from_matrix_basis(S
)
2158 fdeja
= super(QuaternionHermitianEJA
, cls
)
2159 return fdeja
.__classcall
_private
__(cls
,
2165 def inner_product(self
, x
, y
):
2166 # Since a+bi+cj+dk on the diagonal is represented as
2168 # a + bi +cj + dk = [ a b c d]
2173 # we'll quadruple-count the "a" entries if we take the trace of
2175 return _matrix_ip(x
,y
)/4
2178 class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra
):
2180 The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
2181 with the usual inner product and jordan product ``x*y =
2182 (<x_bar,y_bar>, x0*y_bar + y0*x_bar)``. It has dimension `n` over
2187 This multiplication table can be verified by hand::
2189 sage: J = JordanSpinEJA(4)
2190 sage: e0,e1,e2,e3 = J.gens()
2208 def __classcall_private__(cls
, n
, field
=QQ
):
2210 id_matrix
= identity_matrix(field
, n
)
2212 ei
= id_matrix
.column(i
)
2213 Qi
= zero_matrix(field
, n
)
2215 Qi
.set_column(0, ei
)
2216 Qi
+= diagonal_matrix(n
, [ei
[0]]*n
)
2217 # The addition of the diagonal matrix adds an extra ei[0] in the
2218 # upper-left corner of the matrix.
2219 Qi
[0,0] = Qi
[0,0] * ~
field(2)
2222 # The rank of the spin algebra is two, unless we're in a
2223 # one-dimensional ambient space (because the rank is bounded by
2224 # the ambient dimension).
2225 fdeja
= super(JordanSpinEJA
, cls
)
2226 return fdeja
.__classcall
_private
__(cls
, field
, Qs
, rank
=min(n
,2))
2228 def inner_product(self
, x
, y
):
2229 return _usual_ip(x
,y
)