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,
489 The identity in `S^n` is converted to the identity in the EJA::
491 sage: J = RealSymmetricEJA(3)
492 sage: I = matrix.identity(QQ,3)
493 sage: J(I) == J.one()
496 This skew-symmetric matrix can't be represented in the EJA::
498 sage: J = RealSymmetricEJA(3)
499 sage: A = matrix(QQ,3, lambda i,j: i-j)
501 Traceback (most recent call last):
503 ArithmeticError: vector is not in free module
507 Ensure that we can convert any element of the parent's
508 underlying vector space back into an algebra element whose
509 vector representation is what we started with::
511 sage: set_random_seed()
512 sage: J = random_eja()
513 sage: v = J.vector_space().random_element()
514 sage: J(v).vector() == v
518 # Goal: if we're given a matrix, and if it lives in our
519 # parent algebra's "natural ambient space," convert it
520 # into an algebra element.
522 # The catch is, we make a recursive call after converting
523 # the given matrix into a vector that lives in the algebra.
524 # This we need to try the parent class initializer first,
525 # to avoid recursing forever if we're given something that
526 # already fits into the algebra, but also happens to live
527 # in the parent's "natural ambient space" (this happens with
530 FiniteDimensionalAlgebraElement
.__init
__(self
, A
, elt
)
532 natural_basis
= A
.natural_basis()
533 if elt
in natural_basis
[0].matrix_space():
534 # Thanks for nothing! Matrix spaces aren't vector
535 # spaces in Sage, so we have to figure out its
536 # natural-basis coordinates ourselves.
537 V
= VectorSpace(elt
.base_ring(), elt
.nrows()**2)
538 W
= V
.span( _mat2vec(s
) for s
in natural_basis
)
539 coords
= W
.coordinates(_mat2vec(elt
))
540 FiniteDimensionalAlgebraElement
.__init
__(self
, A
, coords
)
542 def __pow__(self
, n
):
544 Return ``self`` raised to the power ``n``.
546 Jordan algebras are always power-associative; see for
547 example Faraut and Koranyi, Proposition II.1.2 (ii).
549 We have to override this because our superclass uses row
550 vectors instead of column vectors! We, on the other hand,
551 assume column vectors everywhere.
555 sage: from mjo.eja.eja_algebra import random_eja
559 The definition of `x^2` is the unambiguous `x*x`::
561 sage: set_random_seed()
562 sage: x = random_eja().random_element()
566 A few examples of power-associativity::
568 sage: set_random_seed()
569 sage: x = random_eja().random_element()
570 sage: x*(x*x)*(x*x) == x^5
572 sage: (x*x)*(x*x*x) == x^5
575 We also know that powers operator-commute (Koecher, Chapter
578 sage: set_random_seed()
579 sage: x = random_eja().random_element()
580 sage: m = ZZ.random_element(0,10)
581 sage: n = ZZ.random_element(0,10)
582 sage: Lxm = (x^m).operator()
583 sage: Lxn = (x^n).operator()
584 sage: Lxm*Lxn == Lxn*Lxm
589 return self
.parent().one()
593 return (self
.operator()**(n
-1))(self
)
596 def apply_univariate_polynomial(self
, p
):
598 Apply the univariate polynomial ``p`` to this element.
600 A priori, SageMath won't allow us to apply a univariate
601 polynomial to an element of an EJA, because we don't know
602 that EJAs are rings (they are usually not associative). Of
603 course, we know that EJAs are power-associative, so the
604 operation is ultimately kosher. This function sidesteps
605 the CAS to get the answer we want and expect.
609 sage: from mjo.eja.eja_algebra import (RealCartesianProductEJA,
614 sage: R = PolynomialRing(QQ, 't')
616 sage: p = t^4 - t^3 + 5*t - 2
617 sage: J = RealCartesianProductEJA(5)
618 sage: J.one().apply_univariate_polynomial(p) == 3*J.one()
623 We should always get back an element of the algebra::
625 sage: set_random_seed()
626 sage: p = PolynomialRing(QQ, 't').random_element()
627 sage: J = random_eja()
628 sage: x = J.random_element()
629 sage: x.apply_univariate_polynomial(p) in J
633 if len(p
.variables()) > 1:
634 raise ValueError("not a univariate polynomial")
637 # Convert the coeficcients to the parent's base ring,
638 # because a priori they might live in an (unnecessarily)
639 # larger ring for which P.sum() would fail below.
640 cs
= [ R(c
) for c
in p
.coefficients(sparse
=False) ]
641 return P
.sum( cs
[k
]*(self
**k
) for k
in range(len(cs
)) )
644 def characteristic_polynomial(self
):
646 Return the characteristic polynomial of this element.
650 sage: from mjo.eja.eja_algebra import RealCartesianProductEJA
654 The rank of `R^3` is three, and the minimal polynomial of
655 the identity element is `(t-1)` from which it follows that
656 the characteristic polynomial should be `(t-1)^3`::
658 sage: J = RealCartesianProductEJA(3)
659 sage: J.one().characteristic_polynomial()
660 t^3 - 3*t^2 + 3*t - 1
662 Likewise, the characteristic of the zero element in the
663 rank-three algebra `R^{n}` should be `t^{3}`::
665 sage: J = RealCartesianProductEJA(3)
666 sage: J.zero().characteristic_polynomial()
671 The characteristic polynomial of an element should evaluate
672 to zero on that element::
674 sage: set_random_seed()
675 sage: x = RealCartesianProductEJA(3).random_element()
676 sage: p = x.characteristic_polynomial()
677 sage: x.apply_univariate_polynomial(p)
681 p
= self
.parent().characteristic_polynomial()
682 return p(*self
.vector())
685 def inner_product(self
, other
):
687 Return the parent algebra's inner product of myself and ``other``.
691 sage: from mjo.eja.eja_algebra import (
692 ....: ComplexHermitianEJA,
694 ....: QuaternionHermitianEJA,
695 ....: RealSymmetricEJA,
700 The inner product in the Jordan spin algebra is the usual
701 inner product on `R^n` (this example only works because the
702 basis for the Jordan algebra is the standard basis in `R^n`)::
704 sage: J = JordanSpinEJA(3)
705 sage: x = vector(QQ,[1,2,3])
706 sage: y = vector(QQ,[4,5,6])
707 sage: x.inner_product(y)
709 sage: J(x).inner_product(J(y))
712 The inner product on `S^n` is `<X,Y> = trace(X*Y)`, where
713 multiplication is the usual matrix multiplication in `S^n`,
714 so the inner product of the identity matrix with itself
717 sage: J = RealSymmetricEJA(3)
718 sage: J.one().inner_product(J.one())
721 Likewise, the inner product on `C^n` is `<X,Y> =
722 Re(trace(X*Y))`, where we must necessarily take the real
723 part because the product of Hermitian matrices may not be
726 sage: J = ComplexHermitianEJA(3)
727 sage: J.one().inner_product(J.one())
730 Ditto for the quaternions::
732 sage: J = QuaternionHermitianEJA(3)
733 sage: J.one().inner_product(J.one())
738 Ensure that we can always compute an inner product, and that
739 it gives us back a real number::
741 sage: set_random_seed()
742 sage: J = random_eja()
743 sage: x = J.random_element()
744 sage: y = J.random_element()
745 sage: x.inner_product(y) in RR
751 raise TypeError("'other' must live in the same algebra")
753 return P
.inner_product(self
, other
)
756 def operator_commutes_with(self
, other
):
758 Return whether or not this element operator-commutes
763 sage: from mjo.eja.eja_algebra import random_eja
767 The definition of a Jordan algebra says that any element
768 operator-commutes with its square::
770 sage: set_random_seed()
771 sage: x = random_eja().random_element()
772 sage: x.operator_commutes_with(x^2)
777 Test Lemma 1 from Chapter III of Koecher::
779 sage: set_random_seed()
780 sage: J = random_eja()
781 sage: u = J.random_element()
782 sage: v = J.random_element()
783 sage: lhs = u.operator_commutes_with(u*v)
784 sage: rhs = v.operator_commutes_with(u^2)
788 Test the first polarization identity from my notes, Koecher
789 Chapter III, or from Baes (2.3)::
791 sage: set_random_seed()
792 sage: J = random_eja()
793 sage: x = J.random_element()
794 sage: y = J.random_element()
795 sage: Lx = x.operator()
796 sage: Ly = y.operator()
797 sage: Lxx = (x*x).operator()
798 sage: Lxy = (x*y).operator()
799 sage: bool(2*Lx*Lxy + Ly*Lxx == 2*Lxy*Lx + Lxx*Ly)
802 Test the second polarization identity from my notes or from
805 sage: set_random_seed()
806 sage: J = random_eja()
807 sage: x = J.random_element()
808 sage: y = J.random_element()
809 sage: z = J.random_element()
810 sage: Lx = x.operator()
811 sage: Ly = y.operator()
812 sage: Lz = z.operator()
813 sage: Lzy = (z*y).operator()
814 sage: Lxy = (x*y).operator()
815 sage: Lxz = (x*z).operator()
816 sage: bool(Lx*Lzy + Lz*Lxy + Ly*Lxz == Lzy*Lx + Lxy*Lz + Lxz*Ly)
819 Test the third polarization identity from my notes or from
822 sage: set_random_seed()
823 sage: J = random_eja()
824 sage: u = J.random_element()
825 sage: y = J.random_element()
826 sage: z = J.random_element()
827 sage: Lu = u.operator()
828 sage: Ly = y.operator()
829 sage: Lz = z.operator()
830 sage: Lzy = (z*y).operator()
831 sage: Luy = (u*y).operator()
832 sage: Luz = (u*z).operator()
833 sage: Luyz = (u*(y*z)).operator()
834 sage: lhs = Lu*Lzy + Lz*Luy + Ly*Luz
835 sage: rhs = Luyz + Ly*Lu*Lz + Lz*Lu*Ly
836 sage: bool(lhs == rhs)
840 if not other
in self
.parent():
841 raise TypeError("'other' must live in the same algebra")
850 Return my determinant, the product of my eigenvalues.
854 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
859 sage: J = JordanSpinEJA(2)
860 sage: e0,e1 = J.gens()
861 sage: x = sum( J.gens() )
867 sage: J = JordanSpinEJA(3)
868 sage: e0,e1,e2 = J.gens()
869 sage: x = sum( J.gens() )
875 An element is invertible if and only if its determinant is
878 sage: set_random_seed()
879 sage: x = random_eja().random_element()
880 sage: x.is_invertible() == (x.det() != 0)
886 p
= P
._charpoly
_coeff
(0)
887 # The _charpoly_coeff function already adds the factor of
888 # -1 to ensure that _charpoly_coeff(0) is really what
889 # appears in front of t^{0} in the charpoly. However,
890 # we want (-1)^r times THAT for the determinant.
891 return ((-1)**r
)*p(*self
.vector())
896 Return the Jordan-multiplicative inverse of this element.
900 We appeal to the quadratic representation as in Koecher's
901 Theorem 12 in Chapter III, Section 5.
905 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
910 The inverse in the spin factor algebra is given in Alizadeh's
913 sage: set_random_seed()
914 sage: n = ZZ.random_element(1,10)
915 sage: J = JordanSpinEJA(n)
916 sage: x = J.random_element()
917 sage: while not x.is_invertible():
918 ....: x = J.random_element()
919 sage: x_vec = x.vector()
921 sage: x_bar = x_vec[1:]
922 sage: coeff = ~(x0^2 - x_bar.inner_product(x_bar))
923 sage: inv_vec = x_vec.parent()([x0] + (-x_bar).list())
924 sage: x_inverse = coeff*inv_vec
925 sage: x.inverse() == J(x_inverse)
930 The identity element is its own inverse::
932 sage: set_random_seed()
933 sage: J = random_eja()
934 sage: J.one().inverse() == J.one()
937 If an element has an inverse, it acts like one::
939 sage: set_random_seed()
940 sage: J = random_eja()
941 sage: x = J.random_element()
942 sage: (not x.is_invertible()) or (x.inverse()*x == J.one())
945 The inverse of the inverse is what we started with::
947 sage: set_random_seed()
948 sage: J = random_eja()
949 sage: x = J.random_element()
950 sage: (not x.is_invertible()) or (x.inverse().inverse() == x)
953 The zero element is never invertible::
955 sage: set_random_seed()
956 sage: J = random_eja().zero().inverse()
957 Traceback (most recent call last):
959 ValueError: element is not invertible
962 if not self
.is_invertible():
963 raise ValueError("element is not invertible")
965 return (~self
.quadratic_representation())(self
)
968 def is_invertible(self
):
970 Return whether or not this element is invertible.
974 The usual way to do this is to check if the determinant is
975 zero, but we need the characteristic polynomial for the
976 determinant. The minimal polynomial is a lot easier to get,
977 so we use Corollary 2 in Chapter V of Koecher to check
978 whether or not the paren't algebra's zero element is a root
979 of this element's minimal polynomial.
981 Beware that we can't use the superclass method, because it
982 relies on the algebra being associative.
986 sage: from mjo.eja.eja_algebra import random_eja
990 The identity element is always invertible::
992 sage: set_random_seed()
993 sage: J = random_eja()
994 sage: J.one().is_invertible()
997 The zero element is never invertible::
999 sage: set_random_seed()
1000 sage: J = random_eja()
1001 sage: J.zero().is_invertible()
1005 zero
= self
.parent().zero()
1006 p
= self
.minimal_polynomial()
1007 return not (p(zero
) == zero
)
1010 def is_nilpotent(self
):
1012 Return whether or not some power of this element is zero.
1016 We use Theorem 5 in Chapter III of Koecher, which says that
1017 an element ``x`` is nilpotent if and only if ``x.operator()``
1018 is nilpotent. And it is a basic fact of linear algebra that
1019 an operator on an `n`-dimensional space is nilpotent if and
1020 only if, when raised to the `n`th power, it equals the zero
1021 operator (for example, see Axler Corollary 8.8).
1025 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
1030 sage: J = JordanSpinEJA(3)
1031 sage: x = sum(J.gens())
1032 sage: x.is_nilpotent()
1037 The identity element is never nilpotent::
1039 sage: set_random_seed()
1040 sage: random_eja().one().is_nilpotent()
1043 The additive identity is always nilpotent::
1045 sage: set_random_seed()
1046 sage: random_eja().zero().is_nilpotent()
1051 zero_operator
= P
.zero().operator()
1052 return self
.operator()**P
.dimension() == zero_operator
1055 def is_regular(self
):
1057 Return whether or not this is a regular element.
1061 sage: from mjo.eja.eja_algebra import JordanSpinEJA
1065 The identity element always has degree one, but any element
1066 linearly-independent from it is regular::
1068 sage: J = JordanSpinEJA(5)
1069 sage: J.one().is_regular()
1071 sage: e0, e1, e2, e3, e4 = J.gens() # e0 is the identity
1072 sage: for x in J.gens():
1073 ....: (J.one() + x).is_regular()
1081 return self
.degree() == self
.parent().rank()
1086 Compute the degree of this element the straightforward way
1087 according to the definition; by appending powers to a list
1088 and figuring out its dimension (that is, whether or not
1089 they're linearly dependent).
1093 sage: from mjo.eja.eja_algebra import JordanSpinEJA
1097 sage: J = JordanSpinEJA(4)
1098 sage: J.one().degree()
1100 sage: e0,e1,e2,e3 = J.gens()
1101 sage: (e0 - e1).degree()
1104 In the spin factor algebra (of rank two), all elements that
1105 aren't multiples of the identity are regular::
1107 sage: set_random_seed()
1108 sage: n = ZZ.random_element(1,10)
1109 sage: J = JordanSpinEJA(n)
1110 sage: x = J.random_element()
1111 sage: x == x.coefficient(0)*J.one() or x.degree() == 2
1115 return self
.span_of_powers().dimension()
1118 def left_matrix(self
):
1120 Our parent class defines ``left_matrix`` and ``matrix``
1121 methods whose names are misleading. We don't want them.
1123 raise NotImplementedError("use operator().matrix() instead")
1125 matrix
= left_matrix
1128 def minimal_polynomial(self
):
1130 Return the minimal polynomial of this element,
1131 as a function of the variable `t`.
1135 We restrict ourselves to the associative subalgebra
1136 generated by this element, and then return the minimal
1137 polynomial of this element's operator matrix (in that
1138 subalgebra). This works by Baes Proposition 2.3.16.
1142 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
1147 The minimal polynomial of the identity and zero elements are
1150 sage: set_random_seed()
1151 sage: J = random_eja()
1152 sage: J.one().minimal_polynomial()
1154 sage: J.zero().minimal_polynomial()
1157 The degree of an element is (by one definition) the degree
1158 of its minimal polynomial::
1160 sage: set_random_seed()
1161 sage: x = random_eja().random_element()
1162 sage: x.degree() == x.minimal_polynomial().degree()
1165 The minimal polynomial and the characteristic polynomial coincide
1166 and are known (see Alizadeh, Example 11.11) for all elements of
1167 the spin factor algebra that aren't scalar multiples of the
1170 sage: set_random_seed()
1171 sage: n = ZZ.random_element(2,10)
1172 sage: J = JordanSpinEJA(n)
1173 sage: y = J.random_element()
1174 sage: while y == y.coefficient(0)*J.one():
1175 ....: y = J.random_element()
1176 sage: y0 = y.vector()[0]
1177 sage: y_bar = y.vector()[1:]
1178 sage: actual = y.minimal_polynomial()
1179 sage: t = PolynomialRing(J.base_ring(),'t').gen(0)
1180 sage: expected = t^2 - 2*y0*t + (y0^2 - norm(y_bar)^2)
1181 sage: bool(actual == expected)
1184 The minimal polynomial should always kill its element::
1186 sage: set_random_seed()
1187 sage: x = random_eja().random_element()
1188 sage: p = x.minimal_polynomial()
1189 sage: x.apply_univariate_polynomial(p)
1193 V
= self
.span_of_powers()
1194 assoc_subalg
= self
.subalgebra_generated_by()
1195 # Mis-design warning: the basis used for span_of_powers()
1196 # and subalgebra_generated_by() must be the same, and in
1198 elt
= assoc_subalg(V
.coordinates(self
.vector()))
1199 return elt
.operator().minimal_polynomial()
1203 def natural_representation(self
):
1205 Return a more-natural representation of this element.
1207 Every finite-dimensional Euclidean Jordan Algebra is a
1208 direct sum of five simple algebras, four of which comprise
1209 Hermitian matrices. This method returns the original
1210 "natural" representation of this element as a Hermitian
1211 matrix, if it has one. If not, you get the usual representation.
1215 sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
1216 ....: QuaternionHermitianEJA)
1220 sage: J = ComplexHermitianEJA(3)
1223 sage: J.one().natural_representation()
1233 sage: J = QuaternionHermitianEJA(3)
1236 sage: J.one().natural_representation()
1237 [1 0 0 0 0 0 0 0 0 0 0 0]
1238 [0 1 0 0 0 0 0 0 0 0 0 0]
1239 [0 0 1 0 0 0 0 0 0 0 0 0]
1240 [0 0 0 1 0 0 0 0 0 0 0 0]
1241 [0 0 0 0 1 0 0 0 0 0 0 0]
1242 [0 0 0 0 0 1 0 0 0 0 0 0]
1243 [0 0 0 0 0 0 1 0 0 0 0 0]
1244 [0 0 0 0 0 0 0 1 0 0 0 0]
1245 [0 0 0 0 0 0 0 0 1 0 0 0]
1246 [0 0 0 0 0 0 0 0 0 1 0 0]
1247 [0 0 0 0 0 0 0 0 0 0 1 0]
1248 [0 0 0 0 0 0 0 0 0 0 0 1]
1251 B
= self
.parent().natural_basis()
1252 W
= B
[0].matrix_space()
1253 return W
.linear_combination(zip(self
.vector(), B
))
1258 Return the left-multiplication-by-this-element
1259 operator on the ambient algebra.
1263 sage: from mjo.eja.eja_algebra import random_eja
1267 sage: set_random_seed()
1268 sage: J = random_eja()
1269 sage: x = J.random_element()
1270 sage: y = J.random_element()
1271 sage: x.operator()(y) == x*y
1273 sage: y.operator()(x) == x*y
1278 fda_elt
= FiniteDimensionalAlgebraElement(P
, self
)
1279 return FiniteDimensionalEuclideanJordanAlgebraOperator(
1282 fda_elt
.matrix().transpose() )
1285 def quadratic_representation(self
, other
=None):
1287 Return the quadratic representation of this element.
1291 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
1296 The explicit form in the spin factor algebra is given by
1297 Alizadeh's Example 11.12::
1299 sage: set_random_seed()
1300 sage: n = ZZ.random_element(1,10)
1301 sage: J = JordanSpinEJA(n)
1302 sage: x = J.random_element()
1303 sage: x_vec = x.vector()
1305 sage: x_bar = x_vec[1:]
1306 sage: A = matrix(QQ, 1, [x_vec.inner_product(x_vec)])
1307 sage: B = 2*x0*x_bar.row()
1308 sage: C = 2*x0*x_bar.column()
1309 sage: D = matrix.identity(QQ, n-1)
1310 sage: D = (x0^2 - x_bar.inner_product(x_bar))*D
1311 sage: D = D + 2*x_bar.tensor_product(x_bar)
1312 sage: Q = matrix.block(2,2,[A,B,C,D])
1313 sage: Q == x.quadratic_representation().matrix()
1316 Test all of the properties from Theorem 11.2 in Alizadeh::
1318 sage: set_random_seed()
1319 sage: J = random_eja()
1320 sage: x = J.random_element()
1321 sage: y = J.random_element()
1322 sage: Lx = x.operator()
1323 sage: Lxx = (x*x).operator()
1324 sage: Qx = x.quadratic_representation()
1325 sage: Qy = y.quadratic_representation()
1326 sage: Qxy = x.quadratic_representation(y)
1327 sage: Qex = J.one().quadratic_representation(x)
1328 sage: n = ZZ.random_element(10)
1329 sage: Qxn = (x^n).quadratic_representation()
1333 sage: 2*Qxy == (x+y).quadratic_representation() - Qx - Qy
1336 Property 2 (multiply on the right for :trac:`28272`):
1338 sage: alpha = QQ.random_element()
1339 sage: (alpha*x).quadratic_representation() == Qx*(alpha^2)
1344 sage: not x.is_invertible() or ( Qx(x.inverse()) == x )
1347 sage: not x.is_invertible() or (
1350 ....: x.inverse().quadratic_representation() )
1353 sage: Qxy(J.one()) == x*y
1358 sage: not x.is_invertible() or (
1359 ....: x.quadratic_representation(x.inverse())*Qx
1360 ....: == Qx*x.quadratic_representation(x.inverse()) )
1363 sage: not x.is_invertible() or (
1364 ....: x.quadratic_representation(x.inverse())*Qx
1366 ....: 2*x.operator()*Qex - Qx )
1369 sage: 2*x.operator()*Qex - Qx == Lxx
1374 sage: Qy(x).quadratic_representation() == Qy*Qx*Qy
1384 sage: not x.is_invertible() or (
1385 ....: Qx*x.inverse().operator() == Lx )
1390 sage: not x.operator_commutes_with(y) or (
1391 ....: Qx(y)^n == Qxn(y^n) )
1397 elif not other
in self
.parent():
1398 raise TypeError("'other' must live in the same algebra")
1401 M
= other
.operator()
1402 return ( L
*M
+ M
*L
- (self
*other
).operator() )
1405 def span_of_powers(self
):
1407 Return the vector space spanned by successive powers of
1410 # The dimension of the subalgebra can't be greater than
1411 # the big algebra, so just put everything into a list
1412 # and let span() get rid of the excess.
1414 # We do the extra ambient_vector_space() in case we're messing
1415 # with polynomials and the direct parent is a module.
1416 V
= self
.parent().vector_space()
1417 return V
.span( (self
**d
).vector() for d
in xrange(V
.dimension()) )
1420 def subalgebra_generated_by(self
):
1422 Return the associative subalgebra of the parent EJA generated
1427 sage: from mjo.eja.eja_algebra import random_eja
1431 sage: set_random_seed()
1432 sage: x = random_eja().random_element()
1433 sage: x.subalgebra_generated_by().is_associative()
1436 Squaring in the subalgebra should work the same as in
1439 sage: set_random_seed()
1440 sage: x = random_eja().random_element()
1441 sage: u = x.subalgebra_generated_by().random_element()
1442 sage: u.operator()(u) == u^2
1446 # First get the subspace spanned by the powers of myself...
1447 V
= self
.span_of_powers()
1448 F
= self
.base_ring()
1450 # Now figure out the entries of the right-multiplication
1451 # matrix for the successive basis elements b0, b1,... of
1454 for b_right
in V
.basis():
1455 eja_b_right
= self
.parent()(b_right
)
1457 # The first row of the right-multiplication matrix by
1458 # b1 is what we get if we apply that matrix to b1. The
1459 # second row of the right multiplication matrix by b1
1460 # is what we get when we apply that matrix to b2...
1462 # IMPORTANT: this assumes that all vectors are COLUMN
1463 # vectors, unlike our superclass (which uses row vectors).
1464 for b_left
in V
.basis():
1465 eja_b_left
= self
.parent()(b_left
)
1466 # Multiply in the original EJA, but then get the
1467 # coordinates from the subalgebra in terms of its
1469 this_row
= V
.coordinates((eja_b_left
*eja_b_right
).vector())
1470 b_right_rows
.append(this_row
)
1471 b_right_matrix
= matrix(F
, b_right_rows
)
1472 mats
.append(b_right_matrix
)
1474 # It's an algebra of polynomials in one element, and EJAs
1475 # are power-associative.
1477 # TODO: choose generator names intelligently.
1479 # The rank is the highest possible degree of a minimal polynomial,
1480 # and is bounded above by the dimension. We know in this case that
1481 # there's an element whose minimal polynomial has the same degree
1482 # as the space's dimension, so that must be its rank too.
1483 return FiniteDimensionalEuclideanJordanAlgebra(
1487 assume_associative
=True,
1491 def subalgebra_idempotent(self
):
1493 Find an idempotent in the associative subalgebra I generate
1494 using Proposition 2.3.5 in Baes.
1498 sage: from mjo.eja.eja_algebra import random_eja
1502 sage: set_random_seed()
1503 sage: J = random_eja()
1504 sage: x = J.random_element()
1505 sage: while x.is_nilpotent():
1506 ....: x = J.random_element()
1507 sage: c = x.subalgebra_idempotent()
1512 if self
.is_nilpotent():
1513 raise ValueError("this only works with non-nilpotent elements!")
1515 V
= self
.span_of_powers()
1516 J
= self
.subalgebra_generated_by()
1517 # Mis-design warning: the basis used for span_of_powers()
1518 # and subalgebra_generated_by() must be the same, and in
1520 u
= J(V
.coordinates(self
.vector()))
1522 # The image of the matrix of left-u^m-multiplication
1523 # will be minimal for some natural number s...
1525 minimal_dim
= V
.dimension()
1526 for i
in xrange(1, V
.dimension()):
1527 this_dim
= (u
**i
).operator().matrix().image().dimension()
1528 if this_dim
< minimal_dim
:
1529 minimal_dim
= this_dim
1532 # Now minimal_matrix should correspond to the smallest
1533 # non-zero subspace in Baes's (or really, Koecher's)
1536 # However, we need to restrict the matrix to work on the
1537 # subspace... or do we? Can't we just solve, knowing that
1538 # A(c) = u^(s+1) should have a solution in the big space,
1541 # Beware, solve_right() means that we're using COLUMN vectors.
1542 # Our FiniteDimensionalAlgebraElement superclass uses rows.
1544 A
= u_next
.operator().matrix()
1545 c_coordinates
= A
.solve_right(u_next
.vector())
1547 # Now c_coordinates is the idempotent we want, but it's in
1548 # the coordinate system of the subalgebra.
1550 # We need the basis for J, but as elements of the parent algebra.
1552 basis
= [self
.parent(v
) for v
in V
.basis()]
1553 return self
.parent().linear_combination(zip(c_coordinates
, basis
))
1558 Return my trace, the sum of my eigenvalues.
1562 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
1563 ....: RealCartesianProductEJA,
1568 sage: J = JordanSpinEJA(3)
1569 sage: x = sum(J.gens())
1575 sage: J = RealCartesianProductEJA(5)
1576 sage: J.one().trace()
1581 The trace of an element is a real number::
1583 sage: set_random_seed()
1584 sage: J = random_eja()
1585 sage: J.random_element().trace() in J.base_ring()
1591 p
= P
._charpoly
_coeff
(r
-1)
1592 # The _charpoly_coeff function already adds the factor of
1593 # -1 to ensure that _charpoly_coeff(r-1) is really what
1594 # appears in front of t^{r-1} in the charpoly. However,
1595 # we want the negative of THAT for the trace.
1596 return -p(*self
.vector())
1599 def trace_inner_product(self
, other
):
1601 Return the trace inner product of myself and ``other``.
1605 sage: from mjo.eja.eja_algebra import random_eja
1609 The trace inner product is commutative::
1611 sage: set_random_seed()
1612 sage: J = random_eja()
1613 sage: x = J.random_element(); y = J.random_element()
1614 sage: x.trace_inner_product(y) == y.trace_inner_product(x)
1617 The trace inner product is bilinear::
1619 sage: set_random_seed()
1620 sage: J = random_eja()
1621 sage: x = J.random_element()
1622 sage: y = J.random_element()
1623 sage: z = J.random_element()
1624 sage: a = QQ.random_element();
1625 sage: actual = (a*(x+z)).trace_inner_product(y)
1626 sage: expected = ( a*x.trace_inner_product(y) +
1627 ....: a*z.trace_inner_product(y) )
1628 sage: actual == expected
1630 sage: actual = x.trace_inner_product(a*(y+z))
1631 sage: expected = ( a*x.trace_inner_product(y) +
1632 ....: a*x.trace_inner_product(z) )
1633 sage: actual == expected
1636 The trace inner product satisfies the compatibility
1637 condition in the definition of a Euclidean Jordan algebra::
1639 sage: set_random_seed()
1640 sage: J = random_eja()
1641 sage: x = J.random_element()
1642 sage: y = J.random_element()
1643 sage: z = J.random_element()
1644 sage: (x*y).trace_inner_product(z) == y.trace_inner_product(x*z)
1648 if not other
in self
.parent():
1649 raise TypeError("'other' must live in the same algebra")
1651 return (self
*other
).trace()
1654 class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1656 Return the Euclidean Jordan Algebra corresponding to the set
1657 `R^n` under the Hadamard product.
1659 Note: this is nothing more than the Cartesian product of ``n``
1660 copies of the spin algebra. Once Cartesian product algebras
1661 are implemented, this can go.
1665 sage: from mjo.eja.eja_algebra import RealCartesianProductEJA
1669 This multiplication table can be verified by hand::
1671 sage: J = RealCartesianProductEJA(3)
1672 sage: e0,e1,e2 = J.gens()
1688 def __classcall_private__(cls
, n
, field
=QQ
):
1689 # The FiniteDimensionalAlgebra constructor takes a list of
1690 # matrices, the ith representing right multiplication by the ith
1691 # basis element in the vector space. So if e_1 = (1,0,0), then
1692 # right (Hadamard) multiplication of x by e_1 picks out the first
1693 # component of x; and likewise for the ith basis element e_i.
1694 Qs
= [ matrix(field
, n
, n
, lambda k
,j
: 1*(k
== j
== i
))
1695 for i
in xrange(n
) ]
1697 fdeja
= super(RealCartesianProductEJA
, cls
)
1698 return fdeja
.__classcall
_private
__(cls
, field
, Qs
, rank
=n
)
1700 def inner_product(self
, x
, y
):
1701 return _usual_ip(x
,y
)
1706 Return a "random" finite-dimensional Euclidean Jordan Algebra.
1710 For now, we choose a random natural number ``n`` (greater than zero)
1711 and then give you back one of the following:
1713 * The cartesian product of the rational numbers ``n`` times; this is
1714 ``QQ^n`` with the Hadamard product.
1716 * The Jordan spin algebra on ``QQ^n``.
1718 * The ``n``-by-``n`` rational symmetric matrices with the symmetric
1721 * The ``n``-by-``n`` complex-rational Hermitian matrices embedded
1722 in the space of ``2n``-by-``2n`` real symmetric matrices.
1724 * The ``n``-by-``n`` quaternion-rational Hermitian matrices embedded
1725 in the space of ``4n``-by-``4n`` real symmetric matrices.
1727 Later this might be extended to return Cartesian products of the
1732 sage: from mjo.eja.eja_algebra import random_eja
1737 Euclidean Jordan algebra of degree...
1741 # The max_n component lets us choose different upper bounds on the
1742 # value "n" that gets passed to the constructor. This is needed
1743 # because e.g. R^{10} is reasonable to test, while the Hermitian
1744 # 10-by-10 quaternion matrices are not.
1745 (constructor
, max_n
) = choice([(RealCartesianProductEJA
, 6),
1747 (RealSymmetricEJA
, 5),
1748 (ComplexHermitianEJA
, 4),
1749 (QuaternionHermitianEJA
, 3)])
1750 n
= ZZ
.random_element(1, max_n
)
1751 return constructor(n
, field
=QQ
)
1755 def _real_symmetric_basis(n
, field
=QQ
):
1757 Return a basis for the space of real symmetric n-by-n matrices.
1759 # The basis of symmetric matrices, as matrices, in their R^(n-by-n)
1763 for j
in xrange(i
+1):
1764 Eij
= matrix(field
, n
, lambda k
,l
: k
==i
and l
==j
)
1768 # Beware, orthogonal but not normalized!
1769 Sij
= Eij
+ Eij
.transpose()
1774 def _complex_hermitian_basis(n
, field
=QQ
):
1776 Returns a basis for the space of complex Hermitian n-by-n matrices.
1780 sage: from mjo.eja.eja_algebra import _complex_hermitian_basis
1784 sage: set_random_seed()
1785 sage: n = ZZ.random_element(1,5)
1786 sage: all( M.is_symmetric() for M in _complex_hermitian_basis(n) )
1790 F
= QuadraticField(-1, 'I')
1793 # This is like the symmetric case, but we need to be careful:
1795 # * We want conjugate-symmetry, not just symmetry.
1796 # * The diagonal will (as a result) be real.
1800 for j
in xrange(i
+1):
1801 Eij
= matrix(field
, n
, lambda k
,l
: k
==i
and l
==j
)
1803 Sij
= _embed_complex_matrix(Eij
)
1806 # Beware, orthogonal but not normalized! The second one
1807 # has a minus because it's conjugated.
1808 Sij_real
= _embed_complex_matrix(Eij
+ Eij
.transpose())
1810 Sij_imag
= _embed_complex_matrix(I
*Eij
- I
*Eij
.transpose())
1815 def _quaternion_hermitian_basis(n
, field
=QQ
):
1817 Returns a basis for the space of quaternion Hermitian n-by-n matrices.
1821 sage: from mjo.eja.eja_algebra import _quaternion_hermitian_basis
1825 sage: set_random_seed()
1826 sage: n = ZZ.random_element(1,5)
1827 sage: all( M.is_symmetric() for M in _quaternion_hermitian_basis(n) )
1831 Q
= QuaternionAlgebra(QQ
,-1,-1)
1834 # This is like the symmetric case, but we need to be careful:
1836 # * We want conjugate-symmetry, not just symmetry.
1837 # * The diagonal will (as a result) be real.
1841 for j
in xrange(i
+1):
1842 Eij
= matrix(Q
, n
, lambda k
,l
: k
==i
and l
==j
)
1844 Sij
= _embed_quaternion_matrix(Eij
)
1847 # Beware, orthogonal but not normalized! The second,
1848 # third, and fourth ones have a minus because they're
1850 Sij_real
= _embed_quaternion_matrix(Eij
+ Eij
.transpose())
1852 Sij_I
= _embed_quaternion_matrix(I
*Eij
- I
*Eij
.transpose())
1854 Sij_J
= _embed_quaternion_matrix(J
*Eij
- J
*Eij
.transpose())
1856 Sij_K
= _embed_quaternion_matrix(K
*Eij
- K
*Eij
.transpose())
1862 return vector(m
.base_ring(), m
.list())
1865 return matrix(v
.base_ring(), sqrt(v
.degree()), v
.list())
1867 def _multiplication_table_from_matrix_basis(basis
):
1869 At least three of the five simple Euclidean Jordan algebras have the
1870 symmetric multiplication (A,B) |-> (AB + BA)/2, where the
1871 multiplication on the right is matrix multiplication. Given a basis
1872 for the underlying matrix space, this function returns a
1873 multiplication table (obtained by looping through the basis
1874 elements) for an algebra of those matrices. A reordered copy
1875 of the basis is also returned to work around the fact that
1876 the ``span()`` in this function will change the order of the basis
1877 from what we think it is, to... something else.
1879 # In S^2, for example, we nominally have four coordinates even
1880 # though the space is of dimension three only. The vector space V
1881 # is supposed to hold the entire long vector, and the subspace W
1882 # of V will be spanned by the vectors that arise from symmetric
1883 # matrices. Thus for S^2, dim(V) == 4 and dim(W) == 3.
1884 field
= basis
[0].base_ring()
1885 dimension
= basis
[0].nrows()
1887 V
= VectorSpace(field
, dimension
**2)
1888 W
= V
.span( _mat2vec(s
) for s
in basis
)
1890 # Taking the span above reorders our basis (thanks, jerk!) so we
1891 # need to put our "matrix basis" in the same order as the
1892 # (reordered) vector basis.
1893 S
= tuple( _vec2mat(b
) for b
in W
.basis() )
1897 # Brute force the multiplication-by-s matrix by looping
1898 # through all elements of the basis and doing the computation
1899 # to find out what the corresponding row should be. BEWARE:
1900 # these multiplication tables won't be symmetric! It therefore
1901 # becomes REALLY IMPORTANT that the underlying algebra
1902 # constructor uses ROW vectors and not COLUMN vectors. That's
1903 # why we're computing rows here and not columns.
1906 this_row
= _mat2vec((s
*t
+ t
*s
)/2)
1907 Q_rows
.append(W
.coordinates(this_row
))
1908 Q
= matrix(field
, W
.dimension(), Q_rows
)
1914 def _embed_complex_matrix(M
):
1916 Embed the n-by-n complex matrix ``M`` into the space of real
1917 matrices of size 2n-by-2n via the map the sends each entry `z = a +
1918 bi` to the block matrix ``[[a,b],[-b,a]]``.
1922 sage: from mjo.eja.eja_algebra import _embed_complex_matrix
1926 sage: F = QuadraticField(-1,'i')
1927 sage: x1 = F(4 - 2*i)
1928 sage: x2 = F(1 + 2*i)
1931 sage: M = matrix(F,2,[[x1,x2],[x3,x4]])
1932 sage: _embed_complex_matrix(M)
1941 Embedding is a homomorphism (isomorphism, in fact)::
1943 sage: set_random_seed()
1944 sage: n = ZZ.random_element(5)
1945 sage: F = QuadraticField(-1, 'i')
1946 sage: X = random_matrix(F, n)
1947 sage: Y = random_matrix(F, n)
1948 sage: actual = _embed_complex_matrix(X) * _embed_complex_matrix(Y)
1949 sage: expected = _embed_complex_matrix(X*Y)
1950 sage: actual == expected
1956 raise ValueError("the matrix 'M' must be square")
1957 field
= M
.base_ring()
1962 blocks
.append(matrix(field
, 2, [[a
,b
],[-b
,a
]]))
1964 # We can drop the imaginaries here.
1965 return matrix
.block(field
.base_ring(), n
, blocks
)
1968 def _unembed_complex_matrix(M
):
1970 The inverse of _embed_complex_matrix().
1974 sage: from mjo.eja.eja_algebra import (_embed_complex_matrix,
1975 ....: _unembed_complex_matrix)
1979 sage: A = matrix(QQ,[ [ 1, 2, 3, 4],
1980 ....: [-2, 1, -4, 3],
1981 ....: [ 9, 10, 11, 12],
1982 ....: [-10, 9, -12, 11] ])
1983 sage: _unembed_complex_matrix(A)
1985 [ 10*i + 9 12*i + 11]
1989 Unembedding is the inverse of embedding::
1991 sage: set_random_seed()
1992 sage: F = QuadraticField(-1, 'i')
1993 sage: M = random_matrix(F, 3)
1994 sage: _unembed_complex_matrix(_embed_complex_matrix(M)) == M
2000 raise ValueError("the matrix 'M' must be square")
2001 if not n
.mod(2).is_zero():
2002 raise ValueError("the matrix 'M' must be a complex embedding")
2004 F
= QuadraticField(-1, 'i')
2007 # Go top-left to bottom-right (reading order), converting every
2008 # 2-by-2 block we see to a single complex element.
2010 for k
in xrange(n
/2):
2011 for j
in xrange(n
/2):
2012 submat
= M
[2*k
:2*k
+2,2*j
:2*j
+2]
2013 if submat
[0,0] != submat
[1,1]:
2014 raise ValueError('bad on-diagonal submatrix')
2015 if submat
[0,1] != -submat
[1,0]:
2016 raise ValueError('bad off-diagonal submatrix')
2017 z
= submat
[0,0] + submat
[0,1]*i
2020 return matrix(F
, n
/2, elements
)
2023 def _embed_quaternion_matrix(M
):
2025 Embed the n-by-n quaternion matrix ``M`` into the space of real
2026 matrices of size 4n-by-4n by first sending each quaternion entry
2027 `z = a + bi + cj + dk` to the block-complex matrix
2028 ``[[a + bi, c+di],[-c + di, a-bi]]`, and then embedding those into
2033 sage: from mjo.eja.eja_algebra import _embed_quaternion_matrix
2037 sage: Q = QuaternionAlgebra(QQ,-1,-1)
2038 sage: i,j,k = Q.gens()
2039 sage: x = 1 + 2*i + 3*j + 4*k
2040 sage: M = matrix(Q, 1, [[x]])
2041 sage: _embed_quaternion_matrix(M)
2047 Embedding is a homomorphism (isomorphism, in fact)::
2049 sage: set_random_seed()
2050 sage: n = ZZ.random_element(5)
2051 sage: Q = QuaternionAlgebra(QQ,-1,-1)
2052 sage: X = random_matrix(Q, n)
2053 sage: Y = random_matrix(Q, n)
2054 sage: actual = _embed_quaternion_matrix(X)*_embed_quaternion_matrix(Y)
2055 sage: expected = _embed_quaternion_matrix(X*Y)
2056 sage: actual == expected
2060 quaternions
= M
.base_ring()
2063 raise ValueError("the matrix 'M' must be square")
2065 F
= QuadraticField(-1, 'i')
2070 t
= z
.coefficient_tuple()
2075 cplx_matrix
= matrix(F
, 2, [[ a
+ b
*i
, c
+ d
*i
],
2076 [-c
+ d
*i
, a
- b
*i
]])
2077 blocks
.append(_embed_complex_matrix(cplx_matrix
))
2079 # We should have real entries by now, so use the realest field
2080 # we've got for the return value.
2081 return matrix
.block(quaternions
.base_ring(), n
, blocks
)
2084 def _unembed_quaternion_matrix(M
):
2086 The inverse of _embed_quaternion_matrix().
2090 sage: from mjo.eja.eja_algebra import (_embed_quaternion_matrix,
2091 ....: _unembed_quaternion_matrix)
2095 sage: M = matrix(QQ, [[ 1, 2, 3, 4],
2096 ....: [-2, 1, -4, 3],
2097 ....: [-3, 4, 1, -2],
2098 ....: [-4, -3, 2, 1]])
2099 sage: _unembed_quaternion_matrix(M)
2100 [1 + 2*i + 3*j + 4*k]
2104 Unembedding is the inverse of embedding::
2106 sage: set_random_seed()
2107 sage: Q = QuaternionAlgebra(QQ, -1, -1)
2108 sage: M = random_matrix(Q, 3)
2109 sage: _unembed_quaternion_matrix(_embed_quaternion_matrix(M)) == M
2115 raise ValueError("the matrix 'M' must be square")
2116 if not n
.mod(4).is_zero():
2117 raise ValueError("the matrix 'M' must be a complex embedding")
2119 Q
= QuaternionAlgebra(QQ
,-1,-1)
2122 # Go top-left to bottom-right (reading order), converting every
2123 # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1
2126 for l
in xrange(n
/4):
2127 for m
in xrange(n
/4):
2128 submat
= _unembed_complex_matrix(M
[4*l
:4*l
+4,4*m
:4*m
+4])
2129 if submat
[0,0] != submat
[1,1].conjugate():
2130 raise ValueError('bad on-diagonal submatrix')
2131 if submat
[0,1] != -submat
[1,0].conjugate():
2132 raise ValueError('bad off-diagonal submatrix')
2133 z
= submat
[0,0].real() + submat
[0,0].imag()*i
2134 z
+= submat
[0,1].real()*j
+ submat
[0,1].imag()*k
2137 return matrix(Q
, n
/4, elements
)
2140 # The usual inner product on R^n.
2142 return x
.vector().inner_product(y
.vector())
2144 # The inner product used for the real symmetric simple EJA.
2145 # We keep it as a separate function because e.g. the complex
2146 # algebra uses the same inner product, except divided by 2.
2147 def _matrix_ip(X
,Y
):
2148 X_mat
= X
.natural_representation()
2149 Y_mat
= Y
.natural_representation()
2150 return (X_mat
*Y_mat
).trace()
2153 class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra
):
2155 The rank-n simple EJA consisting of real symmetric n-by-n
2156 matrices, the usual symmetric Jordan product, and the trace inner
2157 product. It has dimension `(n^2 + n)/2` over the reals.
2161 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
2165 sage: J = RealSymmetricEJA(2)
2166 sage: e0, e1, e2 = J.gens()
2176 The degree of this algebra is `(n^2 + n) / 2`::
2178 sage: set_random_seed()
2179 sage: n = ZZ.random_element(1,5)
2180 sage: J = RealSymmetricEJA(n)
2181 sage: J.degree() == (n^2 + n)/2
2184 The Jordan multiplication is what we think it is::
2186 sage: set_random_seed()
2187 sage: n = ZZ.random_element(1,5)
2188 sage: J = RealSymmetricEJA(n)
2189 sage: x = J.random_element()
2190 sage: y = J.random_element()
2191 sage: actual = (x*y).natural_representation()
2192 sage: X = x.natural_representation()
2193 sage: Y = y.natural_representation()
2194 sage: expected = (X*Y + Y*X)/2
2195 sage: actual == expected
2197 sage: J(expected) == x*y
2202 def __classcall_private__(cls
, n
, field
=QQ
):
2203 S
= _real_symmetric_basis(n
, field
=field
)
2204 (Qs
, T
) = _multiplication_table_from_matrix_basis(S
)
2206 fdeja
= super(RealSymmetricEJA
, cls
)
2207 return fdeja
.__classcall
_private
__(cls
,
2213 def inner_product(self
, x
, y
):
2214 return _matrix_ip(x
,y
)
2217 class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra
):
2219 The rank-n simple EJA consisting of complex Hermitian n-by-n
2220 matrices over the real numbers, the usual symmetric Jordan product,
2221 and the real-part-of-trace inner product. It has dimension `n^2` over
2226 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
2230 The degree of this algebra is `n^2`::
2232 sage: set_random_seed()
2233 sage: n = ZZ.random_element(1,5)
2234 sage: J = ComplexHermitianEJA(n)
2235 sage: J.degree() == n^2
2238 The Jordan multiplication is what we think it is::
2240 sage: set_random_seed()
2241 sage: n = ZZ.random_element(1,5)
2242 sage: J = ComplexHermitianEJA(n)
2243 sage: x = J.random_element()
2244 sage: y = J.random_element()
2245 sage: actual = (x*y).natural_representation()
2246 sage: X = x.natural_representation()
2247 sage: Y = y.natural_representation()
2248 sage: expected = (X*Y + Y*X)/2
2249 sage: actual == expected
2251 sage: J(expected) == x*y
2256 def __classcall_private__(cls
, n
, field
=QQ
):
2257 S
= _complex_hermitian_basis(n
)
2258 (Qs
, T
) = _multiplication_table_from_matrix_basis(S
)
2260 fdeja
= super(ComplexHermitianEJA
, cls
)
2261 return fdeja
.__classcall
_private
__(cls
,
2267 def inner_product(self
, x
, y
):
2268 # Since a+bi on the diagonal is represented as
2273 # we'll double-count the "a" entries if we take the trace of
2275 return _matrix_ip(x
,y
)/2
2278 class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra
):
2280 The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
2281 matrices, the usual symmetric Jordan product, and the
2282 real-part-of-trace inner product. It has dimension `2n^2 - n` over
2287 sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
2291 The degree of this algebra is `n^2`::
2293 sage: set_random_seed()
2294 sage: n = ZZ.random_element(1,5)
2295 sage: J = QuaternionHermitianEJA(n)
2296 sage: J.degree() == 2*(n^2) - n
2299 The Jordan multiplication is what we think it is::
2301 sage: set_random_seed()
2302 sage: n = ZZ.random_element(1,5)
2303 sage: J = QuaternionHermitianEJA(n)
2304 sage: x = J.random_element()
2305 sage: y = J.random_element()
2306 sage: actual = (x*y).natural_representation()
2307 sage: X = x.natural_representation()
2308 sage: Y = y.natural_representation()
2309 sage: expected = (X*Y + Y*X)/2
2310 sage: actual == expected
2312 sage: J(expected) == x*y
2317 def __classcall_private__(cls
, n
, field
=QQ
):
2318 S
= _quaternion_hermitian_basis(n
)
2319 (Qs
, T
) = _multiplication_table_from_matrix_basis(S
)
2321 fdeja
= super(QuaternionHermitianEJA
, cls
)
2322 return fdeja
.__classcall
_private
__(cls
,
2328 def inner_product(self
, x
, y
):
2329 # Since a+bi+cj+dk on the diagonal is represented as
2331 # a + bi +cj + dk = [ a b c d]
2336 # we'll quadruple-count the "a" entries if we take the trace of
2338 return _matrix_ip(x
,y
)/4
2341 class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra
):
2343 The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
2344 with the usual inner product and jordan product ``x*y =
2345 (<x_bar,y_bar>, x0*y_bar + y0*x_bar)``. It has dimension `n` over
2350 sage: from mjo.eja.eja_algebra import JordanSpinEJA
2354 This multiplication table can be verified by hand::
2356 sage: J = JordanSpinEJA(4)
2357 sage: e0,e1,e2,e3 = J.gens()
2375 def __classcall_private__(cls
, n
, field
=QQ
):
2377 id_matrix
= matrix
.identity(field
, n
)
2379 ei
= id_matrix
.column(i
)
2380 Qi
= matrix
.zero(field
, n
)
2382 Qi
.set_column(0, ei
)
2383 Qi
+= matrix
.diagonal(n
, [ei
[0]]*n
)
2384 # The addition of the diagonal matrix adds an extra ei[0] in the
2385 # upper-left corner of the matrix.
2386 Qi
[0,0] = Qi
[0,0] * ~
field(2)
2389 # The rank of the spin algebra is two, unless we're in a
2390 # one-dimensional ambient space (because the rank is bounded by
2391 # the ambient dimension).
2392 fdeja
= super(JordanSpinEJA
, cls
)
2393 return fdeja
.__classcall
_private
__(cls
, field
, Qs
, rank
=min(n
,2))
2395 def inner_product(self
, x
, y
):
2396 return _usual_ip(x
,y
)