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
,
36 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
,
59 assume_associative
=assume_associative
,
63 natural_basis
=natural_basis
)
70 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.
390 if self
._rank
is None:
391 raise ValueError("no rank specified at genesis")
396 def vector_space(self
):
398 Return the vector space that underlies this algebra.
402 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
406 sage: J = RealSymmetricEJA(2)
407 sage: J.vector_space()
408 Vector space of dimension 3 over Rational Field
411 return self
.zero().vector().parent().ambient_vector_space()
414 class Element(FiniteDimensionalAlgebraElement
):
416 An element of a Euclidean Jordan algebra.
421 Oh man, I should not be doing this. This hides the "disabled"
422 methods ``left_matrix`` and ``matrix`` from introspection;
423 in particular it removes them from tab-completion.
425 return filter(lambda s
: s
not in ['left_matrix', 'matrix'],
426 dir(self
.__class
__) )
429 def __init__(self
, A
, elt
=None):
434 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
438 The identity in `S^n` is converted to the identity in the EJA::
440 sage: J = RealSymmetricEJA(3)
441 sage: I = matrix.identity(QQ,3)
442 sage: J(I) == J.one()
445 This skew-symmetric matrix can't be represented in the EJA::
447 sage: J = RealSymmetricEJA(3)
448 sage: A = matrix(QQ,3, lambda i,j: i-j)
450 Traceback (most recent call last):
452 ArithmeticError: vector is not in free module
455 # Goal: if we're given a matrix, and if it lives in our
456 # parent algebra's "natural ambient space," convert it
457 # into an algebra element.
459 # The catch is, we make a recursive call after converting
460 # the given matrix into a vector that lives in the algebra.
461 # This we need to try the parent class initializer first,
462 # to avoid recursing forever if we're given something that
463 # already fits into the algebra, but also happens to live
464 # in the parent's "natural ambient space" (this happens with
467 FiniteDimensionalAlgebraElement
.__init
__(self
, A
, elt
)
469 natural_basis
= A
.natural_basis()
470 if elt
in natural_basis
[0].matrix_space():
471 # Thanks for nothing! Matrix spaces aren't vector
472 # spaces in Sage, so we have to figure out its
473 # natural-basis coordinates ourselves.
474 V
= VectorSpace(elt
.base_ring(), elt
.nrows()**2)
475 W
= V
.span( _mat2vec(s
) for s
in natural_basis
)
476 coords
= W
.coordinates(_mat2vec(elt
))
477 FiniteDimensionalAlgebraElement
.__init
__(self
, A
, coords
)
479 def __pow__(self
, n
):
481 Return ``self`` raised to the power ``n``.
483 Jordan algebras are always power-associative; see for
484 example Faraut and Koranyi, Proposition II.1.2 (ii).
488 We have to override this because our superclass uses row vectors
489 instead of column vectors! We, on the other hand, assume column
494 sage: from mjo.eja.eja_algebra import random_eja
498 sage: set_random_seed()
499 sage: x = random_eja().random_element()
500 sage: x.operator()(x) == (x^2)
503 A few examples of power-associativity::
505 sage: set_random_seed()
506 sage: x = random_eja().random_element()
507 sage: x*(x*x)*(x*x) == x^5
509 sage: (x*x)*(x*x*x) == x^5
512 We also know that powers operator-commute (Koecher, Chapter
515 sage: set_random_seed()
516 sage: x = random_eja().random_element()
517 sage: m = ZZ.random_element(0,10)
518 sage: n = ZZ.random_element(0,10)
519 sage: Lxm = (x^m).operator()
520 sage: Lxn = (x^n).operator()
521 sage: Lxm*Lxn == Lxn*Lxm
526 return self
.parent().one()
530 return (self
.operator()**(n
-1))(self
)
533 def apply_univariate_polynomial(self
, p
):
535 Apply the univariate polynomial ``p`` to this element.
537 A priori, SageMath won't allow us to apply a univariate
538 polynomial to an element of an EJA, because we don't know
539 that EJAs are rings (they are usually not associative). Of
540 course, we know that EJAs are power-associative, so the
541 operation is ultimately kosher. This function sidesteps
542 the CAS to get the answer we want and expect.
546 sage: from mjo.eja.eja_algebra import (RealCartesianProductEJA,
551 sage: R = PolynomialRing(QQ, 't')
553 sage: p = t^4 - t^3 + 5*t - 2
554 sage: J = RealCartesianProductEJA(5)
555 sage: J.one().apply_univariate_polynomial(p) == 3*J.one()
560 We should always get back an element of the algebra::
562 sage: set_random_seed()
563 sage: p = PolynomialRing(QQ, 't').random_element()
564 sage: J = random_eja()
565 sage: x = J.random_element()
566 sage: x.apply_univariate_polynomial(p) in J
570 if len(p
.variables()) > 1:
571 raise ValueError("not a univariate polynomial")
574 # Convert the coeficcients to the parent's base ring,
575 # because a priori they might live in an (unnecessarily)
576 # larger ring for which P.sum() would fail below.
577 cs
= [ R(c
) for c
in p
.coefficients(sparse
=False) ]
578 return P
.sum( cs
[k
]*(self
**k
) for k
in range(len(cs
)) )
581 def characteristic_polynomial(self
):
583 Return the characteristic polynomial of this element.
587 sage: from mjo.eja.eja_algebra import RealCartesianProductEJA
591 The rank of `R^3` is three, and the minimal polynomial of
592 the identity element is `(t-1)` from which it follows that
593 the characteristic polynomial should be `(t-1)^3`::
595 sage: J = RealCartesianProductEJA(3)
596 sage: J.one().characteristic_polynomial()
597 t^3 - 3*t^2 + 3*t - 1
599 Likewise, the characteristic of the zero element in the
600 rank-three algebra `R^{n}` should be `t^{3}`::
602 sage: J = RealCartesianProductEJA(3)
603 sage: J.zero().characteristic_polynomial()
606 The characteristic polynomial of an element should evaluate
607 to zero on that element::
609 sage: set_random_seed()
610 sage: x = RealCartesianProductEJA(3).random_element()
611 sage: p = x.characteristic_polynomial()
612 sage: x.apply_univariate_polynomial(p)
616 p
= self
.parent().characteristic_polynomial()
617 return p(*self
.vector())
620 def inner_product(self
, other
):
622 Return the parent algebra's inner product of myself and ``other``.
626 sage: from mjo.eja.eja_algebra import (
627 ....: ComplexHermitianEJA,
629 ....: QuaternionHermitianEJA,
630 ....: RealSymmetricEJA,
635 The inner product in the Jordan spin algebra is the usual
636 inner product on `R^n` (this example only works because the
637 basis for the Jordan algebra is the standard basis in `R^n`)::
639 sage: J = JordanSpinEJA(3)
640 sage: x = vector(QQ,[1,2,3])
641 sage: y = vector(QQ,[4,5,6])
642 sage: x.inner_product(y)
644 sage: J(x).inner_product(J(y))
647 The inner product on `S^n` is `<X,Y> = trace(X*Y)`, where
648 multiplication is the usual matrix multiplication in `S^n`,
649 so the inner product of the identity matrix with itself
652 sage: J = RealSymmetricEJA(3)
653 sage: J.one().inner_product(J.one())
656 Likewise, the inner product on `C^n` is `<X,Y> =
657 Re(trace(X*Y))`, where we must necessarily take the real
658 part because the product of Hermitian matrices may not be
661 sage: J = ComplexHermitianEJA(3)
662 sage: J.one().inner_product(J.one())
665 Ditto for the quaternions::
667 sage: J = QuaternionHermitianEJA(3)
668 sage: J.one().inner_product(J.one())
673 Ensure that we can always compute an inner product, and that
674 it gives us back a real number::
676 sage: set_random_seed()
677 sage: J = random_eja()
678 sage: x = J.random_element()
679 sage: y = J.random_element()
680 sage: x.inner_product(y) in RR
686 raise TypeError("'other' must live in the same algebra")
688 return P
.inner_product(self
, other
)
691 def operator_commutes_with(self
, other
):
693 Return whether or not this element operator-commutes
698 sage: from mjo.eja.eja_algebra import random_eja
702 The definition of a Jordan algebra says that any element
703 operator-commutes with its square::
705 sage: set_random_seed()
706 sage: x = random_eja().random_element()
707 sage: x.operator_commutes_with(x^2)
712 Test Lemma 1 from Chapter III of Koecher::
714 sage: set_random_seed()
715 sage: J = random_eja()
716 sage: u = J.random_element()
717 sage: v = J.random_element()
718 sage: lhs = u.operator_commutes_with(u*v)
719 sage: rhs = v.operator_commutes_with(u^2)
723 Test the first polarization identity from my notes, Koecher Chapter
724 III, or from Baes (2.3)::
726 sage: set_random_seed()
727 sage: J = random_eja()
728 sage: x = J.random_element()
729 sage: y = J.random_element()
730 sage: Lx = x.operator()
731 sage: Ly = y.operator()
732 sage: Lxx = (x*x).operator()
733 sage: Lxy = (x*y).operator()
734 sage: bool(2*Lx*Lxy + Ly*Lxx == 2*Lxy*Lx + Lxx*Ly)
737 Test the second polarization identity from my notes or from
740 sage: set_random_seed()
741 sage: J = random_eja()
742 sage: x = J.random_element()
743 sage: y = J.random_element()
744 sage: z = J.random_element()
745 sage: Lx = x.operator()
746 sage: Ly = y.operator()
747 sage: Lz = z.operator()
748 sage: Lzy = (z*y).operator()
749 sage: Lxy = (x*y).operator()
750 sage: Lxz = (x*z).operator()
751 sage: bool(Lx*Lzy + Lz*Lxy + Ly*Lxz == Lzy*Lx + Lxy*Lz + Lxz*Ly)
754 Test the third polarization identity from my notes or from
757 sage: set_random_seed()
758 sage: J = random_eja()
759 sage: u = J.random_element()
760 sage: y = J.random_element()
761 sage: z = J.random_element()
762 sage: Lu = u.operator()
763 sage: Ly = y.operator()
764 sage: Lz = z.operator()
765 sage: Lzy = (z*y).operator()
766 sage: Luy = (u*y).operator()
767 sage: Luz = (u*z).operator()
768 sage: Luyz = (u*(y*z)).operator()
769 sage: lhs = Lu*Lzy + Lz*Luy + Ly*Luz
770 sage: rhs = Luyz + Ly*Lu*Lz + Lz*Lu*Ly
771 sage: bool(lhs == rhs)
775 if not other
in self
.parent():
776 raise TypeError("'other' must live in the same algebra")
785 Return my determinant, the product of my eigenvalues.
789 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
794 sage: J = JordanSpinEJA(2)
795 sage: e0,e1 = J.gens()
796 sage: x = sum( J.gens() )
802 sage: J = JordanSpinEJA(3)
803 sage: e0,e1,e2 = J.gens()
804 sage: x = sum( J.gens() )
810 An element is invertible if and only if its determinant is
813 sage: set_random_seed()
814 sage: x = random_eja().random_element()
815 sage: x.is_invertible() == (x.det() != 0)
821 p
= P
._charpoly
_coeff
(0)
822 # The _charpoly_coeff function already adds the factor of
823 # -1 to ensure that _charpoly_coeff(0) is really what
824 # appears in front of t^{0} in the charpoly. However,
825 # we want (-1)^r times THAT for the determinant.
826 return ((-1)**r
)*p(*self
.vector())
831 Return the Jordan-multiplicative inverse of this element.
835 We appeal to the quadratic representation as in Koecher's
836 Theorem 12 in Chapter III, Section 5.
840 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
845 The inverse in the spin factor algebra is given in Alizadeh's
848 sage: set_random_seed()
849 sage: n = ZZ.random_element(1,10)
850 sage: J = JordanSpinEJA(n)
851 sage: x = J.random_element()
852 sage: while not x.is_invertible():
853 ....: x = J.random_element()
854 sage: x_vec = x.vector()
856 sage: x_bar = x_vec[1:]
857 sage: coeff = ~(x0^2 - x_bar.inner_product(x_bar))
858 sage: inv_vec = x_vec.parent()([x0] + (-x_bar).list())
859 sage: x_inverse = coeff*inv_vec
860 sage: x.inverse() == J(x_inverse)
865 The identity element is its own inverse::
867 sage: set_random_seed()
868 sage: J = random_eja()
869 sage: J.one().inverse() == J.one()
872 If an element has an inverse, it acts like one::
874 sage: set_random_seed()
875 sage: J = random_eja()
876 sage: x = J.random_element()
877 sage: (not x.is_invertible()) or (x.inverse()*x == J.one())
880 The inverse of the inverse is what we started with::
882 sage: set_random_seed()
883 sage: J = random_eja()
884 sage: x = J.random_element()
885 sage: (not x.is_invertible()) or (x.inverse().inverse() == x)
888 The zero element is never invertible::
890 sage: set_random_seed()
891 sage: J = random_eja().zero().inverse()
892 Traceback (most recent call last):
894 ValueError: element is not invertible
897 if not self
.is_invertible():
898 raise ValueError("element is not invertible")
900 return (~self
.quadratic_representation())(self
)
903 def is_invertible(self
):
905 Return whether or not this element is invertible.
907 We can't use the superclass method because it relies on
908 the algebra being associative.
912 The usual way to do this is to check if the determinant is
913 zero, but we need the characteristic polynomial for the
914 determinant. The minimal polynomial is a lot easier to get,
915 so we use Corollary 2 in Chapter V of Koecher to check
916 whether or not the paren't algebra's zero element is a root
917 of this element's minimal polynomial.
921 sage: from mjo.eja.eja_algebra import random_eja
925 The identity element is always invertible::
927 sage: set_random_seed()
928 sage: J = random_eja()
929 sage: J.one().is_invertible()
932 The zero element is never invertible::
934 sage: set_random_seed()
935 sage: J = random_eja()
936 sage: J.zero().is_invertible()
940 zero
= self
.parent().zero()
941 p
= self
.minimal_polynomial()
942 return not (p(zero
) == zero
)
945 def is_nilpotent(self
):
947 Return whether or not some power of this element is zero.
949 The superclass method won't work unless we're in an
950 associative algebra, and we aren't. However, we generate
951 an assocoative subalgebra and we're nilpotent there if and
952 only if we're nilpotent here (probably).
956 sage: from mjo.eja.eja_algebra import random_eja
960 The identity element is never nilpotent::
962 sage: set_random_seed()
963 sage: random_eja().one().is_nilpotent()
966 The additive identity is always nilpotent::
968 sage: set_random_seed()
969 sage: random_eja().zero().is_nilpotent()
973 # The element we're going to call "is_nilpotent()" on.
974 # Either myself, interpreted as an element of a finite-
975 # dimensional algebra, or an element of an associative
979 if self
.parent().is_associative():
980 elt
= FiniteDimensionalAlgebraElement(self
.parent(), self
)
982 V
= self
.span_of_powers()
983 assoc_subalg
= self
.subalgebra_generated_by()
984 # Mis-design warning: the basis used for span_of_powers()
985 # and subalgebra_generated_by() must be the same, and in
987 elt
= assoc_subalg(V
.coordinates(self
.vector()))
989 # Recursive call, but should work since elt lives in an
990 # associative algebra.
991 return elt
.is_nilpotent()
994 def is_regular(self
):
996 Return whether or not this is a regular element.
1000 sage: from mjo.eja.eja_algebra import JordanSpinEJA
1004 The identity element always has degree one, but any element
1005 linearly-independent from it is regular::
1007 sage: J = JordanSpinEJA(5)
1008 sage: J.one().is_regular()
1010 sage: e0, e1, e2, e3, e4 = J.gens() # e0 is the identity
1011 sage: for x in J.gens():
1012 ....: (J.one() + x).is_regular()
1020 return self
.degree() == self
.parent().rank()
1025 Compute the degree of this element the straightforward way
1026 according to the definition; by appending powers to a list
1027 and figuring out its dimension (that is, whether or not
1028 they're linearly dependent).
1032 sage: from mjo.eja.eja_algebra import JordanSpinEJA
1036 sage: J = JordanSpinEJA(4)
1037 sage: J.one().degree()
1039 sage: e0,e1,e2,e3 = J.gens()
1040 sage: (e0 - e1).degree()
1043 In the spin factor algebra (of rank two), all elements that
1044 aren't multiples of the identity are regular::
1046 sage: set_random_seed()
1047 sage: n = ZZ.random_element(1,10)
1048 sage: J = JordanSpinEJA(n)
1049 sage: x = J.random_element()
1050 sage: x == x.coefficient(0)*J.one() or x.degree() == 2
1054 return self
.span_of_powers().dimension()
1057 def left_matrix(self
):
1059 Our parent class defines ``left_matrix`` and ``matrix``
1060 methods whose names are misleading. We don't want them.
1062 raise NotImplementedError("use operator().matrix() instead")
1064 matrix
= left_matrix
1067 def minimal_polynomial(self
):
1069 Return the minimal polynomial of this element,
1070 as a function of the variable `t`.
1074 We restrict ourselves to the associative subalgebra
1075 generated by this element, and then return the minimal
1076 polynomial of this element's operator matrix (in that
1077 subalgebra). This works by Baes Proposition 2.3.16.
1081 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
1086 The minimal polynomial of the identity and zero elements are
1089 sage: set_random_seed()
1090 sage: J = random_eja()
1091 sage: J.one().minimal_polynomial()
1093 sage: J.zero().minimal_polynomial()
1096 The degree of an element is (by one definition) the degree
1097 of its minimal polynomial::
1099 sage: set_random_seed()
1100 sage: x = random_eja().random_element()
1101 sage: x.degree() == x.minimal_polynomial().degree()
1104 The minimal polynomial and the characteristic polynomial coincide
1105 and are known (see Alizadeh, Example 11.11) for all elements of
1106 the spin factor algebra that aren't scalar multiples of the
1109 sage: set_random_seed()
1110 sage: n = ZZ.random_element(2,10)
1111 sage: J = JordanSpinEJA(n)
1112 sage: y = J.random_element()
1113 sage: while y == y.coefficient(0)*J.one():
1114 ....: y = J.random_element()
1115 sage: y0 = y.vector()[0]
1116 sage: y_bar = y.vector()[1:]
1117 sage: actual = y.minimal_polynomial()
1118 sage: t = PolynomialRing(J.base_ring(),'t').gen(0)
1119 sage: expected = t^2 - 2*y0*t + (y0^2 - norm(y_bar)^2)
1120 sage: bool(actual == expected)
1123 The minimal polynomial should always kill its element::
1125 sage: set_random_seed()
1126 sage: x = random_eja().random_element()
1127 sage: p = x.minimal_polynomial()
1128 sage: x.apply_univariate_polynomial(p)
1132 V
= self
.span_of_powers()
1133 assoc_subalg
= self
.subalgebra_generated_by()
1134 # Mis-design warning: the basis used for span_of_powers()
1135 # and subalgebra_generated_by() must be the same, and in
1137 elt
= assoc_subalg(V
.coordinates(self
.vector()))
1138 return elt
.operator().minimal_polynomial()
1142 def natural_representation(self
):
1144 Return a more-natural representation of this element.
1146 Every finite-dimensional Euclidean Jordan Algebra is a
1147 direct sum of five simple algebras, four of which comprise
1148 Hermitian matrices. This method returns the original
1149 "natural" representation of this element as a Hermitian
1150 matrix, if it has one. If not, you get the usual representation.
1154 sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA,
1155 ....: QuaternionHermitianEJA)
1159 sage: J = ComplexHermitianEJA(3)
1162 sage: J.one().natural_representation()
1172 sage: J = QuaternionHermitianEJA(3)
1175 sage: J.one().natural_representation()
1176 [1 0 0 0 0 0 0 0 0 0 0 0]
1177 [0 1 0 0 0 0 0 0 0 0 0 0]
1178 [0 0 1 0 0 0 0 0 0 0 0 0]
1179 [0 0 0 1 0 0 0 0 0 0 0 0]
1180 [0 0 0 0 1 0 0 0 0 0 0 0]
1181 [0 0 0 0 0 1 0 0 0 0 0 0]
1182 [0 0 0 0 0 0 1 0 0 0 0 0]
1183 [0 0 0 0 0 0 0 1 0 0 0 0]
1184 [0 0 0 0 0 0 0 0 1 0 0 0]
1185 [0 0 0 0 0 0 0 0 0 1 0 0]
1186 [0 0 0 0 0 0 0 0 0 0 1 0]
1187 [0 0 0 0 0 0 0 0 0 0 0 1]
1190 B
= self
.parent().natural_basis()
1191 W
= B
[0].matrix_space()
1192 return W
.linear_combination(zip(self
.vector(), B
))
1197 Return the left-multiplication-by-this-element
1198 operator on the ambient algebra.
1202 sage: from mjo.eja.eja_algebra import random_eja
1206 sage: set_random_seed()
1207 sage: J = random_eja()
1208 sage: x = J.random_element()
1209 sage: y = J.random_element()
1210 sage: x.operator()(y) == x*y
1212 sage: y.operator()(x) == x*y
1217 fda_elt
= FiniteDimensionalAlgebraElement(P
, self
)
1218 return FiniteDimensionalEuclideanJordanAlgebraOperator(
1221 fda_elt
.matrix().transpose() )
1224 def quadratic_representation(self
, other
=None):
1226 Return the quadratic representation of this element.
1230 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
1235 The explicit form in the spin factor algebra is given by
1236 Alizadeh's Example 11.12::
1238 sage: set_random_seed()
1239 sage: n = ZZ.random_element(1,10)
1240 sage: J = JordanSpinEJA(n)
1241 sage: x = J.random_element()
1242 sage: x_vec = x.vector()
1244 sage: x_bar = x_vec[1:]
1245 sage: A = matrix(QQ, 1, [x_vec.inner_product(x_vec)])
1246 sage: B = 2*x0*x_bar.row()
1247 sage: C = 2*x0*x_bar.column()
1248 sage: D = matrix.identity(QQ, n-1)
1249 sage: D = (x0^2 - x_bar.inner_product(x_bar))*D
1250 sage: D = D + 2*x_bar.tensor_product(x_bar)
1251 sage: Q = matrix.block(2,2,[A,B,C,D])
1252 sage: Q == x.quadratic_representation().matrix()
1255 Test all of the properties from Theorem 11.2 in Alizadeh::
1257 sage: set_random_seed()
1258 sage: J = random_eja()
1259 sage: x = J.random_element()
1260 sage: y = J.random_element()
1261 sage: Lx = x.operator()
1262 sage: Lxx = (x*x).operator()
1263 sage: Qx = x.quadratic_representation()
1264 sage: Qy = y.quadratic_representation()
1265 sage: Qxy = x.quadratic_representation(y)
1266 sage: Qex = J.one().quadratic_representation(x)
1267 sage: n = ZZ.random_element(10)
1268 sage: Qxn = (x^n).quadratic_representation()
1272 sage: 2*Qxy == (x+y).quadratic_representation() - Qx - Qy
1275 Property 2 (multiply on the right for :trac:`28272`):
1277 sage: alpha = QQ.random_element()
1278 sage: (alpha*x).quadratic_representation() == Qx*(alpha^2)
1283 sage: not x.is_invertible() or ( Qx(x.inverse()) == x )
1286 sage: not x.is_invertible() or (
1289 ....: x.inverse().quadratic_representation() )
1292 sage: Qxy(J.one()) == x*y
1297 sage: not x.is_invertible() or (
1298 ....: x.quadratic_representation(x.inverse())*Qx
1299 ....: == Qx*x.quadratic_representation(x.inverse()) )
1302 sage: not x.is_invertible() or (
1303 ....: x.quadratic_representation(x.inverse())*Qx
1305 ....: 2*x.operator()*Qex - Qx )
1308 sage: 2*x.operator()*Qex - Qx == Lxx
1313 sage: Qy(x).quadratic_representation() == Qy*Qx*Qy
1323 sage: not x.is_invertible() or (
1324 ....: Qx*x.inverse().operator() == Lx )
1329 sage: not x.operator_commutes_with(y) or (
1330 ....: Qx(y)^n == Qxn(y^n) )
1336 elif not other
in self
.parent():
1337 raise TypeError("'other' must live in the same algebra")
1340 M
= other
.operator()
1341 return ( L
*M
+ M
*L
- (self
*other
).operator() )
1344 def span_of_powers(self
):
1346 Return the vector space spanned by successive powers of
1349 # The dimension of the subalgebra can't be greater than
1350 # the big algebra, so just put everything into a list
1351 # and let span() get rid of the excess.
1353 # We do the extra ambient_vector_space() in case we're messing
1354 # with polynomials and the direct parent is a module.
1355 V
= self
.parent().vector_space()
1356 return V
.span( (self
**d
).vector() for d
in xrange(V
.dimension()) )
1359 def subalgebra_generated_by(self
):
1361 Return the associative subalgebra of the parent EJA generated
1366 sage: from mjo.eja.eja_algebra import random_eja
1370 sage: set_random_seed()
1371 sage: x = random_eja().random_element()
1372 sage: x.subalgebra_generated_by().is_associative()
1375 Squaring in the subalgebra should work the same as in
1378 sage: set_random_seed()
1379 sage: x = random_eja().random_element()
1380 sage: u = x.subalgebra_generated_by().random_element()
1381 sage: u.operator()(u) == u^2
1385 # First get the subspace spanned by the powers of myself...
1386 V
= self
.span_of_powers()
1387 F
= self
.base_ring()
1389 # Now figure out the entries of the right-multiplication
1390 # matrix for the successive basis elements b0, b1,... of
1393 for b_right
in V
.basis():
1394 eja_b_right
= self
.parent()(b_right
)
1396 # The first row of the right-multiplication matrix by
1397 # b1 is what we get if we apply that matrix to b1. The
1398 # second row of the right multiplication matrix by b1
1399 # is what we get when we apply that matrix to b2...
1401 # IMPORTANT: this assumes that all vectors are COLUMN
1402 # vectors, unlike our superclass (which uses row vectors).
1403 for b_left
in V
.basis():
1404 eja_b_left
= self
.parent()(b_left
)
1405 # Multiply in the original EJA, but then get the
1406 # coordinates from the subalgebra in terms of its
1408 this_row
= V
.coordinates((eja_b_left
*eja_b_right
).vector())
1409 b_right_rows
.append(this_row
)
1410 b_right_matrix
= matrix(F
, b_right_rows
)
1411 mats
.append(b_right_matrix
)
1413 # It's an algebra of polynomials in one element, and EJAs
1414 # are power-associative.
1416 # TODO: choose generator names intelligently.
1417 return FiniteDimensionalEuclideanJordanAlgebra(F
, mats
, assume_associative
=True, names
='f')
1420 def subalgebra_idempotent(self
):
1422 Find an idempotent in the associative subalgebra I generate
1423 using Proposition 2.3.5 in Baes.
1427 sage: from mjo.eja.eja_algebra import random_eja
1431 sage: set_random_seed()
1432 sage: J = random_eja()
1433 sage: x = J.random_element()
1434 sage: while x.is_nilpotent():
1435 ....: x = J.random_element()
1436 sage: c = x.subalgebra_idempotent()
1441 if self
.is_nilpotent():
1442 raise ValueError("this only works with non-nilpotent elements!")
1444 V
= self
.span_of_powers()
1445 J
= self
.subalgebra_generated_by()
1446 # Mis-design warning: the basis used for span_of_powers()
1447 # and subalgebra_generated_by() must be the same, and in
1449 u
= J(V
.coordinates(self
.vector()))
1451 # The image of the matrix of left-u^m-multiplication
1452 # will be minimal for some natural number s...
1454 minimal_dim
= V
.dimension()
1455 for i
in xrange(1, V
.dimension()):
1456 this_dim
= (u
**i
).operator().matrix().image().dimension()
1457 if this_dim
< minimal_dim
:
1458 minimal_dim
= this_dim
1461 # Now minimal_matrix should correspond to the smallest
1462 # non-zero subspace in Baes's (or really, Koecher's)
1465 # However, we need to restrict the matrix to work on the
1466 # subspace... or do we? Can't we just solve, knowing that
1467 # A(c) = u^(s+1) should have a solution in the big space,
1470 # Beware, solve_right() means that we're using COLUMN vectors.
1471 # Our FiniteDimensionalAlgebraElement superclass uses rows.
1473 A
= u_next
.operator().matrix()
1474 c_coordinates
= A
.solve_right(u_next
.vector())
1476 # Now c_coordinates is the idempotent we want, but it's in
1477 # the coordinate system of the subalgebra.
1479 # We need the basis for J, but as elements of the parent algebra.
1481 basis
= [self
.parent(v
) for v
in V
.basis()]
1482 return self
.parent().linear_combination(zip(c_coordinates
, basis
))
1487 Return my trace, the sum of my eigenvalues.
1491 sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
1492 ....: RealCartesianProductEJA,
1497 sage: J = JordanSpinEJA(3)
1498 sage: x = sum(J.gens())
1504 sage: J = RealCartesianProductEJA(5)
1505 sage: J.one().trace()
1510 The trace of an element is a real number::
1512 sage: set_random_seed()
1513 sage: J = random_eja()
1514 sage: J.random_element().trace() in J.base_ring()
1520 p
= P
._charpoly
_coeff
(r
-1)
1521 # The _charpoly_coeff function already adds the factor of
1522 # -1 to ensure that _charpoly_coeff(r-1) is really what
1523 # appears in front of t^{r-1} in the charpoly. However,
1524 # we want the negative of THAT for the trace.
1525 return -p(*self
.vector())
1528 def trace_inner_product(self
, other
):
1530 Return the trace inner product of myself and ``other``.
1534 sage: from mjo.eja.eja_algebra import random_eja
1538 The trace inner product is commutative::
1540 sage: set_random_seed()
1541 sage: J = random_eja()
1542 sage: x = J.random_element(); y = J.random_element()
1543 sage: x.trace_inner_product(y) == y.trace_inner_product(x)
1546 The trace inner product is bilinear::
1548 sage: set_random_seed()
1549 sage: J = random_eja()
1550 sage: x = J.random_element()
1551 sage: y = J.random_element()
1552 sage: z = J.random_element()
1553 sage: a = QQ.random_element();
1554 sage: actual = (a*(x+z)).trace_inner_product(y)
1555 sage: expected = ( a*x.trace_inner_product(y) +
1556 ....: a*z.trace_inner_product(y) )
1557 sage: actual == expected
1559 sage: actual = x.trace_inner_product(a*(y+z))
1560 sage: expected = ( a*x.trace_inner_product(y) +
1561 ....: a*x.trace_inner_product(z) )
1562 sage: actual == expected
1565 The trace inner product satisfies the compatibility
1566 condition in the definition of a Euclidean Jordan algebra::
1568 sage: set_random_seed()
1569 sage: J = random_eja()
1570 sage: x = J.random_element()
1571 sage: y = J.random_element()
1572 sage: z = J.random_element()
1573 sage: (x*y).trace_inner_product(z) == y.trace_inner_product(x*z)
1577 if not other
in self
.parent():
1578 raise TypeError("'other' must live in the same algebra")
1580 return (self
*other
).trace()
1583 class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1585 Return the Euclidean Jordan Algebra corresponding to the set
1586 `R^n` under the Hadamard product.
1588 Note: this is nothing more than the Cartesian product of ``n``
1589 copies of the spin algebra. Once Cartesian product algebras
1590 are implemented, this can go.
1594 sage: from mjo.eja.eja_algebra import RealCartesianProductEJA
1598 This multiplication table can be verified by hand::
1600 sage: J = RealCartesianProductEJA(3)
1601 sage: e0,e1,e2 = J.gens()
1617 def __classcall_private__(cls
, n
, field
=QQ
):
1618 # The FiniteDimensionalAlgebra constructor takes a list of
1619 # matrices, the ith representing right multiplication by the ith
1620 # basis element in the vector space. So if e_1 = (1,0,0), then
1621 # right (Hadamard) multiplication of x by e_1 picks out the first
1622 # component of x; and likewise for the ith basis element e_i.
1623 Qs
= [ matrix(field
, n
, n
, lambda k
,j
: 1*(k
== j
== i
))
1624 for i
in xrange(n
) ]
1626 fdeja
= super(RealCartesianProductEJA
, cls
)
1627 return fdeja
.__classcall
_private
__(cls
, field
, Qs
, rank
=n
)
1629 def inner_product(self
, x
, y
):
1630 return _usual_ip(x
,y
)
1635 Return a "random" finite-dimensional Euclidean Jordan Algebra.
1639 For now, we choose a random natural number ``n`` (greater than zero)
1640 and then give you back one of the following:
1642 * The cartesian product of the rational numbers ``n`` times; this is
1643 ``QQ^n`` with the Hadamard product.
1645 * The Jordan spin algebra on ``QQ^n``.
1647 * The ``n``-by-``n`` rational symmetric matrices with the symmetric
1650 * The ``n``-by-``n`` complex-rational Hermitian matrices embedded
1651 in the space of ``2n``-by-``2n`` real symmetric matrices.
1653 * The ``n``-by-``n`` quaternion-rational Hermitian matrices embedded
1654 in the space of ``4n``-by-``4n`` real symmetric matrices.
1656 Later this might be extended to return Cartesian products of the
1661 sage: from mjo.eja.eja_algebra import random_eja
1666 Euclidean Jordan algebra of degree...
1670 # The max_n component lets us choose different upper bounds on the
1671 # value "n" that gets passed to the constructor. This is needed
1672 # because e.g. R^{10} is reasonable to test, while the Hermitian
1673 # 10-by-10 quaternion matrices are not.
1674 (constructor
, max_n
) = choice([(RealCartesianProductEJA
, 6),
1676 (RealSymmetricEJA
, 5),
1677 (ComplexHermitianEJA
, 4),
1678 (QuaternionHermitianEJA
, 3)])
1679 n
= ZZ
.random_element(1, max_n
)
1680 return constructor(n
, field
=QQ
)
1684 def _real_symmetric_basis(n
, field
=QQ
):
1686 Return a basis for the space of real symmetric n-by-n matrices.
1688 # The basis of symmetric matrices, as matrices, in their R^(n-by-n)
1692 for j
in xrange(i
+1):
1693 Eij
= matrix(field
, n
, lambda k
,l
: k
==i
and l
==j
)
1697 # Beware, orthogonal but not normalized!
1698 Sij
= Eij
+ Eij
.transpose()
1703 def _complex_hermitian_basis(n
, field
=QQ
):
1705 Returns a basis for the space of complex Hermitian n-by-n matrices.
1709 sage: from mjo.eja.eja_algebra import _complex_hermitian_basis
1713 sage: set_random_seed()
1714 sage: n = ZZ.random_element(1,5)
1715 sage: all( M.is_symmetric() for M in _complex_hermitian_basis(n) )
1719 F
= QuadraticField(-1, 'I')
1722 # This is like the symmetric case, but we need to be careful:
1724 # * We want conjugate-symmetry, not just symmetry.
1725 # * The diagonal will (as a result) be real.
1729 for j
in xrange(i
+1):
1730 Eij
= matrix(field
, n
, lambda k
,l
: k
==i
and l
==j
)
1732 Sij
= _embed_complex_matrix(Eij
)
1735 # Beware, orthogonal but not normalized! The second one
1736 # has a minus because it's conjugated.
1737 Sij_real
= _embed_complex_matrix(Eij
+ Eij
.transpose())
1739 Sij_imag
= _embed_complex_matrix(I
*Eij
- I
*Eij
.transpose())
1744 def _quaternion_hermitian_basis(n
, field
=QQ
):
1746 Returns a basis for the space of quaternion Hermitian n-by-n matrices.
1750 sage: from mjo.eja.eja_algebra import _quaternion_hermitian_basis
1754 sage: set_random_seed()
1755 sage: n = ZZ.random_element(1,5)
1756 sage: all( M.is_symmetric() for M in _quaternion_hermitian_basis(n) )
1760 Q
= QuaternionAlgebra(QQ
,-1,-1)
1763 # This is like the symmetric case, but we need to be careful:
1765 # * We want conjugate-symmetry, not just symmetry.
1766 # * The diagonal will (as a result) be real.
1770 for j
in xrange(i
+1):
1771 Eij
= matrix(Q
, n
, lambda k
,l
: k
==i
and l
==j
)
1773 Sij
= _embed_quaternion_matrix(Eij
)
1776 # Beware, orthogonal but not normalized! The second,
1777 # third, and fourth ones have a minus because they're
1779 Sij_real
= _embed_quaternion_matrix(Eij
+ Eij
.transpose())
1781 Sij_I
= _embed_quaternion_matrix(I
*Eij
- I
*Eij
.transpose())
1783 Sij_J
= _embed_quaternion_matrix(J
*Eij
- J
*Eij
.transpose())
1785 Sij_K
= _embed_quaternion_matrix(K
*Eij
- K
*Eij
.transpose())
1791 return vector(m
.base_ring(), m
.list())
1794 return matrix(v
.base_ring(), sqrt(v
.degree()), v
.list())
1796 def _multiplication_table_from_matrix_basis(basis
):
1798 At least three of the five simple Euclidean Jordan algebras have the
1799 symmetric multiplication (A,B) |-> (AB + BA)/2, where the
1800 multiplication on the right is matrix multiplication. Given a basis
1801 for the underlying matrix space, this function returns a
1802 multiplication table (obtained by looping through the basis
1803 elements) for an algebra of those matrices. A reordered copy
1804 of the basis is also returned to work around the fact that
1805 the ``span()`` in this function will change the order of the basis
1806 from what we think it is, to... something else.
1808 # In S^2, for example, we nominally have four coordinates even
1809 # though the space is of dimension three only. The vector space V
1810 # is supposed to hold the entire long vector, and the subspace W
1811 # of V will be spanned by the vectors that arise from symmetric
1812 # matrices. Thus for S^2, dim(V) == 4 and dim(W) == 3.
1813 field
= basis
[0].base_ring()
1814 dimension
= basis
[0].nrows()
1816 V
= VectorSpace(field
, dimension
**2)
1817 W
= V
.span( _mat2vec(s
) for s
in basis
)
1819 # Taking the span above reorders our basis (thanks, jerk!) so we
1820 # need to put our "matrix basis" in the same order as the
1821 # (reordered) vector basis.
1822 S
= tuple( _vec2mat(b
) for b
in W
.basis() )
1826 # Brute force the multiplication-by-s matrix by looping
1827 # through all elements of the basis and doing the computation
1828 # to find out what the corresponding row should be. BEWARE:
1829 # these multiplication tables won't be symmetric! It therefore
1830 # becomes REALLY IMPORTANT that the underlying algebra
1831 # constructor uses ROW vectors and not COLUMN vectors. That's
1832 # why we're computing rows here and not columns.
1835 this_row
= _mat2vec((s
*t
+ t
*s
)/2)
1836 Q_rows
.append(W
.coordinates(this_row
))
1837 Q
= matrix(field
, W
.dimension(), Q_rows
)
1843 def _embed_complex_matrix(M
):
1845 Embed the n-by-n complex matrix ``M`` into the space of real
1846 matrices of size 2n-by-2n via the map the sends each entry `z = a +
1847 bi` to the block matrix ``[[a,b],[-b,a]]``.
1851 sage: from mjo.eja.eja_algebra import _embed_complex_matrix
1855 sage: F = QuadraticField(-1,'i')
1856 sage: x1 = F(4 - 2*i)
1857 sage: x2 = F(1 + 2*i)
1860 sage: M = matrix(F,2,[[x1,x2],[x3,x4]])
1861 sage: _embed_complex_matrix(M)
1870 Embedding is a homomorphism (isomorphism, in fact)::
1872 sage: set_random_seed()
1873 sage: n = ZZ.random_element(5)
1874 sage: F = QuadraticField(-1, 'i')
1875 sage: X = random_matrix(F, n)
1876 sage: Y = random_matrix(F, n)
1877 sage: actual = _embed_complex_matrix(X) * _embed_complex_matrix(Y)
1878 sage: expected = _embed_complex_matrix(X*Y)
1879 sage: actual == expected
1885 raise ValueError("the matrix 'M' must be square")
1886 field
= M
.base_ring()
1891 blocks
.append(matrix(field
, 2, [[a
,b
],[-b
,a
]]))
1893 # We can drop the imaginaries here.
1894 return matrix
.block(field
.base_ring(), n
, blocks
)
1897 def _unembed_complex_matrix(M
):
1899 The inverse of _embed_complex_matrix().
1903 sage: from mjo.eja.eja_algebra import (_embed_complex_matrix,
1904 ....: _unembed_complex_matrix)
1908 sage: A = matrix(QQ,[ [ 1, 2, 3, 4],
1909 ....: [-2, 1, -4, 3],
1910 ....: [ 9, 10, 11, 12],
1911 ....: [-10, 9, -12, 11] ])
1912 sage: _unembed_complex_matrix(A)
1914 [ 10*i + 9 12*i + 11]
1918 Unembedding is the inverse of embedding::
1920 sage: set_random_seed()
1921 sage: F = QuadraticField(-1, 'i')
1922 sage: M = random_matrix(F, 3)
1923 sage: _unembed_complex_matrix(_embed_complex_matrix(M)) == M
1929 raise ValueError("the matrix 'M' must be square")
1930 if not n
.mod(2).is_zero():
1931 raise ValueError("the matrix 'M' must be a complex embedding")
1933 F
= QuadraticField(-1, 'i')
1936 # Go top-left to bottom-right (reading order), converting every
1937 # 2-by-2 block we see to a single complex element.
1939 for k
in xrange(n
/2):
1940 for j
in xrange(n
/2):
1941 submat
= M
[2*k
:2*k
+2,2*j
:2*j
+2]
1942 if submat
[0,0] != submat
[1,1]:
1943 raise ValueError('bad on-diagonal submatrix')
1944 if submat
[0,1] != -submat
[1,0]:
1945 raise ValueError('bad off-diagonal submatrix')
1946 z
= submat
[0,0] + submat
[0,1]*i
1949 return matrix(F
, n
/2, elements
)
1952 def _embed_quaternion_matrix(M
):
1954 Embed the n-by-n quaternion matrix ``M`` into the space of real
1955 matrices of size 4n-by-4n by first sending each quaternion entry
1956 `z = a + bi + cj + dk` to the block-complex matrix
1957 ``[[a + bi, c+di],[-c + di, a-bi]]`, and then embedding those into
1962 sage: from mjo.eja.eja_algebra import _embed_quaternion_matrix
1966 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1967 sage: i,j,k = Q.gens()
1968 sage: x = 1 + 2*i + 3*j + 4*k
1969 sage: M = matrix(Q, 1, [[x]])
1970 sage: _embed_quaternion_matrix(M)
1976 Embedding is a homomorphism (isomorphism, in fact)::
1978 sage: set_random_seed()
1979 sage: n = ZZ.random_element(5)
1980 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1981 sage: X = random_matrix(Q, n)
1982 sage: Y = random_matrix(Q, n)
1983 sage: actual = _embed_quaternion_matrix(X)*_embed_quaternion_matrix(Y)
1984 sage: expected = _embed_quaternion_matrix(X*Y)
1985 sage: actual == expected
1989 quaternions
= M
.base_ring()
1992 raise ValueError("the matrix 'M' must be square")
1994 F
= QuadraticField(-1, 'i')
1999 t
= z
.coefficient_tuple()
2004 cplx_matrix
= matrix(F
, 2, [[ a
+ b
*i
, c
+ d
*i
],
2005 [-c
+ d
*i
, a
- b
*i
]])
2006 blocks
.append(_embed_complex_matrix(cplx_matrix
))
2008 # We should have real entries by now, so use the realest field
2009 # we've got for the return value.
2010 return matrix
.block(quaternions
.base_ring(), n
, blocks
)
2013 def _unembed_quaternion_matrix(M
):
2015 The inverse of _embed_quaternion_matrix().
2019 sage: from mjo.eja.eja_algebra import (_embed_quaternion_matrix,
2020 ....: _unembed_quaternion_matrix)
2024 sage: M = matrix(QQ, [[ 1, 2, 3, 4],
2025 ....: [-2, 1, -4, 3],
2026 ....: [-3, 4, 1, -2],
2027 ....: [-4, -3, 2, 1]])
2028 sage: _unembed_quaternion_matrix(M)
2029 [1 + 2*i + 3*j + 4*k]
2033 Unembedding is the inverse of embedding::
2035 sage: set_random_seed()
2036 sage: Q = QuaternionAlgebra(QQ, -1, -1)
2037 sage: M = random_matrix(Q, 3)
2038 sage: _unembed_quaternion_matrix(_embed_quaternion_matrix(M)) == M
2044 raise ValueError("the matrix 'M' must be square")
2045 if not n
.mod(4).is_zero():
2046 raise ValueError("the matrix 'M' must be a complex embedding")
2048 Q
= QuaternionAlgebra(QQ
,-1,-1)
2051 # Go top-left to bottom-right (reading order), converting every
2052 # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1
2055 for l
in xrange(n
/4):
2056 for m
in xrange(n
/4):
2057 submat
= _unembed_complex_matrix(M
[4*l
:4*l
+4,4*m
:4*m
+4])
2058 if submat
[0,0] != submat
[1,1].conjugate():
2059 raise ValueError('bad on-diagonal submatrix')
2060 if submat
[0,1] != -submat
[1,0].conjugate():
2061 raise ValueError('bad off-diagonal submatrix')
2062 z
= submat
[0,0].real() + submat
[0,0].imag()*i
2063 z
+= submat
[0,1].real()*j
+ submat
[0,1].imag()*k
2066 return matrix(Q
, n
/4, elements
)
2069 # The usual inner product on R^n.
2071 return x
.vector().inner_product(y
.vector())
2073 # The inner product used for the real symmetric simple EJA.
2074 # We keep it as a separate function because e.g. the complex
2075 # algebra uses the same inner product, except divided by 2.
2076 def _matrix_ip(X
,Y
):
2077 X_mat
= X
.natural_representation()
2078 Y_mat
= Y
.natural_representation()
2079 return (X_mat
*Y_mat
).trace()
2082 class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra
):
2084 The rank-n simple EJA consisting of real symmetric n-by-n
2085 matrices, the usual symmetric Jordan product, and the trace inner
2086 product. It has dimension `(n^2 + n)/2` over the reals.
2090 sage: from mjo.eja.eja_algebra import RealSymmetricEJA
2094 sage: J = RealSymmetricEJA(2)
2095 sage: e0, e1, e2 = J.gens()
2105 The degree of this algebra is `(n^2 + n) / 2`::
2107 sage: set_random_seed()
2108 sage: n = ZZ.random_element(1,5)
2109 sage: J = RealSymmetricEJA(n)
2110 sage: J.degree() == (n^2 + n)/2
2113 The Jordan multiplication is what we think it is::
2115 sage: set_random_seed()
2116 sage: n = ZZ.random_element(1,5)
2117 sage: J = RealSymmetricEJA(n)
2118 sage: x = J.random_element()
2119 sage: y = J.random_element()
2120 sage: actual = (x*y).natural_representation()
2121 sage: X = x.natural_representation()
2122 sage: Y = y.natural_representation()
2123 sage: expected = (X*Y + Y*X)/2
2124 sage: actual == expected
2126 sage: J(expected) == x*y
2131 def __classcall_private__(cls
, n
, field
=QQ
):
2132 S
= _real_symmetric_basis(n
, field
=field
)
2133 (Qs
, T
) = _multiplication_table_from_matrix_basis(S
)
2135 fdeja
= super(RealSymmetricEJA
, cls
)
2136 return fdeja
.__classcall
_private
__(cls
,
2142 def inner_product(self
, x
, y
):
2143 return _matrix_ip(x
,y
)
2146 class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra
):
2148 The rank-n simple EJA consisting of complex Hermitian n-by-n
2149 matrices over the real numbers, the usual symmetric Jordan product,
2150 and the real-part-of-trace inner product. It has dimension `n^2` over
2155 sage: from mjo.eja.eja_algebra import ComplexHermitianEJA
2159 The degree of this algebra is `n^2`::
2161 sage: set_random_seed()
2162 sage: n = ZZ.random_element(1,5)
2163 sage: J = ComplexHermitianEJA(n)
2164 sage: J.degree() == n^2
2167 The Jordan multiplication is what we think it is::
2169 sage: set_random_seed()
2170 sage: n = ZZ.random_element(1,5)
2171 sage: J = ComplexHermitianEJA(n)
2172 sage: x = J.random_element()
2173 sage: y = J.random_element()
2174 sage: actual = (x*y).natural_representation()
2175 sage: X = x.natural_representation()
2176 sage: Y = y.natural_representation()
2177 sage: expected = (X*Y + Y*X)/2
2178 sage: actual == expected
2180 sage: J(expected) == x*y
2185 def __classcall_private__(cls
, n
, field
=QQ
):
2186 S
= _complex_hermitian_basis(n
)
2187 (Qs
, T
) = _multiplication_table_from_matrix_basis(S
)
2189 fdeja
= super(ComplexHermitianEJA
, cls
)
2190 return fdeja
.__classcall
_private
__(cls
,
2196 def inner_product(self
, x
, y
):
2197 # Since a+bi on the diagonal is represented as
2202 # we'll double-count the "a" entries if we take the trace of
2204 return _matrix_ip(x
,y
)/2
2207 class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra
):
2209 The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
2210 matrices, the usual symmetric Jordan product, and the
2211 real-part-of-trace inner product. It has dimension `2n^2 - n` over
2216 sage: from mjo.eja.eja_algebra import QuaternionHermitianEJA
2220 The degree of this algebra is `n^2`::
2222 sage: set_random_seed()
2223 sage: n = ZZ.random_element(1,5)
2224 sage: J = QuaternionHermitianEJA(n)
2225 sage: J.degree() == 2*(n^2) - n
2228 The Jordan multiplication is what we think it is::
2230 sage: set_random_seed()
2231 sage: n = ZZ.random_element(1,5)
2232 sage: J = QuaternionHermitianEJA(n)
2233 sage: x = J.random_element()
2234 sage: y = J.random_element()
2235 sage: actual = (x*y).natural_representation()
2236 sage: X = x.natural_representation()
2237 sage: Y = y.natural_representation()
2238 sage: expected = (X*Y + Y*X)/2
2239 sage: actual == expected
2241 sage: J(expected) == x*y
2246 def __classcall_private__(cls
, n
, field
=QQ
):
2247 S
= _quaternion_hermitian_basis(n
)
2248 (Qs
, T
) = _multiplication_table_from_matrix_basis(S
)
2250 fdeja
= super(QuaternionHermitianEJA
, cls
)
2251 return fdeja
.__classcall
_private
__(cls
,
2257 def inner_product(self
, x
, y
):
2258 # Since a+bi+cj+dk on the diagonal is represented as
2260 # a + bi +cj + dk = [ a b c d]
2265 # we'll quadruple-count the "a" entries if we take the trace of
2267 return _matrix_ip(x
,y
)/4
2270 class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra
):
2272 The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
2273 with the usual inner product and jordan product ``x*y =
2274 (<x_bar,y_bar>, x0*y_bar + y0*x_bar)``. It has dimension `n` over
2279 sage: from mjo.eja.eja_algebra import JordanSpinEJA
2283 This multiplication table can be verified by hand::
2285 sage: J = JordanSpinEJA(4)
2286 sage: e0,e1,e2,e3 = J.gens()
2304 def __classcall_private__(cls
, n
, field
=QQ
):
2306 id_matrix
= matrix
.identity(field
, n
)
2308 ei
= id_matrix
.column(i
)
2309 Qi
= matrix
.zero(field
, n
)
2311 Qi
.set_column(0, ei
)
2312 Qi
+= matrix
.diagonal(n
, [ei
[0]]*n
)
2313 # The addition of the diagonal matrix adds an extra ei[0] in the
2314 # upper-left corner of the matrix.
2315 Qi
[0,0] = Qi
[0,0] * ~
field(2)
2318 # The rank of the spin algebra is two, unless we're in a
2319 # one-dimensional ambient space (because the rank is bounded by
2320 # the ambient dimension).
2321 fdeja
= super(JordanSpinEJA
, cls
)
2322 return fdeja
.__classcall
_private
__(cls
, field
, Qs
, rank
=min(n
,2))
2324 def inner_product(self
, x
, y
):
2325 return _usual_ip(x
,y
)