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.
10 from sage
.algebras
.finite_dimensional_algebras
.finite_dimensional_algebra
import FiniteDimensionalAlgebra
11 from sage
.algebras
.finite_dimensional_algebras
.finite_dimensional_algebra_element
import FiniteDimensionalAlgebraElement
12 from sage
.algebras
.quatalg
.quaternion_algebra
import QuaternionAlgebra
13 from sage
.categories
.finite_dimensional_algebras_with_basis
import FiniteDimensionalAlgebrasWithBasis
14 from sage
.functions
.other
import sqrt
15 from sage
.matrix
.constructor
import matrix
16 from sage
.misc
.cachefunc
import cached_method
17 from sage
.misc
.prandom
import choice
18 from sage
.modules
.free_module
import VectorSpace
19 from sage
.modules
.free_module_element
import vector
20 from sage
.rings
.integer_ring
import ZZ
21 from sage
.rings
.number_field
.number_field
import QuadraticField
22 from sage
.rings
.polynomial
.polynomial_ring_constructor
import PolynomialRing
23 from sage
.rings
.rational_field
import QQ
24 from sage
.structure
.element
import is_Matrix
25 from sage
.structure
.category_object
import normalize_names
27 from mjo
.eja
.eja_operator
import FiniteDimensionalEuclideanJordanAlgebraOperator
30 class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra
):
32 def __classcall_private__(cls
,
37 assume_associative
=False,
41 mult_table
= [b
.base_extend(field
) for b
in mult_table
]
44 if not (is_Matrix(b
) and b
.dimensions() == (n
, n
)):
45 raise ValueError("input is not a multiplication table")
46 mult_table
= tuple(mult_table
)
48 cat
= FiniteDimensionalAlgebrasWithBasis(field
)
49 cat
.or_subcategory(category
)
50 if assume_associative
:
51 cat
= cat
.Associative()
53 names
= normalize_names(n
, names
)
55 fda
= super(FiniteDimensionalEuclideanJordanAlgebra
, cls
)
56 return fda
.__classcall
__(cls
,
60 assume_associative
=assume_associative
,
63 natural_basis
=natural_basis
)
71 assume_associative
=False,
77 sage: from mjo.eja.eja_algebra import random_eja
81 By definition, Jordan multiplication commutes::
83 sage: set_random_seed()
84 sage: J = random_eja()
85 sage: x = J.random_element()
86 sage: y = J.random_element()
92 self
._natural
_basis
= natural_basis
93 self
._multiplication
_table
= mult_table
94 fda
= super(FiniteDimensionalEuclideanJordanAlgebra
, self
)
103 Return a string representation of ``self``.
107 sage: from mjo.eja.eja_algebra import JordanSpinEJA
111 Ensure that it says what we think it says::
113 sage: JordanSpinEJA(2, field=QQ)
114 Euclidean Jordan algebra of degree 2 over Rational Field
115 sage: JordanSpinEJA(3, field=RDF)
116 Euclidean Jordan algebra of degree 3 over Real Double Field
119 fmt
= "Euclidean Jordan algebra of degree {} over {}"
120 return fmt
.format(self
.degree(), self
.base_ring())
123 def _a_regular_element(self
):
125 Guess a regular element. Needed to compute the basis for our
126 characteristic polynomial coefficients.
130 sage: from mjo.eja.eja_algebra import random_eja
134 Ensure that this hacky method succeeds for every algebra that we
135 know how to construct::
137 sage: set_random_seed()
138 sage: J = random_eja()
139 sage: J._a_regular_element().is_regular()
144 z
= self
.sum( (i
+1)*gs
[i
] for i
in range(len(gs
)) )
145 if not z
.is_regular():
146 raise ValueError("don't know a regular element")
151 def _charpoly_basis_space(self
):
153 Return the vector space spanned by the basis used in our
154 characteristic polynomial coefficients. This is used not only to
155 compute those coefficients, but also any time we need to
156 evaluate the coefficients (like when we compute the trace or
159 z
= self
._a
_regular
_element
()
160 V
= self
.vector_space()
161 V1
= V
.span_of_basis( (z
**k
).vector() for k
in range(self
.rank()) )
162 b
= (V1
.basis() + V1
.complement().basis())
163 return V
.span_of_basis(b
)
167 def _charpoly_coeff(self
, i
):
169 Return the coefficient polynomial "a_{i}" of this algebra's
170 general characteristic polynomial.
172 Having this be a separate cached method lets us compute and
173 store the trace/determinant (a_{r-1} and a_{0} respectively)
174 separate from the entire characteristic polynomial.
176 (A_of_x
, x
, xr
, detA
) = self
._charpoly
_matrix
_system
()
177 R
= A_of_x
.base_ring()
179 # Guaranteed by theory
182 # Danger: the in-place modification is done for performance
183 # reasons (reconstructing a matrix with huge polynomial
184 # entries is slow), but I don't know how cached_method works,
185 # so it's highly possible that we're modifying some global
186 # list variable by reference, here. In other words, you
187 # probably shouldn't call this method twice on the same
188 # algebra, at the same time, in two threads
189 Ai_orig
= A_of_x
.column(i
)
190 A_of_x
.set_column(i
,xr
)
191 numerator
= A_of_x
.det()
192 A_of_x
.set_column(i
,Ai_orig
)
194 # We're relying on the theory here to ensure that each a_i is
195 # indeed back in R, and the added negative signs are to make
196 # the whole charpoly expression sum to zero.
197 return R(-numerator
/detA
)
201 def _charpoly_matrix_system(self
):
203 Compute the matrix whose entries A_ij are polynomials in
204 X1,...,XN, the vector ``x`` of variables X1,...,XN, the vector
205 corresponding to `x^r` and the determinent of the matrix A =
206 [A_ij]. In other words, all of the fixed (cachable) data needed
207 to compute the coefficients of the characteristic polynomial.
212 # Construct a new algebra over a multivariate polynomial ring...
213 names
= ['X' + str(i
) for i
in range(1,n
+1)]
214 R
= PolynomialRing(self
.base_ring(), names
)
215 J
= FiniteDimensionalEuclideanJordanAlgebra(R
,
216 self
._multiplication
_table
,
219 idmat
= matrix
.identity(J
.base_ring(), n
)
221 W
= self
._charpoly
_basis
_space
()
222 W
= W
.change_ring(R
.fraction_field())
224 # Starting with the standard coordinates x = (X1,X2,...,Xn)
225 # and then converting the entries to W-coordinates allows us
226 # to pass in the standard coordinates to the charpoly and get
227 # back the right answer. Specifically, with x = (X1,X2,...,Xn),
230 # W.coordinates(x^2) eval'd at (standard z-coords)
234 # W-coords of (standard coords of x^2 eval'd at std-coords of z)
236 # We want the middle equivalent thing in our matrix, but use
237 # the first equivalent thing instead so that we can pass in
238 # standard coordinates.
240 l1
= [matrix
.column(W
.coordinates((x
**k
).vector())) for k
in range(r
)]
241 l2
= [idmat
.column(k
-1).column() for k
in range(r
+1, n
+1)]
242 A_of_x
= matrix
.block(R
, 1, n
, (l1
+ l2
))
243 xr
= W
.coordinates((x
**r
).vector())
244 return (A_of_x
, x
, xr
, A_of_x
.det())
248 def characteristic_polynomial(self
):
250 Return a characteristic polynomial that works for all elements
253 The resulting polynomial has `n+1` variables, where `n` is the
254 dimension of this algebra. The first `n` variables correspond to
255 the coordinates of an algebra element: when evaluated at the
256 coordinates of an algebra element with respect to a certain
257 basis, the result is a univariate polynomial (in the one
258 remaining variable ``t``), namely the characteristic polynomial
263 sage: from mjo.eja.eja_algebra import JordanSpinEJA
267 The characteristic polynomial in the spin algebra is given in
268 Alizadeh, Example 11.11::
270 sage: J = JordanSpinEJA(3)
271 sage: p = J.characteristic_polynomial(); p
272 X1^2 - X2^2 - X3^2 + (-2*t)*X1 + t^2
273 sage: xvec = J.one().vector()
281 # The list of coefficient polynomials a_1, a_2, ..., a_n.
282 a
= [ self
._charpoly
_coeff
(i
) for i
in range(n
) ]
284 # We go to a bit of trouble here to reorder the
285 # indeterminates, so that it's easier to evaluate the
286 # characteristic polynomial at x's coordinates and get back
287 # something in terms of t, which is what we want.
289 S
= PolynomialRing(self
.base_ring(),'t')
291 S
= PolynomialRing(S
, R
.variable_names())
294 # Note: all entries past the rth should be zero. The
295 # coefficient of the highest power (x^r) is 1, but it doesn't
296 # appear in the solution vector which contains coefficients
297 # for the other powers (to make them sum to x^r).
299 a
[r
] = 1 # corresponds to x^r
301 # When the rank is equal to the dimension, trying to
302 # assign a[r] goes out-of-bounds.
303 a
.append(1) # corresponds to x^r
305 return sum( a
[k
]*(t
**k
) for k
in range(len(a
)) )
308 def inner_product(self
, x
, y
):
310 The inner product associated with this Euclidean Jordan algebra.
312 Defaults to the trace inner product, but can be overridden by
313 subclasses if they are sure that the necessary properties are
318 sage: from mjo.eja.eja_algebra import random_eja
322 The inner product must satisfy its axiom for this algebra to truly
323 be a Euclidean Jordan Algebra::
325 sage: set_random_seed()
326 sage: J = random_eja()
327 sage: x = J.random_element()
328 sage: y = J.random_element()
329 sage: z = J.random_element()
330 sage: (x*y).inner_product(z) == y.inner_product(x*z)
334 if (not x
in self
) or (not y
in self
):
335 raise TypeError("arguments must live in this algebra")
336 return x
.trace_inner_product(y
)
339 def natural_basis(self
):
341 Return a more-natural representation of this algebra's basis.
343 Every finite-dimensional Euclidean Jordan Algebra is a direct
344 sum of five simple algebras, four of which comprise Hermitian
345 matrices. This method returns the original "natural" basis
346 for our underlying vector space. (Typically, the natural basis
347 is used to construct the multiplication table in the first place.)
349 Note that this will always return a matrix. The standard basis
350 in `R^n` will be returned as `n`-by-`1` column matrices.
354 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
355 ....: RealSymmetricEJA)
359 sage: J = RealSymmetricEJA(2)
362 sage: J.natural_basis()
370 sage: J = JordanSpinEJA(2)
373 sage: J.natural_basis()
380 if self
._natural
_basis
is None:
381 return tuple( b
.vector().column() for b
in self
.basis() )
383 return self
._natural
_basis
388 Return the rank of this EJA.
392 The author knows of no algorithm to compute the rank of an EJA
393 where only the multiplication table is known. In lieu of one, we
394 require the rank to be specified when the algebra is created,
395 and simply pass along that number here.
399 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
400 ....: RealSymmetricEJA,
401 ....: ComplexHermitianEJA,
402 ....: QuaternionHermitianEJA,
407 The rank of the Jordan spin algebra is always two::
409 sage: JordanSpinEJA(2).rank()
411 sage: JordanSpinEJA(3).rank()
413 sage: JordanSpinEJA(4).rank()
416 The rank of the `n`-by-`n` Hermitian real, complex, or
417 quaternion matrices is `n`::
419 sage: RealSymmetricEJA(2).rank()
421 sage: ComplexHermitianEJA(2).rank()
423 sage: QuaternionHermitianEJA(2).rank()
425 sage: RealSymmetricEJA(5).rank()
427 sage: ComplexHermitianEJA(5).rank()
429 sage: QuaternionHermitianEJA(5).rank()
434 Ensure that every EJA that we know how to construct has a
435 positive integer rank::
437 sage: set_random_seed()
438 sage: r = random_eja().rank()
439 sage: r in ZZ and r > 0
446 def vector_space(self
):
448 Return the vector space that underlies this algebra.
452 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
456 sage: J = RealSymmetricEJA(2)
457 sage: J.vector_space()
458 Vector space of dimension 3 over Rational Field
461 return self
.zero().vector().parent().ambient_vector_space()
464 class Element(FiniteDimensionalAlgebraElement
):
466 An element of a Euclidean Jordan algebra.
471 Oh man, I should not be doing this. This hides the "disabled"
472 methods ``left_matrix`` and ``matrix`` from introspection;
473 in particular it removes them from tab-completion.
475 return filter(lambda s
: s
not in ['left_matrix', 'matrix'],
476 dir(self
.__class
__) )
479 def __init__(self
, A
, elt
=None):
484 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
488 The identity in `S^n` is converted to the identity in the EJA::
490 sage: J = RealSymmetricEJA(3)
491 sage: I = matrix.identity(QQ,3)
492 sage: J(I) == J.one()
495 This skew-symmetric matrix can't be represented in the EJA::
497 sage: J = RealSymmetricEJA(3)
498 sage: A = matrix(QQ,3, lambda i,j: i-j)
500 Traceback (most recent call last):
502 ArithmeticError: vector is not in free module
505 # Goal: if we're given a matrix, and if it lives in our
506 # parent algebra's "natural ambient space," convert it
507 # into an algebra element.
509 # The catch is, we make a recursive call after converting
510 # the given matrix into a vector that lives in the algebra.
511 # This we need to try the parent class initializer first,
512 # to avoid recursing forever if we're given something that
513 # already fits into the algebra, but also happens to live
514 # in the parent's "natural ambient space" (this happens with
517 FiniteDimensionalAlgebraElement
.__init
__(self
, A
, elt
)
519 natural_basis
= A
.natural_basis()
520 if elt
in natural_basis
[0].matrix_space():
521 # Thanks for nothing! Matrix spaces aren't vector
522 # spaces in Sage, so we have to figure out its
523 # natural-basis coordinates ourselves.
524 V
= VectorSpace(elt
.base_ring(), elt
.nrows()**2)
525 W
= V
.span( _mat2vec(s
) for s
in natural_basis
)
526 coords
= W
.coordinates(_mat2vec(elt
))
527 FiniteDimensionalAlgebraElement
.__init
__(self
, A
, coords
)
529 def __pow__(self
, n
):
531 Return ``self`` raised to the power ``n``.
533 Jordan algebras are always power-associative; see for
534 example Faraut and Koranyi, Proposition II.1.2 (ii).
538 We have to override this because our superclass uses row vectors
539 instead of column vectors! We, on the other hand, assume column
544 sage: from mjo.eja.eja_algebra import random_eja
548 sage: set_random_seed()
549 sage: x = random_eja().random_element()
550 sage: x.operator()(x) == (x^2)
553 A few examples of power-associativity::
555 sage: set_random_seed()
556 sage: x = random_eja().random_element()
557 sage: x*(x*x)*(x*x) == x^5
559 sage: (x*x)*(x*x*x) == x^5
562 We also know that powers operator-commute (Koecher, Chapter
565 sage: set_random_seed()
566 sage: x = random_eja().random_element()
567 sage: m = ZZ.random_element(0,10)
568 sage: n = ZZ.random_element(0,10)
569 sage: Lxm = (x^m).operator()
570 sage: Lxn = (x^n).operator()
571 sage: Lxm*Lxn == Lxn*Lxm
576 return self
.parent().one()
580 return (self
.operator()**(n
-1))(self
)
583 def apply_univariate_polynomial(self
, p
):
585 Apply the univariate polynomial ``p`` to this element.
587 A priori, SageMath won't allow us to apply a univariate
588 polynomial to an element of an EJA, because we don't know
589 that EJAs are rings (they are usually not associative). Of
590 course, we know that EJAs are power-associative, so the
591 operation is ultimately kosher. This function sidesteps
592 the CAS to get the answer we want and expect.
596 sage: from mjo.eja.eja_algebra import (RealCartesianProductEJA,
601 sage: R = PolynomialRing(QQ, 't')
603 sage: p = t^4 - t^3 + 5*t - 2
604 sage: J = RealCartesianProductEJA(5)
605 sage: J.one().apply_univariate_polynomial(p) == 3*J.one()
610 We should always get back an element of the algebra::
612 sage: set_random_seed()
613 sage: p = PolynomialRing(QQ, 't').random_element()
614 sage: J = random_eja()
615 sage: x = J.random_element()
616 sage: x.apply_univariate_polynomial(p) in J
620 if len(p
.variables()) > 1:
621 raise ValueError("not a univariate polynomial")
624 # Convert the coeficcients to the parent's base ring,
625 # because a priori they might live in an (unnecessarily)
626 # larger ring for which P.sum() would fail below.
627 cs
= [ R(c
) for c
in p
.coefficients(sparse
=False) ]
628 return P
.sum( cs
[k
]*(self
**k
) for k
in range(len(cs
)) )
631 def characteristic_polynomial(self
):
633 Return the characteristic polynomial of this element.
637 sage: from mjo.eja.eja_algebra import RealCartesianProductEJA
641 The rank of `R^3` is three, and the minimal polynomial of
642 the identity element is `(t-1)` from which it follows that
643 the characteristic polynomial should be `(t-1)^3`::
645 sage: J = RealCartesianProductEJA(3)
646 sage: J.one().characteristic_polynomial()
647 t^3 - 3*t^2 + 3*t - 1
649 Likewise, the characteristic of the zero element in the
650 rank-three algebra `R^{n}` should be `t^{3}`::
652 sage: J = RealCartesianProductEJA(3)
653 sage: J.zero().characteristic_polynomial()
656 The characteristic polynomial of an element should evaluate
657 to zero on that element::
659 sage: set_random_seed()
660 sage: x = RealCartesianProductEJA(3).random_element()
661 sage: p = x.characteristic_polynomial()
662 sage: x.apply_univariate_polynomial(p)
666 p
= self
.parent().characteristic_polynomial()
667 return p(*self
.vector())
670 def inner_product(self
, other
):
672 Return the parent algebra's inner product of myself and ``other``.
676 sage: from mjo.eja.eja_algebra import (
677 ....: ComplexHermitianEJA,
679 ....: QuaternionHermitianEJA,
680 ....: RealSymmetricEJA,
685 The inner product in the Jordan spin algebra is the usual
686 inner product on `R^n` (this example only works because the
687 basis for the Jordan algebra is the standard basis in `R^n`)::
689 sage: J = JordanSpinEJA(3)
690 sage: x = vector(QQ,[1,2,3])
691 sage: y = vector(QQ,[4,5,6])
692 sage: x.inner_product(y)
694 sage: J(x).inner_product(J(y))
697 The inner product on `S^n` is `<X,Y> = trace(X*Y)`, where
698 multiplication is the usual matrix multiplication in `S^n`,
699 so the inner product of the identity matrix with itself
702 sage: J = RealSymmetricEJA(3)
703 sage: J.one().inner_product(J.one())
706 Likewise, the inner product on `C^n` is `<X,Y> =
707 Re(trace(X*Y))`, where we must necessarily take the real
708 part because the product of Hermitian matrices may not be
711 sage: J = ComplexHermitianEJA(3)
712 sage: J.one().inner_product(J.one())
715 Ditto for the quaternions::
717 sage: J = QuaternionHermitianEJA(3)
718 sage: J.one().inner_product(J.one())
723 Ensure that we can always compute an inner product, and that
724 it gives us back a real number::
726 sage: set_random_seed()
727 sage: J = random_eja()
728 sage: x = J.random_element()
729 sage: y = J.random_element()
730 sage: x.inner_product(y) in RR
736 raise TypeError("'other' must live in the same algebra")
738 return P
.inner_product(self
, other
)
741 def operator_commutes_with(self
, other
):
743 Return whether or not this element operator-commutes
748 sage: from mjo.eja.eja_algebra import random_eja
752 The definition of a Jordan algebra says that any element
753 operator-commutes with its square::
755 sage: set_random_seed()
756 sage: x = random_eja().random_element()
757 sage: x.operator_commutes_with(x^2)
762 Test Lemma 1 from Chapter III of Koecher::
764 sage: set_random_seed()
765 sage: J = random_eja()
766 sage: u = J.random_element()
767 sage: v = J.random_element()
768 sage: lhs = u.operator_commutes_with(u*v)
769 sage: rhs = v.operator_commutes_with(u^2)
773 Test the first polarization identity from my notes, Koecher Chapter
774 III, or from Baes (2.3)::
776 sage: set_random_seed()
777 sage: J = random_eja()
778 sage: x = J.random_element()
779 sage: y = J.random_element()
780 sage: Lx = x.operator()
781 sage: Ly = y.operator()
782 sage: Lxx = (x*x).operator()
783 sage: Lxy = (x*y).operator()
784 sage: bool(2*Lx*Lxy + Ly*Lxx == 2*Lxy*Lx + Lxx*Ly)
787 Test the second polarization identity from my notes or from
790 sage: set_random_seed()
791 sage: J = random_eja()
792 sage: x = J.random_element()
793 sage: y = J.random_element()
794 sage: z = J.random_element()
795 sage: Lx = x.operator()
796 sage: Ly = y.operator()
797 sage: Lz = z.operator()
798 sage: Lzy = (z*y).operator()
799 sage: Lxy = (x*y).operator()
800 sage: Lxz = (x*z).operator()
801 sage: bool(Lx*Lzy + Lz*Lxy + Ly*Lxz == Lzy*Lx + Lxy*Lz + Lxz*Ly)
804 Test the third polarization identity from my notes or from
807 sage: set_random_seed()
808 sage: J = random_eja()
809 sage: u = J.random_element()
810 sage: y = J.random_element()
811 sage: z = J.random_element()
812 sage: Lu = u.operator()
813 sage: Ly = y.operator()
814 sage: Lz = z.operator()
815 sage: Lzy = (z*y).operator()
816 sage: Luy = (u*y).operator()
817 sage: Luz = (u*z).operator()
818 sage: Luyz = (u*(y*z)).operator()
819 sage: lhs = Lu*Lzy + Lz*Luy + Ly*Luz
820 sage: rhs = Luyz + Ly*Lu*Lz + Lz*Lu*Ly
821 sage: bool(lhs == rhs)
825 if not other
in self
.parent():
826 raise TypeError("'other' must live in the same algebra")
835 Return my determinant, the product of my eigenvalues.
839 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
844 sage: J = JordanSpinEJA(2)
845 sage: e0,e1 = J.gens()
846 sage: x = sum( J.gens() )
852 sage: J = JordanSpinEJA(3)
853 sage: e0,e1,e2 = J.gens()
854 sage: x = sum( J.gens() )
860 An element is invertible if and only if its determinant is
863 sage: set_random_seed()
864 sage: x = random_eja().random_element()
865 sage: x.is_invertible() == (x.det() != 0)
871 p
= P
._charpoly
_coeff
(0)
872 # The _charpoly_coeff function already adds the factor of
873 # -1 to ensure that _charpoly_coeff(0) is really what
874 # appears in front of t^{0} in the charpoly. However,
875 # we want (-1)^r times THAT for the determinant.
876 return ((-1)**r
)*p(*self
.vector())
881 Return the Jordan-multiplicative inverse of this element.
885 We appeal to the quadratic representation as in Koecher's
886 Theorem 12 in Chapter III, Section 5.
890 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
895 The inverse in the spin factor algebra is given in Alizadeh's
898 sage: set_random_seed()
899 sage: n = ZZ.random_element(1,10)
900 sage: J = JordanSpinEJA(n)
901 sage: x = J.random_element()
902 sage: while not x.is_invertible():
903 ....: x = J.random_element()
904 sage: x_vec = x.vector()
906 sage: x_bar = x_vec[1:]
907 sage: coeff = ~(x0^2 - x_bar.inner_product(x_bar))
908 sage: inv_vec = x_vec.parent()([x0] + (-x_bar).list())
909 sage: x_inverse = coeff*inv_vec
910 sage: x.inverse() == J(x_inverse)
915 The identity element is its own inverse::
917 sage: set_random_seed()
918 sage: J = random_eja()
919 sage: J.one().inverse() == J.one()
922 If an element has an inverse, it acts like one::
924 sage: set_random_seed()
925 sage: J = random_eja()
926 sage: x = J.random_element()
927 sage: (not x.is_invertible()) or (x.inverse()*x == J.one())
930 The inverse of the inverse is what we started with::
932 sage: set_random_seed()
933 sage: J = random_eja()
934 sage: x = J.random_element()
935 sage: (not x.is_invertible()) or (x.inverse().inverse() == x)
938 The zero element is never invertible::
940 sage: set_random_seed()
941 sage: J = random_eja().zero().inverse()
942 Traceback (most recent call last):
944 ValueError: element is not invertible
947 if not self
.is_invertible():
948 raise ValueError("element is not invertible")
950 return (~self
.quadratic_representation())(self
)
953 def is_invertible(self
):
955 Return whether or not this element is invertible.
957 We can't use the superclass method because it relies on
958 the algebra being associative.
962 The usual way to do this is to check if the determinant is
963 zero, but we need the characteristic polynomial for the
964 determinant. The minimal polynomial is a lot easier to get,
965 so we use Corollary 2 in Chapter V of Koecher to check
966 whether or not the paren't algebra's zero element is a root
967 of this element's minimal polynomial.
971 sage: from mjo.eja.eja_algebra import random_eja
975 The identity element is always invertible::
977 sage: set_random_seed()
978 sage: J = random_eja()
979 sage: J.one().is_invertible()
982 The zero element is never invertible::
984 sage: set_random_seed()
985 sage: J = random_eja()
986 sage: J.zero().is_invertible()
990 zero
= self
.parent().zero()
991 p
= self
.minimal_polynomial()
992 return not (p(zero
) == zero
)
995 def is_nilpotent(self
):
997 Return whether or not some power of this element is zero.
999 The superclass method won't work unless we're in an
1000 associative algebra, and we aren't. However, we generate
1001 an assocoative subalgebra and we're nilpotent there if and
1002 only if we're nilpotent here (probably).
1006 sage: from mjo.eja.eja_algebra import random_eja
1010 The identity element is never nilpotent::
1012 sage: set_random_seed()
1013 sage: random_eja().one().is_nilpotent()
1016 The additive identity is always nilpotent::
1018 sage: set_random_seed()
1019 sage: random_eja().zero().is_nilpotent()
1023 # The element we're going to call "is_nilpotent()" on.
1024 # Either myself, interpreted as an element of a finite-
1025 # dimensional algebra, or an element of an associative
1029 if self
.parent().is_associative():
1030 elt
= FiniteDimensionalAlgebraElement(self
.parent(), self
)
1032 V
= self
.span_of_powers()
1033 assoc_subalg
= self
.subalgebra_generated_by()
1034 # Mis-design warning: the basis used for span_of_powers()
1035 # and subalgebra_generated_by() must be the same, and in
1037 elt
= assoc_subalg(V
.coordinates(self
.vector()))
1039 # Recursive call, but should work since elt lives in an
1040 # associative algebra.
1041 return elt
.is_nilpotent()
1044 def is_regular(self
):
1046 Return whether or not this is a regular element.
1050 sage: from mjo.eja.eja_algebra import JordanSpinEJA
1054 The identity element always has degree one, but any element
1055 linearly-independent from it is regular::
1057 sage: J = JordanSpinEJA(5)
1058 sage: J.one().is_regular()
1060 sage: e0, e1, e2, e3, e4 = J.gens() # e0 is the identity
1061 sage: for x in J.gens():
1062 ....: (J.one() + x).is_regular()
1070 return self
.degree() == self
.parent().rank()
1075 Compute the degree of this element the straightforward way
1076 according to the definition; by appending powers to a list
1077 and figuring out its dimension (that is, whether or not
1078 they're linearly dependent).
1082 sage: from mjo.eja.eja_algebra import JordanSpinEJA
1086 sage: J = JordanSpinEJA(4)
1087 sage: J.one().degree()
1089 sage: e0,e1,e2,e3 = J.gens()
1090 sage: (e0 - e1).degree()
1093 In the spin factor algebra (of rank two), all elements that
1094 aren't multiples of the identity are regular::
1096 sage: set_random_seed()
1097 sage: n = ZZ.random_element(1,10)
1098 sage: J = JordanSpinEJA(n)
1099 sage: x = J.random_element()
1100 sage: x == x.coefficient(0)*J.one() or x.degree() == 2
1104 return self
.span_of_powers().dimension()
1107 def left_matrix(self
):
1109 Our parent class defines ``left_matrix`` and ``matrix``
1110 methods whose names are misleading. We don't want them.
1112 raise NotImplementedError("use operator().matrix() instead")
1114 matrix
= left_matrix
1117 def minimal_polynomial(self
):
1119 Return the minimal polynomial of this element,
1120 as a function of the variable `t`.
1124 We restrict ourselves to the associative subalgebra
1125 generated by this element, and then return the minimal
1126 polynomial of this element's operator matrix (in that
1127 subalgebra). This works by Baes Proposition 2.3.16.
1131 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
1136 The minimal polynomial of the identity and zero elements are
1139 sage: set_random_seed()
1140 sage: J = random_eja()
1141 sage: J.one().minimal_polynomial()
1143 sage: J.zero().minimal_polynomial()
1146 The degree of an element is (by one definition) the degree
1147 of its minimal polynomial::
1149 sage: set_random_seed()
1150 sage: x = random_eja().random_element()
1151 sage: x.degree() == x.minimal_polynomial().degree()
1154 The minimal polynomial and the characteristic polynomial coincide
1155 and are known (see Alizadeh, Example 11.11) for all elements of
1156 the spin factor algebra that aren't scalar multiples of the
1159 sage: set_random_seed()
1160 sage: n = ZZ.random_element(2,10)
1161 sage: J = JordanSpinEJA(n)
1162 sage: y = J.random_element()
1163 sage: while y == y.coefficient(0)*J.one():
1164 ....: y = J.random_element()
1165 sage: y0 = y.vector()[0]
1166 sage: y_bar = y.vector()[1:]
1167 sage: actual = y.minimal_polynomial()
1168 sage: t = PolynomialRing(J.base_ring(),'t').gen(0)
1169 sage: expected = t^2 - 2*y0*t + (y0^2 - norm(y_bar)^2)
1170 sage: bool(actual == expected)
1173 The minimal polynomial should always kill its element::
1175 sage: set_random_seed()
1176 sage: x = random_eja().random_element()
1177 sage: p = x.minimal_polynomial()
1178 sage: x.apply_univariate_polynomial(p)
1182 V
= self
.span_of_powers()
1183 assoc_subalg
= self
.subalgebra_generated_by()
1184 # Mis-design warning: the basis used for span_of_powers()
1185 # and subalgebra_generated_by() must be the same, and in
1187 elt
= assoc_subalg(V
.coordinates(self
.vector()))
1188 return elt
.operator().minimal_polynomial()
1192 def natural_representation(self
):
1194 Return a more-natural representation of this element.
1196 Every finite-dimensional Euclidean Jordan Algebra is a
1197 direct sum of five simple algebras, four of which comprise
1198 Hermitian matrices. This method returns the original
1199 "natural" representation of this element as a Hermitian
1200 matrix, if it has one. If not, you get the usual representation.
1204 sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
1205 ....: QuaternionHermitianEJA)
1209 sage: J = ComplexHermitianEJA(3)
1212 sage: J.one().natural_representation()
1222 sage: J = QuaternionHermitianEJA(3)
1225 sage: J.one().natural_representation()
1226 [1 0 0 0 0 0 0 0 0 0 0 0]
1227 [0 1 0 0 0 0 0 0 0 0 0 0]
1228 [0 0 1 0 0 0 0 0 0 0 0 0]
1229 [0 0 0 1 0 0 0 0 0 0 0 0]
1230 [0 0 0 0 1 0 0 0 0 0 0 0]
1231 [0 0 0 0 0 1 0 0 0 0 0 0]
1232 [0 0 0 0 0 0 1 0 0 0 0 0]
1233 [0 0 0 0 0 0 0 1 0 0 0 0]
1234 [0 0 0 0 0 0 0 0 1 0 0 0]
1235 [0 0 0 0 0 0 0 0 0 1 0 0]
1236 [0 0 0 0 0 0 0 0 0 0 1 0]
1237 [0 0 0 0 0 0 0 0 0 0 0 1]
1240 B
= self
.parent().natural_basis()
1241 W
= B
[0].matrix_space()
1242 return W
.linear_combination(zip(self
.vector(), B
))
1247 Return the left-multiplication-by-this-element
1248 operator on the ambient algebra.
1252 sage: from mjo.eja.eja_algebra import random_eja
1256 sage: set_random_seed()
1257 sage: J = random_eja()
1258 sage: x = J.random_element()
1259 sage: y = J.random_element()
1260 sage: x.operator()(y) == x*y
1262 sage: y.operator()(x) == x*y
1267 fda_elt
= FiniteDimensionalAlgebraElement(P
, self
)
1268 return FiniteDimensionalEuclideanJordanAlgebraOperator(
1271 fda_elt
.matrix().transpose() )
1274 def quadratic_representation(self
, other
=None):
1276 Return the quadratic representation of this element.
1280 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
1285 The explicit form in the spin factor algebra is given by
1286 Alizadeh's Example 11.12::
1288 sage: set_random_seed()
1289 sage: n = ZZ.random_element(1,10)
1290 sage: J = JordanSpinEJA(n)
1291 sage: x = J.random_element()
1292 sage: x_vec = x.vector()
1294 sage: x_bar = x_vec[1:]
1295 sage: A = matrix(QQ, 1, [x_vec.inner_product(x_vec)])
1296 sage: B = 2*x0*x_bar.row()
1297 sage: C = 2*x0*x_bar.column()
1298 sage: D = matrix.identity(QQ, n-1)
1299 sage: D = (x0^2 - x_bar.inner_product(x_bar))*D
1300 sage: D = D + 2*x_bar.tensor_product(x_bar)
1301 sage: Q = matrix.block(2,2,[A,B,C,D])
1302 sage: Q == x.quadratic_representation().matrix()
1305 Test all of the properties from Theorem 11.2 in Alizadeh::
1307 sage: set_random_seed()
1308 sage: J = random_eja()
1309 sage: x = J.random_element()
1310 sage: y = J.random_element()
1311 sage: Lx = x.operator()
1312 sage: Lxx = (x*x).operator()
1313 sage: Qx = x.quadratic_representation()
1314 sage: Qy = y.quadratic_representation()
1315 sage: Qxy = x.quadratic_representation(y)
1316 sage: Qex = J.one().quadratic_representation(x)
1317 sage: n = ZZ.random_element(10)
1318 sage: Qxn = (x^n).quadratic_representation()
1322 sage: 2*Qxy == (x+y).quadratic_representation() - Qx - Qy
1325 Property 2 (multiply on the right for :trac:`28272`):
1327 sage: alpha = QQ.random_element()
1328 sage: (alpha*x).quadratic_representation() == Qx*(alpha^2)
1333 sage: not x.is_invertible() or ( Qx(x.inverse()) == x )
1336 sage: not x.is_invertible() or (
1339 ....: x.inverse().quadratic_representation() )
1342 sage: Qxy(J.one()) == x*y
1347 sage: not x.is_invertible() or (
1348 ....: x.quadratic_representation(x.inverse())*Qx
1349 ....: == Qx*x.quadratic_representation(x.inverse()) )
1352 sage: not x.is_invertible() or (
1353 ....: x.quadratic_representation(x.inverse())*Qx
1355 ....: 2*x.operator()*Qex - Qx )
1358 sage: 2*x.operator()*Qex - Qx == Lxx
1363 sage: Qy(x).quadratic_representation() == Qy*Qx*Qy
1373 sage: not x.is_invertible() or (
1374 ....: Qx*x.inverse().operator() == Lx )
1379 sage: not x.operator_commutes_with(y) or (
1380 ....: Qx(y)^n == Qxn(y^n) )
1386 elif not other
in self
.parent():
1387 raise TypeError("'other' must live in the same algebra")
1390 M
= other
.operator()
1391 return ( L
*M
+ M
*L
- (self
*other
).operator() )
1394 def span_of_powers(self
):
1396 Return the vector space spanned by successive powers of
1399 # The dimension of the subalgebra can't be greater than
1400 # the big algebra, so just put everything into a list
1401 # and let span() get rid of the excess.
1403 # We do the extra ambient_vector_space() in case we're messing
1404 # with polynomials and the direct parent is a module.
1405 V
= self
.parent().vector_space()
1406 return V
.span( (self
**d
).vector() for d
in xrange(V
.dimension()) )
1409 def subalgebra_generated_by(self
):
1411 Return the associative subalgebra of the parent EJA generated
1416 sage: from mjo.eja.eja_algebra import random_eja
1420 sage: set_random_seed()
1421 sage: x = random_eja().random_element()
1422 sage: x.subalgebra_generated_by().is_associative()
1425 Squaring in the subalgebra should work the same as in
1428 sage: set_random_seed()
1429 sage: x = random_eja().random_element()
1430 sage: u = x.subalgebra_generated_by().random_element()
1431 sage: u.operator()(u) == u^2
1435 # First get the subspace spanned by the powers of myself...
1436 V
= self
.span_of_powers()
1437 F
= self
.base_ring()
1439 # Now figure out the entries of the right-multiplication
1440 # matrix for the successive basis elements b0, b1,... of
1443 for b_right
in V
.basis():
1444 eja_b_right
= self
.parent()(b_right
)
1446 # The first row of the right-multiplication matrix by
1447 # b1 is what we get if we apply that matrix to b1. The
1448 # second row of the right multiplication matrix by b1
1449 # is what we get when we apply that matrix to b2...
1451 # IMPORTANT: this assumes that all vectors are COLUMN
1452 # vectors, unlike our superclass (which uses row vectors).
1453 for b_left
in V
.basis():
1454 eja_b_left
= self
.parent()(b_left
)
1455 # Multiply in the original EJA, but then get the
1456 # coordinates from the subalgebra in terms of its
1458 this_row
= V
.coordinates((eja_b_left
*eja_b_right
).vector())
1459 b_right_rows
.append(this_row
)
1460 b_right_matrix
= matrix(F
, b_right_rows
)
1461 mats
.append(b_right_matrix
)
1463 # It's an algebra of polynomials in one element, and EJAs
1464 # are power-associative.
1466 # TODO: choose generator names intelligently.
1468 # The rank is the highest possible degree of a minimal polynomial,
1469 # and is bounded above by the dimension. We know in this case that
1470 # there's an element whose minimal polynomial has the same degree
1471 # as the space's dimension, so that must be its rank too.
1472 return FiniteDimensionalEuclideanJordanAlgebra(
1476 assume_associative
=True,
1480 def subalgebra_idempotent(self
):
1482 Find an idempotent in the associative subalgebra I generate
1483 using Proposition 2.3.5 in Baes.
1487 sage: from mjo.eja.eja_algebra import random_eja
1491 sage: set_random_seed()
1492 sage: J = random_eja()
1493 sage: x = J.random_element()
1494 sage: while x.is_nilpotent():
1495 ....: x = J.random_element()
1496 sage: c = x.subalgebra_idempotent()
1501 if self
.is_nilpotent():
1502 raise ValueError("this only works with non-nilpotent elements!")
1504 V
= self
.span_of_powers()
1505 J
= self
.subalgebra_generated_by()
1506 # Mis-design warning: the basis used for span_of_powers()
1507 # and subalgebra_generated_by() must be the same, and in
1509 u
= J(V
.coordinates(self
.vector()))
1511 # The image of the matrix of left-u^m-multiplication
1512 # will be minimal for some natural number s...
1514 minimal_dim
= V
.dimension()
1515 for i
in xrange(1, V
.dimension()):
1516 this_dim
= (u
**i
).operator().matrix().image().dimension()
1517 if this_dim
< minimal_dim
:
1518 minimal_dim
= this_dim
1521 # Now minimal_matrix should correspond to the smallest
1522 # non-zero subspace in Baes's (or really, Koecher's)
1525 # However, we need to restrict the matrix to work on the
1526 # subspace... or do we? Can't we just solve, knowing that
1527 # A(c) = u^(s+1) should have a solution in the big space,
1530 # Beware, solve_right() means that we're using COLUMN vectors.
1531 # Our FiniteDimensionalAlgebraElement superclass uses rows.
1533 A
= u_next
.operator().matrix()
1534 c_coordinates
= A
.solve_right(u_next
.vector())
1536 # Now c_coordinates is the idempotent we want, but it's in
1537 # the coordinate system of the subalgebra.
1539 # We need the basis for J, but as elements of the parent algebra.
1541 basis
= [self
.parent(v
) for v
in V
.basis()]
1542 return self
.parent().linear_combination(zip(c_coordinates
, basis
))
1547 Return my trace, the sum of my eigenvalues.
1551 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
1552 ....: RealCartesianProductEJA,
1557 sage: J = JordanSpinEJA(3)
1558 sage: x = sum(J.gens())
1564 sage: J = RealCartesianProductEJA(5)
1565 sage: J.one().trace()
1570 The trace of an element is a real number::
1572 sage: set_random_seed()
1573 sage: J = random_eja()
1574 sage: J.random_element().trace() in J.base_ring()
1580 p
= P
._charpoly
_coeff
(r
-1)
1581 # The _charpoly_coeff function already adds the factor of
1582 # -1 to ensure that _charpoly_coeff(r-1) is really what
1583 # appears in front of t^{r-1} in the charpoly. However,
1584 # we want the negative of THAT for the trace.
1585 return -p(*self
.vector())
1588 def trace_inner_product(self
, other
):
1590 Return the trace inner product of myself and ``other``.
1594 sage: from mjo.eja.eja_algebra import random_eja
1598 The trace inner product is commutative::
1600 sage: set_random_seed()
1601 sage: J = random_eja()
1602 sage: x = J.random_element(); y = J.random_element()
1603 sage: x.trace_inner_product(y) == y.trace_inner_product(x)
1606 The trace inner product is bilinear::
1608 sage: set_random_seed()
1609 sage: J = random_eja()
1610 sage: x = J.random_element()
1611 sage: y = J.random_element()
1612 sage: z = J.random_element()
1613 sage: a = QQ.random_element();
1614 sage: actual = (a*(x+z)).trace_inner_product(y)
1615 sage: expected = ( a*x.trace_inner_product(y) +
1616 ....: a*z.trace_inner_product(y) )
1617 sage: actual == expected
1619 sage: actual = x.trace_inner_product(a*(y+z))
1620 sage: expected = ( a*x.trace_inner_product(y) +
1621 ....: a*x.trace_inner_product(z) )
1622 sage: actual == expected
1625 The trace inner product satisfies the compatibility
1626 condition in the definition of a Euclidean Jordan algebra::
1628 sage: set_random_seed()
1629 sage: J = random_eja()
1630 sage: x = J.random_element()
1631 sage: y = J.random_element()
1632 sage: z = J.random_element()
1633 sage: (x*y).trace_inner_product(z) == y.trace_inner_product(x*z)
1637 if not other
in self
.parent():
1638 raise TypeError("'other' must live in the same algebra")
1640 return (self
*other
).trace()
1643 class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1645 Return the Euclidean Jordan Algebra corresponding to the set
1646 `R^n` under the Hadamard product.
1648 Note: this is nothing more than the Cartesian product of ``n``
1649 copies of the spin algebra. Once Cartesian product algebras
1650 are implemented, this can go.
1654 sage: from mjo.eja.eja_algebra import RealCartesianProductEJA
1658 This multiplication table can be verified by hand::
1660 sage: J = RealCartesianProductEJA(3)
1661 sage: e0,e1,e2 = J.gens()
1677 def __classcall_private__(cls
, n
, field
=QQ
):
1678 # The FiniteDimensionalAlgebra constructor takes a list of
1679 # matrices, the ith representing right multiplication by the ith
1680 # basis element in the vector space. So if e_1 = (1,0,0), then
1681 # right (Hadamard) multiplication of x by e_1 picks out the first
1682 # component of x; and likewise for the ith basis element e_i.
1683 Qs
= [ matrix(field
, n
, n
, lambda k
,j
: 1*(k
== j
== i
))
1684 for i
in xrange(n
) ]
1686 fdeja
= super(RealCartesianProductEJA
, cls
)
1687 return fdeja
.__classcall
_private
__(cls
, field
, Qs
, rank
=n
)
1689 def inner_product(self
, x
, y
):
1690 return _usual_ip(x
,y
)
1695 Return a "random" finite-dimensional Euclidean Jordan Algebra.
1699 For now, we choose a random natural number ``n`` (greater than zero)
1700 and then give you back one of the following:
1702 * The cartesian product of the rational numbers ``n`` times; this is
1703 ``QQ^n`` with the Hadamard product.
1705 * The Jordan spin algebra on ``QQ^n``.
1707 * The ``n``-by-``n`` rational symmetric matrices with the symmetric
1710 * The ``n``-by-``n`` complex-rational Hermitian matrices embedded
1711 in the space of ``2n``-by-``2n`` real symmetric matrices.
1713 * The ``n``-by-``n`` quaternion-rational Hermitian matrices embedded
1714 in the space of ``4n``-by-``4n`` real symmetric matrices.
1716 Later this might be extended to return Cartesian products of the
1721 sage: from mjo.eja.eja_algebra import random_eja
1726 Euclidean Jordan algebra of degree...
1730 # The max_n component lets us choose different upper bounds on the
1731 # value "n" that gets passed to the constructor. This is needed
1732 # because e.g. R^{10} is reasonable to test, while the Hermitian
1733 # 10-by-10 quaternion matrices are not.
1734 (constructor
, max_n
) = choice([(RealCartesianProductEJA
, 6),
1736 (RealSymmetricEJA
, 5),
1737 (ComplexHermitianEJA
, 4),
1738 (QuaternionHermitianEJA
, 3)])
1739 n
= ZZ
.random_element(1, max_n
)
1740 return constructor(n
, field
=QQ
)
1744 def _real_symmetric_basis(n
, field
=QQ
):
1746 Return a basis for the space of real symmetric n-by-n matrices.
1748 # The basis of symmetric matrices, as matrices, in their R^(n-by-n)
1752 for j
in xrange(i
+1):
1753 Eij
= matrix(field
, n
, lambda k
,l
: k
==i
and l
==j
)
1757 # Beware, orthogonal but not normalized!
1758 Sij
= Eij
+ Eij
.transpose()
1763 def _complex_hermitian_basis(n
, field
=QQ
):
1765 Returns a basis for the space of complex Hermitian n-by-n matrices.
1769 sage: from mjo.eja.eja_algebra import _complex_hermitian_basis
1773 sage: set_random_seed()
1774 sage: n = ZZ.random_element(1,5)
1775 sage: all( M.is_symmetric() for M in _complex_hermitian_basis(n) )
1779 F
= QuadraticField(-1, 'I')
1782 # This is like the symmetric case, but we need to be careful:
1784 # * We want conjugate-symmetry, not just symmetry.
1785 # * The diagonal will (as a result) be real.
1789 for j
in xrange(i
+1):
1790 Eij
= matrix(field
, n
, lambda k
,l
: k
==i
and l
==j
)
1792 Sij
= _embed_complex_matrix(Eij
)
1795 # Beware, orthogonal but not normalized! The second one
1796 # has a minus because it's conjugated.
1797 Sij_real
= _embed_complex_matrix(Eij
+ Eij
.transpose())
1799 Sij_imag
= _embed_complex_matrix(I
*Eij
- I
*Eij
.transpose())
1804 def _quaternion_hermitian_basis(n
, field
=QQ
):
1806 Returns a basis for the space of quaternion Hermitian n-by-n matrices.
1810 sage: from mjo.eja.eja_algebra import _quaternion_hermitian_basis
1814 sage: set_random_seed()
1815 sage: n = ZZ.random_element(1,5)
1816 sage: all( M.is_symmetric() for M in _quaternion_hermitian_basis(n) )
1820 Q
= QuaternionAlgebra(QQ
,-1,-1)
1823 # This is like the symmetric case, but we need to be careful:
1825 # * We want conjugate-symmetry, not just symmetry.
1826 # * The diagonal will (as a result) be real.
1830 for j
in xrange(i
+1):
1831 Eij
= matrix(Q
, n
, lambda k
,l
: k
==i
and l
==j
)
1833 Sij
= _embed_quaternion_matrix(Eij
)
1836 # Beware, orthogonal but not normalized! The second,
1837 # third, and fourth ones have a minus because they're
1839 Sij_real
= _embed_quaternion_matrix(Eij
+ Eij
.transpose())
1841 Sij_I
= _embed_quaternion_matrix(I
*Eij
- I
*Eij
.transpose())
1843 Sij_J
= _embed_quaternion_matrix(J
*Eij
- J
*Eij
.transpose())
1845 Sij_K
= _embed_quaternion_matrix(K
*Eij
- K
*Eij
.transpose())
1851 return vector(m
.base_ring(), m
.list())
1854 return matrix(v
.base_ring(), sqrt(v
.degree()), v
.list())
1856 def _multiplication_table_from_matrix_basis(basis
):
1858 At least three of the five simple Euclidean Jordan algebras have the
1859 symmetric multiplication (A,B) |-> (AB + BA)/2, where the
1860 multiplication on the right is matrix multiplication. Given a basis
1861 for the underlying matrix space, this function returns a
1862 multiplication table (obtained by looping through the basis
1863 elements) for an algebra of those matrices. A reordered copy
1864 of the basis is also returned to work around the fact that
1865 the ``span()`` in this function will change the order of the basis
1866 from what we think it is, to... something else.
1868 # In S^2, for example, we nominally have four coordinates even
1869 # though the space is of dimension three only. The vector space V
1870 # is supposed to hold the entire long vector, and the subspace W
1871 # of V will be spanned by the vectors that arise from symmetric
1872 # matrices. Thus for S^2, dim(V) == 4 and dim(W) == 3.
1873 field
= basis
[0].base_ring()
1874 dimension
= basis
[0].nrows()
1876 V
= VectorSpace(field
, dimension
**2)
1877 W
= V
.span( _mat2vec(s
) for s
in basis
)
1879 # Taking the span above reorders our basis (thanks, jerk!) so we
1880 # need to put our "matrix basis" in the same order as the
1881 # (reordered) vector basis.
1882 S
= tuple( _vec2mat(b
) for b
in W
.basis() )
1886 # Brute force the multiplication-by-s matrix by looping
1887 # through all elements of the basis and doing the computation
1888 # to find out what the corresponding row should be. BEWARE:
1889 # these multiplication tables won't be symmetric! It therefore
1890 # becomes REALLY IMPORTANT that the underlying algebra
1891 # constructor uses ROW vectors and not COLUMN vectors. That's
1892 # why we're computing rows here and not columns.
1895 this_row
= _mat2vec((s
*t
+ t
*s
)/2)
1896 Q_rows
.append(W
.coordinates(this_row
))
1897 Q
= matrix(field
, W
.dimension(), Q_rows
)
1903 def _embed_complex_matrix(M
):
1905 Embed the n-by-n complex matrix ``M`` into the space of real
1906 matrices of size 2n-by-2n via the map the sends each entry `z = a +
1907 bi` to the block matrix ``[[a,b],[-b,a]]``.
1911 sage: from mjo.eja.eja_algebra import _embed_complex_matrix
1915 sage: F = QuadraticField(-1,'i')
1916 sage: x1 = F(4 - 2*i)
1917 sage: x2 = F(1 + 2*i)
1920 sage: M = matrix(F,2,[[x1,x2],[x3,x4]])
1921 sage: _embed_complex_matrix(M)
1930 Embedding is a homomorphism (isomorphism, in fact)::
1932 sage: set_random_seed()
1933 sage: n = ZZ.random_element(5)
1934 sage: F = QuadraticField(-1, 'i')
1935 sage: X = random_matrix(F, n)
1936 sage: Y = random_matrix(F, n)
1937 sage: actual = _embed_complex_matrix(X) * _embed_complex_matrix(Y)
1938 sage: expected = _embed_complex_matrix(X*Y)
1939 sage: actual == expected
1945 raise ValueError("the matrix 'M' must be square")
1946 field
= M
.base_ring()
1951 blocks
.append(matrix(field
, 2, [[a
,b
],[-b
,a
]]))
1953 # We can drop the imaginaries here.
1954 return matrix
.block(field
.base_ring(), n
, blocks
)
1957 def _unembed_complex_matrix(M
):
1959 The inverse of _embed_complex_matrix().
1963 sage: from mjo.eja.eja_algebra import (_embed_complex_matrix,
1964 ....: _unembed_complex_matrix)
1968 sage: A = matrix(QQ,[ [ 1, 2, 3, 4],
1969 ....: [-2, 1, -4, 3],
1970 ....: [ 9, 10, 11, 12],
1971 ....: [-10, 9, -12, 11] ])
1972 sage: _unembed_complex_matrix(A)
1974 [ 10*i + 9 12*i + 11]
1978 Unembedding is the inverse of embedding::
1980 sage: set_random_seed()
1981 sage: F = QuadraticField(-1, 'i')
1982 sage: M = random_matrix(F, 3)
1983 sage: _unembed_complex_matrix(_embed_complex_matrix(M)) == M
1989 raise ValueError("the matrix 'M' must be square")
1990 if not n
.mod(2).is_zero():
1991 raise ValueError("the matrix 'M' must be a complex embedding")
1993 F
= QuadraticField(-1, 'i')
1996 # Go top-left to bottom-right (reading order), converting every
1997 # 2-by-2 block we see to a single complex element.
1999 for k
in xrange(n
/2):
2000 for j
in xrange(n
/2):
2001 submat
= M
[2*k
:2*k
+2,2*j
:2*j
+2]
2002 if submat
[0,0] != submat
[1,1]:
2003 raise ValueError('bad on-diagonal submatrix')
2004 if submat
[0,1] != -submat
[1,0]:
2005 raise ValueError('bad off-diagonal submatrix')
2006 z
= submat
[0,0] + submat
[0,1]*i
2009 return matrix(F
, n
/2, elements
)
2012 def _embed_quaternion_matrix(M
):
2014 Embed the n-by-n quaternion matrix ``M`` into the space of real
2015 matrices of size 4n-by-4n by first sending each quaternion entry
2016 `z = a + bi + cj + dk` to the block-complex matrix
2017 ``[[a + bi, c+di],[-c + di, a-bi]]`, and then embedding those into
2022 sage: from mjo.eja.eja_algebra import _embed_quaternion_matrix
2026 sage: Q = QuaternionAlgebra(QQ,-1,-1)
2027 sage: i,j,k = Q.gens()
2028 sage: x = 1 + 2*i + 3*j + 4*k
2029 sage: M = matrix(Q, 1, [[x]])
2030 sage: _embed_quaternion_matrix(M)
2036 Embedding is a homomorphism (isomorphism, in fact)::
2038 sage: set_random_seed()
2039 sage: n = ZZ.random_element(5)
2040 sage: Q = QuaternionAlgebra(QQ,-1,-1)
2041 sage: X = random_matrix(Q, n)
2042 sage: Y = random_matrix(Q, n)
2043 sage: actual = _embed_quaternion_matrix(X)*_embed_quaternion_matrix(Y)
2044 sage: expected = _embed_quaternion_matrix(X*Y)
2045 sage: actual == expected
2049 quaternions
= M
.base_ring()
2052 raise ValueError("the matrix 'M' must be square")
2054 F
= QuadraticField(-1, 'i')
2059 t
= z
.coefficient_tuple()
2064 cplx_matrix
= matrix(F
, 2, [[ a
+ b
*i
, c
+ d
*i
],
2065 [-c
+ d
*i
, a
- b
*i
]])
2066 blocks
.append(_embed_complex_matrix(cplx_matrix
))
2068 # We should have real entries by now, so use the realest field
2069 # we've got for the return value.
2070 return matrix
.block(quaternions
.base_ring(), n
, blocks
)
2073 def _unembed_quaternion_matrix(M
):
2075 The inverse of _embed_quaternion_matrix().
2079 sage: from mjo.eja.eja_algebra import (_embed_quaternion_matrix,
2080 ....: _unembed_quaternion_matrix)
2084 sage: M = matrix(QQ, [[ 1, 2, 3, 4],
2085 ....: [-2, 1, -4, 3],
2086 ....: [-3, 4, 1, -2],
2087 ....: [-4, -3, 2, 1]])
2088 sage: _unembed_quaternion_matrix(M)
2089 [1 + 2*i + 3*j + 4*k]
2093 Unembedding is the inverse of embedding::
2095 sage: set_random_seed()
2096 sage: Q = QuaternionAlgebra(QQ, -1, -1)
2097 sage: M = random_matrix(Q, 3)
2098 sage: _unembed_quaternion_matrix(_embed_quaternion_matrix(M)) == M
2104 raise ValueError("the matrix 'M' must be square")
2105 if not n
.mod(4).is_zero():
2106 raise ValueError("the matrix 'M' must be a complex embedding")
2108 Q
= QuaternionAlgebra(QQ
,-1,-1)
2111 # Go top-left to bottom-right (reading order), converting every
2112 # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1
2115 for l
in xrange(n
/4):
2116 for m
in xrange(n
/4):
2117 submat
= _unembed_complex_matrix(M
[4*l
:4*l
+4,4*m
:4*m
+4])
2118 if submat
[0,0] != submat
[1,1].conjugate():
2119 raise ValueError('bad on-diagonal submatrix')
2120 if submat
[0,1] != -submat
[1,0].conjugate():
2121 raise ValueError('bad off-diagonal submatrix')
2122 z
= submat
[0,0].real() + submat
[0,0].imag()*i
2123 z
+= submat
[0,1].real()*j
+ submat
[0,1].imag()*k
2126 return matrix(Q
, n
/4, elements
)
2129 # The usual inner product on R^n.
2131 return x
.vector().inner_product(y
.vector())
2133 # The inner product used for the real symmetric simple EJA.
2134 # We keep it as a separate function because e.g. the complex
2135 # algebra uses the same inner product, except divided by 2.
2136 def _matrix_ip(X
,Y
):
2137 X_mat
= X
.natural_representation()
2138 Y_mat
= Y
.natural_representation()
2139 return (X_mat
*Y_mat
).trace()
2142 class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra
):
2144 The rank-n simple EJA consisting of real symmetric n-by-n
2145 matrices, the usual symmetric Jordan product, and the trace inner
2146 product. It has dimension `(n^2 + n)/2` over the reals.
2150 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
2154 sage: J = RealSymmetricEJA(2)
2155 sage: e0, e1, e2 = J.gens()
2165 The degree of this algebra is `(n^2 + n) / 2`::
2167 sage: set_random_seed()
2168 sage: n = ZZ.random_element(1,5)
2169 sage: J = RealSymmetricEJA(n)
2170 sage: J.degree() == (n^2 + n)/2
2173 The Jordan multiplication is what we think it is::
2175 sage: set_random_seed()
2176 sage: n = ZZ.random_element(1,5)
2177 sage: J = RealSymmetricEJA(n)
2178 sage: x = J.random_element()
2179 sage: y = J.random_element()
2180 sage: actual = (x*y).natural_representation()
2181 sage: X = x.natural_representation()
2182 sage: Y = y.natural_representation()
2183 sage: expected = (X*Y + Y*X)/2
2184 sage: actual == expected
2186 sage: J(expected) == x*y
2191 def __classcall_private__(cls
, n
, field
=QQ
):
2192 S
= _real_symmetric_basis(n
, field
=field
)
2193 (Qs
, T
) = _multiplication_table_from_matrix_basis(S
)
2195 fdeja
= super(RealSymmetricEJA
, cls
)
2196 return fdeja
.__classcall
_private
__(cls
,
2202 def inner_product(self
, x
, y
):
2203 return _matrix_ip(x
,y
)
2206 class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra
):
2208 The rank-n simple EJA consisting of complex Hermitian n-by-n
2209 matrices over the real numbers, the usual symmetric Jordan product,
2210 and the real-part-of-trace inner product. It has dimension `n^2` over
2215 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
2219 The degree of this algebra is `n^2`::
2221 sage: set_random_seed()
2222 sage: n = ZZ.random_element(1,5)
2223 sage: J = ComplexHermitianEJA(n)
2224 sage: J.degree() == n^2
2227 The Jordan multiplication is what we think it is::
2229 sage: set_random_seed()
2230 sage: n = ZZ.random_element(1,5)
2231 sage: J = ComplexHermitianEJA(n)
2232 sage: x = J.random_element()
2233 sage: y = J.random_element()
2234 sage: actual = (x*y).natural_representation()
2235 sage: X = x.natural_representation()
2236 sage: Y = y.natural_representation()
2237 sage: expected = (X*Y + Y*X)/2
2238 sage: actual == expected
2240 sage: J(expected) == x*y
2245 def __classcall_private__(cls
, n
, field
=QQ
):
2246 S
= _complex_hermitian_basis(n
)
2247 (Qs
, T
) = _multiplication_table_from_matrix_basis(S
)
2249 fdeja
= super(ComplexHermitianEJA
, cls
)
2250 return fdeja
.__classcall
_private
__(cls
,
2256 def inner_product(self
, x
, y
):
2257 # Since a+bi on the diagonal is represented as
2262 # we'll double-count the "a" entries if we take the trace of
2264 return _matrix_ip(x
,y
)/2
2267 class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra
):
2269 The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
2270 matrices, the usual symmetric Jordan product, and the
2271 real-part-of-trace inner product. It has dimension `2n^2 - n` over
2276 sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
2280 The degree of this algebra is `n^2`::
2282 sage: set_random_seed()
2283 sage: n = ZZ.random_element(1,5)
2284 sage: J = QuaternionHermitianEJA(n)
2285 sage: J.degree() == 2*(n^2) - n
2288 The Jordan multiplication is what we think it is::
2290 sage: set_random_seed()
2291 sage: n = ZZ.random_element(1,5)
2292 sage: J = QuaternionHermitianEJA(n)
2293 sage: x = J.random_element()
2294 sage: y = J.random_element()
2295 sage: actual = (x*y).natural_representation()
2296 sage: X = x.natural_representation()
2297 sage: Y = y.natural_representation()
2298 sage: expected = (X*Y + Y*X)/2
2299 sage: actual == expected
2301 sage: J(expected) == x*y
2306 def __classcall_private__(cls
, n
, field
=QQ
):
2307 S
= _quaternion_hermitian_basis(n
)
2308 (Qs
, T
) = _multiplication_table_from_matrix_basis(S
)
2310 fdeja
= super(QuaternionHermitianEJA
, cls
)
2311 return fdeja
.__classcall
_private
__(cls
,
2317 def inner_product(self
, x
, y
):
2318 # Since a+bi+cj+dk on the diagonal is represented as
2320 # a + bi +cj + dk = [ a b c d]
2325 # we'll quadruple-count the "a" entries if we take the trace of
2327 return _matrix_ip(x
,y
)/4
2330 class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra
):
2332 The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
2333 with the usual inner product and jordan product ``x*y =
2334 (<x_bar,y_bar>, x0*y_bar + y0*x_bar)``. It has dimension `n` over
2339 sage: from mjo.eja.eja_algebra import JordanSpinEJA
2343 This multiplication table can be verified by hand::
2345 sage: J = JordanSpinEJA(4)
2346 sage: e0,e1,e2,e3 = J.gens()
2364 def __classcall_private__(cls
, n
, field
=QQ
):
2366 id_matrix
= matrix
.identity(field
, n
)
2368 ei
= id_matrix
.column(i
)
2369 Qi
= matrix
.zero(field
, n
)
2371 Qi
.set_column(0, ei
)
2372 Qi
+= matrix
.diagonal(n
, [ei
[0]]*n
)
2373 # The addition of the diagonal matrix adds an extra ei[0] in the
2374 # upper-left corner of the matrix.
2375 Qi
[0,0] = Qi
[0,0] * ~
field(2)
2378 # The rank of the spin algebra is two, unless we're in a
2379 # one-dimensional ambient space (because the rank is bounded by
2380 # the ambient dimension).
2381 fdeja
= super(JordanSpinEJA
, cls
)
2382 return fdeja
.__classcall
_private
__(cls
, field
, Qs
, rank
=min(n
,2))
2384 def inner_product(self
, x
, y
):
2385 return _usual_ip(x
,y
)