2 Euclidean Jordan Algebras. These are formally-real Jordan Algebras;
3 specifically those where u^2 + v^2 = 0 implies that u = v = 0. They
4 are used in optimization, and have some additional nice methods beyond
5 what can be supported in a general Jordan Algebra.
8 from sage
.categories
.magmatic_algebras
import MagmaticAlgebras
9 from sage
.structure
.element
import is_Matrix
10 from sage
.structure
.category_object
import normalize_names
12 from sage
.algebras
.finite_dimensional_algebras
.finite_dimensional_algebra
import FiniteDimensionalAlgebra
13 from sage
.algebras
.finite_dimensional_algebras
.finite_dimensional_algebra_element
import FiniteDimensionalAlgebraElement
15 class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra
):
17 def __classcall_private__(cls
,
21 assume_associative
=False,
25 mult_table
= [b
.base_extend(field
) for b
in mult_table
]
28 if not (is_Matrix(b
) and b
.dimensions() == (n
, n
)):
29 raise ValueError("input is not a multiplication table")
30 mult_table
= tuple(mult_table
)
32 cat
= MagmaticAlgebras(field
).FiniteDimensional().WithBasis()
33 cat
.or_subcategory(category
)
34 if assume_associative
:
35 cat
= cat
.Associative()
37 names
= normalize_names(n
, names
)
39 fda
= super(FiniteDimensionalEuclideanJordanAlgebra
, cls
)
40 return fda
.__classcall
__(cls
,
43 assume_associative
=assume_associative
,
49 def __init__(self
, field
,
52 assume_associative
=False,
56 fda
= super(FiniteDimensionalEuclideanJordanAlgebra
, self
)
65 Return a string representation of ``self``.
67 fmt
= "Euclidean Jordan algebra of degree {} over {}"
68 return fmt
.format(self
.degree(), self
.base_ring())
72 Return the rank of this EJA.
74 if self
._rank
is None:
75 raise ValueError("no rank specified at genesis")
80 class Element(FiniteDimensionalAlgebraElement
):
82 An element of a Euclidean Jordan algebra.
87 Return ``self`` raised to the power ``n``.
89 Jordan algebras are always power-associative; see for
90 example Faraut and Koranyi, Proposition II.1.2 (ii).
94 We have to override this because our superclass uses row vectors
95 instead of column vectors! We, on the other hand, assume column
100 sage: set_random_seed()
101 sage: x = random_eja().random_element()
102 sage: x.matrix()*x.vector() == (x**2).vector()
112 return A
.element_class(A
, (self
.matrix()**(n
-1))*self
.vector())
115 def characteristic_polynomial(self
):
117 Return my characteristic polynomial (if I'm a regular
120 Eventually this should be implemented in terms of the parent
121 algebra's characteristic polynomial that works for ALL
124 if self
.is_regular():
125 return self
.minimal_polynomial()
127 raise NotImplementedError('irregular element')
132 Return my determinant, the product of my eigenvalues.
137 sage: e0,e1 = J.gens()
142 sage: e0,e1,e2 = J.gens()
143 sage: x = e0 + e1 + e2
148 cs
= self
.characteristic_polynomial().coefficients(sparse
=False)
151 return cs
[0] * (-1)**r
153 raise ValueError('charpoly had no coefficients')
156 def is_nilpotent(self
):
158 Return whether or not some power of this element is zero.
160 The superclass method won't work unless we're in an
161 associative algebra, and we aren't. However, we generate
162 an assocoative subalgebra and we're nilpotent there if and
163 only if we're nilpotent here (probably).
167 The identity element is never nilpotent::
169 sage: set_random_seed()
170 sage: random_eja().one().is_nilpotent()
173 The additive identity is always nilpotent::
175 sage: set_random_seed()
176 sage: random_eja().zero().is_nilpotent()
180 # The element we're going to call "is_nilpotent()" on.
181 # Either myself, interpreted as an element of a finite-
182 # dimensional algebra, or an element of an associative
186 if self
.parent().is_associative():
187 elt
= FiniteDimensionalAlgebraElement(self
.parent(), self
)
189 V
= self
.span_of_powers()
190 assoc_subalg
= self
.subalgebra_generated_by()
191 # Mis-design warning: the basis used for span_of_powers()
192 # and subalgebra_generated_by() must be the same, and in
194 elt
= assoc_subalg(V
.coordinates(self
.vector()))
196 # Recursive call, but should work since elt lives in an
197 # associative algebra.
198 return elt
.is_nilpotent()
201 def is_regular(self
):
203 Return whether or not this is a regular element.
207 The identity element always has degree one, but any element
208 linearly-independent from it is regular::
211 sage: J.one().is_regular()
213 sage: e0, e1, e2, e3, e4 = J.gens() # e0 is the identity
214 sage: for x in J.gens():
215 ....: (J.one() + x).is_regular()
223 return self
.degree() == self
.parent().rank()
228 Compute the degree of this element the straightforward way
229 according to the definition; by appending powers to a list
230 and figuring out its dimension (that is, whether or not
231 they're linearly dependent).
236 sage: J.one().degree()
238 sage: e0,e1,e2,e3 = J.gens()
239 sage: (e0 - e1).degree()
242 In the spin factor algebra (of rank two), all elements that
243 aren't multiples of the identity are regular::
245 sage: set_random_seed()
246 sage: n = ZZ.random_element(1,10).abs()
248 sage: x = J.random_element()
249 sage: x == x.coefficient(0)*J.one() or x.degree() == 2
253 return self
.span_of_powers().dimension()
258 Return the matrix that represents left- (or right-)
259 multiplication by this element in the parent algebra.
261 We have to override this because the superclass method
262 returns a matrix that acts on row vectors (that is, on
265 fda_elt
= FiniteDimensionalAlgebraElement(self
.parent(), self
)
266 return fda_elt
.matrix().transpose()
269 def minimal_polynomial(self
):
273 sage: set_random_seed()
274 sage: x = random_eja().random_element()
275 sage: x.degree() == x.minimal_polynomial().degree()
280 sage: set_random_seed()
281 sage: x = random_eja().random_element()
282 sage: x.degree() == x.minimal_polynomial().degree()
285 The minimal polynomial and the characteristic polynomial coincide
286 and are known (see Alizadeh, Example 11.11) for all elements of
287 the spin factor algebra that aren't scalar multiples of the
290 sage: set_random_seed()
291 sage: n = ZZ.random_element(2,10).abs()
293 sage: y = J.random_element()
294 sage: while y == y.coefficient(0)*J.one():
295 ....: y = J.random_element()
296 sage: y0 = y.vector()[0]
297 sage: y_bar = y.vector()[1:]
298 sage: actual = y.minimal_polynomial()
299 sage: x = SR.symbol('x', domain='real')
300 sage: expected = x^2 - 2*y0*x + (y0^2 - norm(y_bar)^2)
301 sage: bool(actual == expected)
305 # The element we're going to call "minimal_polynomial()" on.
306 # Either myself, interpreted as an element of a finite-
307 # dimensional algebra, or an element of an associative
311 if self
.parent().is_associative():
312 elt
= FiniteDimensionalAlgebraElement(self
.parent(), self
)
314 V
= self
.span_of_powers()
315 assoc_subalg
= self
.subalgebra_generated_by()
316 # Mis-design warning: the basis used for span_of_powers()
317 # and subalgebra_generated_by() must be the same, and in
319 elt
= assoc_subalg(V
.coordinates(self
.vector()))
321 # Recursive call, but should work since elt lives in an
322 # associative algebra.
323 return elt
.minimal_polynomial()
326 def quadratic_representation(self
):
328 Return the quadratic representation of this element.
332 The explicit form in the spin factor algebra is given by
333 Alizadeh's Example 11.12::
335 sage: n = ZZ.random_element(1,10).abs()
337 sage: x = J.random_element()
338 sage: x_vec = x.vector()
340 sage: x_bar = x_vec[1:]
341 sage: A = matrix(QQ, 1, [x_vec.inner_product(x_vec)])
342 sage: B = 2*x0*x_bar.row()
343 sage: C = 2*x0*x_bar.column()
344 sage: D = identity_matrix(QQ, n-1)
345 sage: D = (x0^2 - x_bar.inner_product(x_bar))*D
346 sage: D = D + 2*x_bar.tensor_product(x_bar)
347 sage: Q = block_matrix(2,2,[A,B,C,D])
348 sage: Q == x.quadratic_representation()
352 return 2*(self
.matrix()**2) - (self
**2).matrix()
355 def span_of_powers(self
):
357 Return the vector space spanned by successive powers of
360 # The dimension of the subalgebra can't be greater than
361 # the big algebra, so just put everything into a list
362 # and let span() get rid of the excess.
363 V
= self
.vector().parent()
364 return V
.span( (self
**d
).vector() for d
in xrange(V
.dimension()) )
367 def subalgebra_generated_by(self
):
369 Return the associative subalgebra of the parent EJA generated
374 sage: set_random_seed()
375 sage: x = random_eja().random_element()
376 sage: x.subalgebra_generated_by().is_associative()
379 Squaring in the subalgebra should be the same thing as
380 squaring in the superalgebra::
382 sage: set_random_seed()
383 sage: x = random_eja().random_element()
384 sage: u = x.subalgebra_generated_by().random_element()
385 sage: u.matrix()*u.vector() == (u**2).vector()
389 # First get the subspace spanned by the powers of myself...
390 V
= self
.span_of_powers()
393 # Now figure out the entries of the right-multiplication
394 # matrix for the successive basis elements b0, b1,... of
397 for b_right
in V
.basis():
398 eja_b_right
= self
.parent()(b_right
)
400 # The first row of the right-multiplication matrix by
401 # b1 is what we get if we apply that matrix to b1. The
402 # second row of the right multiplication matrix by b1
403 # is what we get when we apply that matrix to b2...
405 # IMPORTANT: this assumes that all vectors are COLUMN
406 # vectors, unlike our superclass (which uses row vectors).
407 for b_left
in V
.basis():
408 eja_b_left
= self
.parent()(b_left
)
409 # Multiply in the original EJA, but then get the
410 # coordinates from the subalgebra in terms of its
412 this_row
= V
.coordinates((eja_b_left
*eja_b_right
).vector())
413 b_right_rows
.append(this_row
)
414 b_right_matrix
= matrix(F
, b_right_rows
)
415 mats
.append(b_right_matrix
)
417 # It's an algebra of polynomials in one element, and EJAs
418 # are power-associative.
420 # TODO: choose generator names intelligently.
421 return FiniteDimensionalEuclideanJordanAlgebra(F
, mats
, assume_associative
=True, names
='f')
424 def subalgebra_idempotent(self
):
426 Find an idempotent in the associative subalgebra I generate
427 using Proposition 2.3.5 in Baes.
431 sage: set_random_seed()
433 sage: c = J.random_element().subalgebra_idempotent()
437 sage: c = J.random_element().subalgebra_idempotent()
442 if self
.is_nilpotent():
443 raise ValueError("this only works with non-nilpotent elements!")
445 V
= self
.span_of_powers()
446 J
= self
.subalgebra_generated_by()
447 # Mis-design warning: the basis used for span_of_powers()
448 # and subalgebra_generated_by() must be the same, and in
450 u
= J(V
.coordinates(self
.vector()))
452 # The image of the matrix of left-u^m-multiplication
453 # will be minimal for some natural number s...
455 minimal_dim
= V
.dimension()
456 for i
in xrange(1, V
.dimension()):
457 this_dim
= (u
**i
).matrix().image().dimension()
458 if this_dim
< minimal_dim
:
459 minimal_dim
= this_dim
462 # Now minimal_matrix should correspond to the smallest
463 # non-zero subspace in Baes's (or really, Koecher's)
466 # However, we need to restrict the matrix to work on the
467 # subspace... or do we? Can't we just solve, knowing that
468 # A(c) = u^(s+1) should have a solution in the big space,
471 # Beware, solve_right() means that we're using COLUMN vectors.
472 # Our FiniteDimensionalAlgebraElement superclass uses rows.
475 c_coordinates
= A
.solve_right(u_next
.vector())
477 # Now c_coordinates is the idempotent we want, but it's in
478 # the coordinate system of the subalgebra.
480 # We need the basis for J, but as elements of the parent algebra.
482 basis
= [self
.parent(v
) for v
in V
.basis()]
483 return self
.parent().linear_combination(zip(c_coordinates
, basis
))
488 Return my trace, the sum of my eigenvalues.
493 sage: e0,e1,e2 = J.gens()
494 sage: x = e0 + e1 + e2
499 cs
= self
.characteristic_polynomial().coefficients(sparse
=False)
503 raise ValueError('charpoly had fewer than 2 coefficients')
506 def eja_rn(dimension
, field
=QQ
):
508 Return the Euclidean Jordan Algebra corresponding to the set
509 `R^n` under the Hadamard product.
513 This multiplication table can be verified by hand::
516 sage: e0,e1,e2 = J.gens()
531 # The FiniteDimensionalAlgebra constructor takes a list of
532 # matrices, the ith representing right multiplication by the ith
533 # basis element in the vector space. So if e_1 = (1,0,0), then
534 # right (Hadamard) multiplication of x by e_1 picks out the first
535 # component of x; and likewise for the ith basis element e_i.
536 Qs
= [ matrix(field
, dimension
, dimension
, lambda k
,j
: 1*(k
== j
== i
))
537 for i
in xrange(dimension
) ]
539 return FiniteDimensionalEuclideanJordanAlgebra(field
,Qs
,rank
=dimension
)
542 def eja_ln(dimension
, field
=QQ
):
544 Return the Jordan algebra corresponding to the Lorentz "ice cream"
545 cone of the given ``dimension``.
549 This multiplication table can be verified by hand::
552 sage: e0,e1,e2,e3 = J.gens()
568 In one dimension, this is the reals under multiplication::
577 id_matrix
= identity_matrix(field
,dimension
)
578 for i
in xrange(dimension
):
579 ei
= id_matrix
.column(i
)
580 Qi
= zero_matrix(field
,dimension
)
583 Qi
+= diagonal_matrix(dimension
, [ei
[0]]*dimension
)
584 # The addition of the diagonal matrix adds an extra ei[0] in the
585 # upper-left corner of the matrix.
586 Qi
[0,0] = Qi
[0,0] * ~
field(2)
589 # The rank of the spin factor algebra is two, UNLESS we're in a
590 # one-dimensional ambient space (the rank is bounded by the
591 # ambient dimension).
592 rank
= min(dimension
,2)
593 return FiniteDimensionalEuclideanJordanAlgebra(field
,Qs
,rank
=rank
)
596 def eja_sn(dimension
, field
=QQ
):
598 Return the simple Jordan algebra of ``dimension``-by-``dimension``
599 symmetric matrices over ``field``.
604 sage: e0, e1, e2 = J.gens()
615 # In S^2, for example, we nominally have four coordinates even
616 # though the space is of dimension three only. The vector space V
617 # is supposed to hold the entire long vector, and the subspace W
618 # of V will be spanned by the vectors that arise from symmetric
619 # matrices. Thus for S^2, dim(V) == 4 and dim(W) == 3.
620 V
= VectorSpace(field
, dimension
**2)
622 # The basis of symmetric matrices, as matrices, in their R^(n-by-n)
626 for i
in xrange(dimension
):
627 for j
in xrange(i
+1):
628 Eij
= matrix(field
, dimension
, lambda k
,l
: k
==i
and l
==j
)
632 Sij
= Eij
+ Eij
.transpose()
636 return vector(field
, m
.list())
639 return matrix(field
, dimension
, v
.list())
641 W
= V
.span( mat2vec(s
) for s
in S
)
643 # Taking the span above reorders our basis (thanks, jerk!) so we
644 # need to put our "matrix basis" in the same order as the
645 # (reordered) vector basis.
646 S
= [ vec2mat(b
) for b
in W
.basis() ]
649 # Brute force the multiplication-by-s matrix by looping
650 # through all elements of the basis and doing the computation
651 # to find out what the corresponding row should be. BEWARE:
652 # these multiplication tables won't be symmetric! It therefore
653 # becomes REALLY IMPORTANT that the underlying algebra
654 # constructor uses ROW vectors and not COLUMN vectors. That's
655 # why we're computing rows here and not columns.
658 this_row
= mat2vec((s
*t
+ t
*s
)/2)
659 Q_rows
.append(W
.coordinates(this_row
))
660 Q
= matrix(field
,Q_rows
)
663 return FiniteDimensionalEuclideanJordanAlgebra(field
,Qs
,rank
=dimension
)
668 Return a "random" finite-dimensional Euclidean Jordan Algebra.
672 For now, we choose a random natural number ``n`` (greater than zero)
673 and then give you back one of the following:
675 * The cartesian product of the rational numbers ``n`` times; this is
676 ``QQ^n`` with the Hadamard product.
678 * The Jordan spin algebra on ``QQ^n``.
680 * The ``n``-by-``n`` rational symmetric matrices with the symmetric
683 Later this might be extended to return Cartesian products of the
689 Euclidean Jordan algebra of degree...
692 n
= ZZ
.random_element(1,10).abs()
693 constructor
= choice([eja_rn
, eja_ln
, eja_sn
])
694 return constructor(dimension
=n
, field
=QQ
)