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,
26 mult_table
= [b
.base_extend(field
) for b
in mult_table
]
29 if not (is_Matrix(b
) and b
.dimensions() == (n
, n
)):
30 raise ValueError("input is not a multiplication table")
31 mult_table
= tuple(mult_table
)
33 cat
= MagmaticAlgebras(field
).FiniteDimensional().WithBasis()
34 cat
.or_subcategory(category
)
35 if assume_associative
:
36 cat
= cat
.Associative()
38 names
= normalize_names(n
, names
)
40 fda
= super(FiniteDimensionalEuclideanJordanAlgebra
, cls
)
41 return fda
.__classcall
__(cls
,
44 assume_associative
=assume_associative
,
48 natural_basis
=natural_basis
)
55 assume_associative
=False,
62 By definition, Jordan multiplication commutes::
64 sage: set_random_seed()
65 sage: J = random_eja()
66 sage: x = J.random_element()
67 sage: y = J.random_element()
73 self
._natural
_basis
= natural_basis
74 self
._multiplication
_table
= mult_table
75 fda
= super(FiniteDimensionalEuclideanJordanAlgebra
, self
)
84 Return a string representation of ``self``.
86 fmt
= "Euclidean Jordan algebra of degree {} over {}"
87 return fmt
.format(self
.degree(), self
.base_ring())
90 def _a_regular_element(self
):
92 Guess a regular element. Needed to compute the basis for our
93 characteristic polynomial coefficients.
96 z
= self
.sum( (i
+1)*gs
[i
] for i
in range(len(gs
)) )
97 if not z
.is_regular():
98 raise ValueError("don't know a regular element")
103 def _charpoly_basis_space(self
):
105 Return the vector space spanned by the basis used in our
106 characteristic polynomial coefficients. This is used not only to
107 compute those coefficients, but also any time we need to
108 evaluate the coefficients (like when we compute the trace or
111 z
= self
._a
_regular
_element
()
112 V
= z
.vector().parent().ambient_vector_space()
113 V1
= V
.span_of_basis( (z
**k
).vector() for k
in range(self
.rank()) )
114 b
= (V1
.basis() + V1
.complement().basis())
115 return V
.span_of_basis(b
)
119 def _charpoly_coeff(self
, i
):
121 Return the coefficient polynomial "a_{i}" of this algebra's
122 general characteristic polynomial.
124 Having this be a separate cached method lets us compute and
125 store the trace/determinant (a_{r-1} and a_{0} respectively)
126 separate from the entire characteristic polynomial.
128 (A_of_x
, x
, xr
, detA
) = self
._charpoly
_matrix
_system
()
129 R
= A_of_x
.base_ring()
131 # Guaranteed by theory
134 # Danger: the in-place modification is done for performance
135 # reasons (reconstructing a matrix with huge polynomial
136 # entries is slow), but I don't know how cached_method works,
137 # so it's highly possible that we're modifying some global
138 # list variable by reference, here. In other words, you
139 # probably shouldn't call this method twice on the same
140 # algebra, at the same time, in two threads
141 Ai_orig
= A_of_x
.column(i
)
142 A_of_x
.set_column(i
,xr
)
143 numerator
= A_of_x
.det()
144 A_of_x
.set_column(i
,Ai_orig
)
146 # We're relying on the theory here to ensure that each a_i is
147 # indeed back in R, and the added negative signs are to make
148 # the whole charpoly expression sum to zero.
149 return R(-numerator
/detA
)
153 def _charpoly_matrix_system(self
):
155 Compute the matrix whose entries A_ij are polynomials in
156 X1,...,XN, the vector ``x`` of variables X1,...,XN, the vector
157 corresponding to `x^r` and the determinent of the matrix A =
158 [A_ij]. In other words, all of the fixed (cachable) data needed
159 to compute the coefficients of the characteristic polynomial.
164 # Construct a new algebra over a multivariate polynomial ring...
165 names
= ['X' + str(i
) for i
in range(1,n
+1)]
166 R
= PolynomialRing(self
.base_ring(), names
)
167 J
= FiniteDimensionalEuclideanJordanAlgebra(R
,
168 self
._multiplication
_table
,
171 idmat
= identity_matrix(J
.base_ring(), n
)
173 W
= self
._charpoly
_basis
_space
()
174 W
= W
.change_ring(R
.fraction_field())
176 # Starting with the standard coordinates x = (X1,X2,...,Xn)
177 # and then converting the entries to W-coordinates allows us
178 # to pass in the standard coordinates to the charpoly and get
179 # back the right answer. Specifically, with x = (X1,X2,...,Xn),
182 # W.coordinates(x^2) eval'd at (standard z-coords)
186 # W-coords of (standard coords of x^2 eval'd at std-coords of z)
188 # We want the middle equivalent thing in our matrix, but use
189 # the first equivalent thing instead so that we can pass in
190 # standard coordinates.
191 x
= J(vector(R
, R
.gens()))
192 l1
= [column_matrix(W
.coordinates((x
**k
).vector())) for k
in range(r
)]
193 l2
= [idmat
.column(k
-1).column() for k
in range(r
+1, n
+1)]
194 A_of_x
= block_matrix(R
, 1, n
, (l1
+ l2
))
195 xr
= W
.coordinates((x
**r
).vector())
196 return (A_of_x
, x
, xr
, A_of_x
.det())
200 def characteristic_polynomial(self
):
205 This implementation doesn't guarantee that the polynomial
206 denominator in the coefficients is not identically zero, so
207 theoretically it could crash. The way that this is handled
208 in e.g. Faraut and Koranyi is to use a basis that guarantees
209 the denominator is non-zero. But, doing so requires knowledge
210 of at least one regular element, and we don't even know how
211 to do that. The trade-off is that, if we use the standard basis,
212 the resulting polynomial will accept the "usual" coordinates. In
213 other words, we don't have to do a change of basis before e.g.
214 computing the trace or determinant.
218 The characteristic polynomial in the spin algebra is given in
219 Alizadeh, Example 11.11::
221 sage: J = JordanSpinEJA(3)
222 sage: p = J.characteristic_polynomial(); p
223 X1^2 - X2^2 - X3^2 + (-2*t)*X1 + t^2
224 sage: xvec = J.one().vector()
232 # The list of coefficient polynomials a_1, a_2, ..., a_n.
233 a
= [ self
._charpoly
_coeff
(i
) for i
in range(n
) ]
235 # We go to a bit of trouble here to reorder the
236 # indeterminates, so that it's easier to evaluate the
237 # characteristic polynomial at x's coordinates and get back
238 # something in terms of t, which is what we want.
240 S
= PolynomialRing(self
.base_ring(),'t')
242 S
= PolynomialRing(S
, R
.variable_names())
245 # Note: all entries past the rth should be zero. The
246 # coefficient of the highest power (x^r) is 1, but it doesn't
247 # appear in the solution vector which contains coefficients
248 # for the other powers (to make them sum to x^r).
250 a
[r
] = 1 # corresponds to x^r
252 # When the rank is equal to the dimension, trying to
253 # assign a[r] goes out-of-bounds.
254 a
.append(1) # corresponds to x^r
256 return sum( a
[k
]*(t
**k
) for k
in range(len(a
)) )
259 def inner_product(self
, x
, y
):
261 The inner product associated with this Euclidean Jordan algebra.
263 Defaults to the trace inner product, but can be overridden by
264 subclasses if they are sure that the necessary properties are
269 The inner product must satisfy its axiom for this algebra to truly
270 be a Euclidean Jordan Algebra::
272 sage: set_random_seed()
273 sage: J = random_eja()
274 sage: x = J.random_element()
275 sage: y = J.random_element()
276 sage: z = J.random_element()
277 sage: (x*y).inner_product(z) == y.inner_product(x*z)
281 if (not x
in self
) or (not y
in self
):
282 raise TypeError("arguments must live in this algebra")
283 return x
.trace_inner_product(y
)
286 def natural_basis(self
):
288 Return a more-natural representation of this algebra's basis.
290 Every finite-dimensional Euclidean Jordan Algebra is a direct
291 sum of five simple algebras, four of which comprise Hermitian
292 matrices. This method returns the original "natural" basis
293 for our underlying vector space. (Typically, the natural basis
294 is used to construct the multiplication table in the first place.)
296 Note that this will always return a matrix. The standard basis
297 in `R^n` will be returned as `n`-by-`1` column matrices.
301 sage: J = RealSymmetricEJA(2)
304 sage: J.natural_basis()
312 sage: J = JordanSpinEJA(2)
315 sage: J.natural_basis()
322 if self
._natural
_basis
is None:
323 return tuple( b
.vector().column() for b
in self
.basis() )
325 return self
._natural
_basis
330 Return the rank of this EJA.
332 if self
._rank
is None:
333 raise ValueError("no rank specified at genesis")
338 class Element(FiniteDimensionalAlgebraElement
):
340 An element of a Euclidean Jordan algebra.
345 Oh man, I should not be doing this. This hides the "disabled"
346 methods ``left_matrix`` and ``matrix`` from introspection;
347 in particular it removes them from tab-completion.
349 return filter(lambda s
: s
not in ['left_matrix', 'matrix'],
350 dir(self
.__class
__) )
353 def __init__(self
, A
, elt
=None):
357 The identity in `S^n` is converted to the identity in the EJA::
359 sage: J = RealSymmetricEJA(3)
360 sage: I = identity_matrix(QQ,3)
361 sage: J(I) == J.one()
364 This skew-symmetric matrix can't be represented in the EJA::
366 sage: J = RealSymmetricEJA(3)
367 sage: A = matrix(QQ,3, lambda i,j: i-j)
369 Traceback (most recent call last):
371 ArithmeticError: vector is not in free module
374 # Goal: if we're given a matrix, and if it lives in our
375 # parent algebra's "natural ambient space," convert it
376 # into an algebra element.
378 # The catch is, we make a recursive call after converting
379 # the given matrix into a vector that lives in the algebra.
380 # This we need to try the parent class initializer first,
381 # to avoid recursing forever if we're given something that
382 # already fits into the algebra, but also happens to live
383 # in the parent's "natural ambient space" (this happens with
386 FiniteDimensionalAlgebraElement
.__init
__(self
, A
, elt
)
388 natural_basis
= A
.natural_basis()
389 if elt
in natural_basis
[0].matrix_space():
390 # Thanks for nothing! Matrix spaces aren't vector
391 # spaces in Sage, so we have to figure out its
392 # natural-basis coordinates ourselves.
393 V
= VectorSpace(elt
.base_ring(), elt
.nrows()**2)
394 W
= V
.span( _mat2vec(s
) for s
in natural_basis
)
395 coords
= W
.coordinates(_mat2vec(elt
))
396 FiniteDimensionalAlgebraElement
.__init
__(self
, A
, coords
)
398 def __pow__(self
, n
):
400 Return ``self`` raised to the power ``n``.
402 Jordan algebras are always power-associative; see for
403 example Faraut and Koranyi, Proposition II.1.2 (ii).
407 We have to override this because our superclass uses row vectors
408 instead of column vectors! We, on the other hand, assume column
413 sage: set_random_seed()
414 sage: x = random_eja().random_element()
415 sage: x.operator_matrix()*x.vector() == (x^2).vector()
418 A few examples of power-associativity::
420 sage: set_random_seed()
421 sage: x = random_eja().random_element()
422 sage: x*(x*x)*(x*x) == x^5
424 sage: (x*x)*(x*x*x) == x^5
427 We also know that powers operator-commute (Koecher, Chapter
430 sage: set_random_seed()
431 sage: x = random_eja().random_element()
432 sage: m = ZZ.random_element(0,10)
433 sage: n = ZZ.random_element(0,10)
434 sage: Lxm = (x^m).operator_matrix()
435 sage: Lxn = (x^n).operator_matrix()
436 sage: Lxm*Lxn == Lxn*Lxm
446 return A( (self
.operator_matrix()**(n
-1))*self
.vector() )
449 def apply_univariate_polynomial(self
, p
):
451 Apply the univariate polynomial ``p`` to this element.
453 A priori, SageMath won't allow us to apply a univariate
454 polynomial to an element of an EJA, because we don't know
455 that EJAs are rings (they are usually not associative). Of
456 course, we know that EJAs are power-associative, so the
457 operation is ultimately kosher. This function sidesteps
458 the CAS to get the answer we want and expect.
462 sage: R = PolynomialRing(QQ, 't')
464 sage: p = t^4 - t^3 + 5*t - 2
465 sage: J = RealCartesianProductEJA(5)
466 sage: J.one().apply_univariate_polynomial(p) == 3*J.one()
471 We should always get back an element of the algebra::
473 sage: set_random_seed()
474 sage: p = PolynomialRing(QQ, 't').random_element()
475 sage: J = random_eja()
476 sage: x = J.random_element()
477 sage: x.apply_univariate_polynomial(p) in J
481 if len(p
.variables()) > 1:
482 raise ValueError("not a univariate polynomial")
485 # Convert the coeficcients to the parent's base ring,
486 # because a priori they might live in an (unnecessarily)
487 # larger ring for which P.sum() would fail below.
488 cs
= [ R(c
) for c
in p
.coefficients(sparse
=False) ]
489 return P
.sum( cs
[k
]*(self
**k
) for k
in range(len(cs
)) )
492 def characteristic_polynomial(self
):
494 Return the characteristic polynomial of this element.
498 The rank of `R^3` is three, and the minimal polynomial of
499 the identity element is `(t-1)` from which it follows that
500 the characteristic polynomial should be `(t-1)^3`::
502 sage: J = RealCartesianProductEJA(3)
503 sage: J.one().characteristic_polynomial()
504 t^3 - 3*t^2 + 3*t - 1
506 Likewise, the characteristic of the zero element in the
507 rank-three algebra `R^{n}` should be `t^{3}`::
509 sage: J = RealCartesianProductEJA(3)
510 sage: J.zero().characteristic_polynomial()
513 The characteristic polynomial of an element should evaluate
514 to zero on that element::
516 sage: set_random_seed()
517 sage: x = RealCartesianProductEJA(3).random_element()
518 sage: p = x.characteristic_polynomial()
519 sage: x.apply_univariate_polynomial(p)
523 p
= self
.parent().characteristic_polynomial()
524 return p(*self
.vector())
527 def inner_product(self
, other
):
529 Return the parent algebra's inner product of myself and ``other``.
533 The inner product in the Jordan spin algebra is the usual
534 inner product on `R^n` (this example only works because the
535 basis for the Jordan algebra is the standard basis in `R^n`)::
537 sage: J = JordanSpinEJA(3)
538 sage: x = vector(QQ,[1,2,3])
539 sage: y = vector(QQ,[4,5,6])
540 sage: x.inner_product(y)
542 sage: J(x).inner_product(J(y))
545 The inner product on `S^n` is `<X,Y> = trace(X*Y)`, where
546 multiplication is the usual matrix multiplication in `S^n`,
547 so the inner product of the identity matrix with itself
550 sage: J = RealSymmetricEJA(3)
551 sage: J.one().inner_product(J.one())
554 Likewise, the inner product on `C^n` is `<X,Y> =
555 Re(trace(X*Y))`, where we must necessarily take the real
556 part because the product of Hermitian matrices may not be
559 sage: J = ComplexHermitianEJA(3)
560 sage: J.one().inner_product(J.one())
563 Ditto for the quaternions::
565 sage: J = QuaternionHermitianEJA(3)
566 sage: J.one().inner_product(J.one())
571 Ensure that we can always compute an inner product, and that
572 it gives us back a real number::
574 sage: set_random_seed()
575 sage: J = random_eja()
576 sage: x = J.random_element()
577 sage: y = J.random_element()
578 sage: x.inner_product(y) in RR
584 raise TypeError("'other' must live in the same algebra")
586 return P
.inner_product(self
, other
)
589 def operator_commutes_with(self
, other
):
591 Return whether or not this element operator-commutes
596 The definition of a Jordan algebra says that any element
597 operator-commutes with its square::
599 sage: set_random_seed()
600 sage: x = random_eja().random_element()
601 sage: x.operator_commutes_with(x^2)
606 Test Lemma 1 from Chapter III of Koecher::
608 sage: set_random_seed()
609 sage: J = random_eja()
610 sage: u = J.random_element()
611 sage: v = J.random_element()
612 sage: lhs = u.operator_commutes_with(u*v)
613 sage: rhs = v.operator_commutes_with(u^2)
618 if not other
in self
.parent():
619 raise TypeError("'other' must live in the same algebra")
621 A
= self
.operator_matrix()
622 B
= other
.operator_matrix()
628 Return my determinant, the product of my eigenvalues.
632 sage: J = JordanSpinEJA(2)
633 sage: e0,e1 = J.gens()
634 sage: x = sum( J.gens() )
640 sage: J = JordanSpinEJA(3)
641 sage: e0,e1,e2 = J.gens()
642 sage: x = sum( J.gens() )
648 An element is invertible if and only if its determinant is
651 sage: set_random_seed()
652 sage: x = random_eja().random_element()
653 sage: x.is_invertible() == (x.det() != 0)
659 p
= P
._charpoly
_coeff
(0)
660 # The _charpoly_coeff function already adds the factor of
661 # -1 to ensure that _charpoly_coeff(0) is really what
662 # appears in front of t^{0} in the charpoly. However,
663 # we want (-1)^r times THAT for the determinant.
664 return ((-1)**r
)*p(*self
.vector())
669 Return the Jordan-multiplicative inverse of this element.
673 We appeal to the quadratic representation as in Koecher's
674 Theorem 12 in Chapter III, Section 5.
678 The inverse in the spin factor algebra is given in Alizadeh's
681 sage: set_random_seed()
682 sage: n = ZZ.random_element(1,10)
683 sage: J = JordanSpinEJA(n)
684 sage: x = J.random_element()
685 sage: while x.is_zero():
686 ....: x = J.random_element()
687 sage: x_vec = x.vector()
689 sage: x_bar = x_vec[1:]
690 sage: coeff = 1/(x0^2 - x_bar.inner_product(x_bar))
691 sage: inv_vec = x_vec.parent()([x0] + (-x_bar).list())
692 sage: x_inverse = coeff*inv_vec
693 sage: x.inverse() == J(x_inverse)
698 The identity element is its own inverse::
700 sage: set_random_seed()
701 sage: J = random_eja()
702 sage: J.one().inverse() == J.one()
705 If an element has an inverse, it acts like one::
707 sage: set_random_seed()
708 sage: J = random_eja()
709 sage: x = J.random_element()
710 sage: (not x.is_invertible()) or (x.inverse()*x == J.one())
713 The inverse of the inverse is what we started with::
715 sage: set_random_seed()
716 sage: J = random_eja()
717 sage: x = J.random_element()
718 sage: (not x.is_invertible()) or (x.inverse().inverse() == x)
721 The zero element is never invertible::
723 sage: set_random_seed()
724 sage: J = random_eja().zero().inverse()
725 Traceback (most recent call last):
727 ValueError: element is not invertible
730 if not self
.is_invertible():
731 raise ValueError("element is not invertible")
734 return P(self
.quadratic_representation().inverse()*self
.vector())
737 def is_invertible(self
):
739 Return whether or not this element is invertible.
741 We can't use the superclass method because it relies on
742 the algebra being associative.
746 The usual way to do this is to check if the determinant is
747 zero, but we need the characteristic polynomial for the
748 determinant. The minimal polynomial is a lot easier to get,
749 so we use Corollary 2 in Chapter V of Koecher to check
750 whether or not the paren't algebra's zero element is a root
751 of this element's minimal polynomial.
755 The identity element is always invertible::
757 sage: set_random_seed()
758 sage: J = random_eja()
759 sage: J.one().is_invertible()
762 The zero element is never invertible::
764 sage: set_random_seed()
765 sage: J = random_eja()
766 sage: J.zero().is_invertible()
770 zero
= self
.parent().zero()
771 p
= self
.minimal_polynomial()
772 return not (p(zero
) == zero
)
775 def is_nilpotent(self
):
777 Return whether or not some power of this element is zero.
779 The superclass method won't work unless we're in an
780 associative algebra, and we aren't. However, we generate
781 an assocoative subalgebra and we're nilpotent there if and
782 only if we're nilpotent here (probably).
786 The identity element is never nilpotent::
788 sage: set_random_seed()
789 sage: random_eja().one().is_nilpotent()
792 The additive identity is always nilpotent::
794 sage: set_random_seed()
795 sage: random_eja().zero().is_nilpotent()
799 # The element we're going to call "is_nilpotent()" on.
800 # Either myself, interpreted as an element of a finite-
801 # dimensional algebra, or an element of an associative
805 if self
.parent().is_associative():
806 elt
= FiniteDimensionalAlgebraElement(self
.parent(), self
)
808 V
= self
.span_of_powers()
809 assoc_subalg
= self
.subalgebra_generated_by()
810 # Mis-design warning: the basis used for span_of_powers()
811 # and subalgebra_generated_by() must be the same, and in
813 elt
= assoc_subalg(V
.coordinates(self
.vector()))
815 # Recursive call, but should work since elt lives in an
816 # associative algebra.
817 return elt
.is_nilpotent()
820 def is_regular(self
):
822 Return whether or not this is a regular element.
826 The identity element always has degree one, but any element
827 linearly-independent from it is regular::
829 sage: J = JordanSpinEJA(5)
830 sage: J.one().is_regular()
832 sage: e0, e1, e2, e3, e4 = J.gens() # e0 is the identity
833 sage: for x in J.gens():
834 ....: (J.one() + x).is_regular()
842 return self
.degree() == self
.parent().rank()
847 Compute the degree of this element the straightforward way
848 according to the definition; by appending powers to a list
849 and figuring out its dimension (that is, whether or not
850 they're linearly dependent).
854 sage: J = JordanSpinEJA(4)
855 sage: J.one().degree()
857 sage: e0,e1,e2,e3 = J.gens()
858 sage: (e0 - e1).degree()
861 In the spin factor algebra (of rank two), all elements that
862 aren't multiples of the identity are regular::
864 sage: set_random_seed()
865 sage: n = ZZ.random_element(1,10)
866 sage: J = JordanSpinEJA(n)
867 sage: x = J.random_element()
868 sage: x == x.coefficient(0)*J.one() or x.degree() == 2
872 return self
.span_of_powers().dimension()
875 def left_matrix(self
):
877 Our parent class defines ``left_matrix`` and ``matrix``
878 methods whose names are misleading. We don't want them.
880 raise NotImplementedError("use operator_matrix() instead")
885 def minimal_polynomial(self
):
887 Return the minimal polynomial of this element,
888 as a function of the variable `t`.
892 We restrict ourselves to the associative subalgebra
893 generated by this element, and then return the minimal
894 polynomial of this element's operator matrix (in that
895 subalgebra). This works by Baes Proposition 2.3.16.
899 The minimal polynomial of the identity and zero elements are
902 sage: set_random_seed()
903 sage: J = random_eja()
904 sage: J.one().minimal_polynomial()
906 sage: J.zero().minimal_polynomial()
909 The degree of an element is (by one definition) the degree
910 of its minimal polynomial::
912 sage: set_random_seed()
913 sage: x = random_eja().random_element()
914 sage: x.degree() == x.minimal_polynomial().degree()
917 The minimal polynomial and the characteristic polynomial coincide
918 and are known (see Alizadeh, Example 11.11) for all elements of
919 the spin factor algebra that aren't scalar multiples of the
922 sage: set_random_seed()
923 sage: n = ZZ.random_element(2,10)
924 sage: J = JordanSpinEJA(n)
925 sage: y = J.random_element()
926 sage: while y == y.coefficient(0)*J.one():
927 ....: y = J.random_element()
928 sage: y0 = y.vector()[0]
929 sage: y_bar = y.vector()[1:]
930 sage: actual = y.minimal_polynomial()
931 sage: t = PolynomialRing(J.base_ring(),'t').gen(0)
932 sage: expected = t^2 - 2*y0*t + (y0^2 - norm(y_bar)^2)
933 sage: bool(actual == expected)
936 The minimal polynomial should always kill its element::
938 sage: set_random_seed()
939 sage: x = random_eja().random_element()
940 sage: p = x.minimal_polynomial()
941 sage: x.apply_univariate_polynomial(p)
945 V
= self
.span_of_powers()
946 assoc_subalg
= self
.subalgebra_generated_by()
947 # Mis-design warning: the basis used for span_of_powers()
948 # and subalgebra_generated_by() must be the same, and in
950 elt
= assoc_subalg(V
.coordinates(self
.vector()))
952 # We get back a symbolic polynomial in 'x' but want a real
954 p_of_x
= elt
.operator_matrix().minimal_polynomial()
955 return p_of_x
.change_variable_name('t')
958 def natural_representation(self
):
960 Return a more-natural representation of this element.
962 Every finite-dimensional Euclidean Jordan Algebra is a
963 direct sum of five simple algebras, four of which comprise
964 Hermitian matrices. This method returns the original
965 "natural" representation of this element as a Hermitian
966 matrix, if it has one. If not, you get the usual representation.
970 sage: J = ComplexHermitianEJA(3)
973 sage: J.one().natural_representation()
983 sage: J = QuaternionHermitianEJA(3)
986 sage: J.one().natural_representation()
987 [1 0 0 0 0 0 0 0 0 0 0 0]
988 [0 1 0 0 0 0 0 0 0 0 0 0]
989 [0 0 1 0 0 0 0 0 0 0 0 0]
990 [0 0 0 1 0 0 0 0 0 0 0 0]
991 [0 0 0 0 1 0 0 0 0 0 0 0]
992 [0 0 0 0 0 1 0 0 0 0 0 0]
993 [0 0 0 0 0 0 1 0 0 0 0 0]
994 [0 0 0 0 0 0 0 1 0 0 0 0]
995 [0 0 0 0 0 0 0 0 1 0 0 0]
996 [0 0 0 0 0 0 0 0 0 1 0 0]
997 [0 0 0 0 0 0 0 0 0 0 1 0]
998 [0 0 0 0 0 0 0 0 0 0 0 1]
1001 B
= self
.parent().natural_basis()
1002 W
= B
[0].matrix_space()
1003 return W
.linear_combination(zip(self
.vector(), B
))
1006 def operator_matrix(self
):
1008 Return the matrix that represents left- (or right-)
1009 multiplication by this element in the parent algebra.
1011 We implement this ourselves to work around the fact that
1012 our parent class represents everything with row vectors.
1016 Test the first polarization identity from my notes, Koecher Chapter
1017 III, or from Baes (2.3)::
1019 sage: set_random_seed()
1020 sage: J = random_eja()
1021 sage: x = J.random_element()
1022 sage: y = J.random_element()
1023 sage: Lx = x.operator_matrix()
1024 sage: Ly = y.operator_matrix()
1025 sage: Lxx = (x*x).operator_matrix()
1026 sage: Lxy = (x*y).operator_matrix()
1027 sage: bool(2*Lx*Lxy + Ly*Lxx == 2*Lxy*Lx + Lxx*Ly)
1030 Test the second polarization identity from my notes or from
1033 sage: set_random_seed()
1034 sage: J = random_eja()
1035 sage: x = J.random_element()
1036 sage: y = J.random_element()
1037 sage: z = J.random_element()
1038 sage: Lx = x.operator_matrix()
1039 sage: Ly = y.operator_matrix()
1040 sage: Lz = z.operator_matrix()
1041 sage: Lzy = (z*y).operator_matrix()
1042 sage: Lxy = (x*y).operator_matrix()
1043 sage: Lxz = (x*z).operator_matrix()
1044 sage: bool(Lx*Lzy + Lz*Lxy + Ly*Lxz == Lzy*Lx + Lxy*Lz + Lxz*Ly)
1047 Test the third polarization identity from my notes or from
1050 sage: set_random_seed()
1051 sage: J = random_eja()
1052 sage: u = J.random_element()
1053 sage: y = J.random_element()
1054 sage: z = J.random_element()
1055 sage: Lu = u.operator_matrix()
1056 sage: Ly = y.operator_matrix()
1057 sage: Lz = z.operator_matrix()
1058 sage: Lzy = (z*y).operator_matrix()
1059 sage: Luy = (u*y).operator_matrix()
1060 sage: Luz = (u*z).operator_matrix()
1061 sage: Luyz = (u*(y*z)).operator_matrix()
1062 sage: lhs = Lu*Lzy + Lz*Luy + Ly*Luz
1063 sage: rhs = Luyz + Ly*Lu*Lz + Lz*Lu*Ly
1064 sage: bool(lhs == rhs)
1068 fda_elt
= FiniteDimensionalAlgebraElement(self
.parent(), self
)
1069 return fda_elt
.matrix().transpose()
1072 def quadratic_representation(self
, other
=None):
1074 Return the quadratic representation of this element.
1078 The explicit form in the spin factor algebra is given by
1079 Alizadeh's Example 11.12::
1081 sage: set_random_seed()
1082 sage: n = ZZ.random_element(1,10)
1083 sage: J = JordanSpinEJA(n)
1084 sage: x = J.random_element()
1085 sage: x_vec = x.vector()
1087 sage: x_bar = x_vec[1:]
1088 sage: A = matrix(QQ, 1, [x_vec.inner_product(x_vec)])
1089 sage: B = 2*x0*x_bar.row()
1090 sage: C = 2*x0*x_bar.column()
1091 sage: D = identity_matrix(QQ, n-1)
1092 sage: D = (x0^2 - x_bar.inner_product(x_bar))*D
1093 sage: D = D + 2*x_bar.tensor_product(x_bar)
1094 sage: Q = block_matrix(2,2,[A,B,C,D])
1095 sage: Q == x.quadratic_representation()
1098 Test all of the properties from Theorem 11.2 in Alizadeh::
1100 sage: set_random_seed()
1101 sage: J = random_eja()
1102 sage: x = J.random_element()
1103 sage: y = J.random_element()
1104 sage: Lx = x.operator_matrix()
1105 sage: Lxx = (x*x).operator_matrix()
1106 sage: Qx = x.quadratic_representation()
1107 sage: Qy = y.quadratic_representation()
1108 sage: Qxy = x.quadratic_representation(y)
1109 sage: Qex = J.one().quadratic_representation(x)
1110 sage: n = ZZ.random_element(10)
1111 sage: Qxn = (x^n).quadratic_representation()
1115 sage: 2*Qxy == (x+y).quadratic_representation() - Qx - Qy
1120 sage: alpha = QQ.random_element()
1121 sage: (alpha*x).quadratic_representation() == (alpha^2)*Qx
1126 sage: not x.is_invertible() or (
1127 ....: Qx*x.inverse().vector() == x.vector() )
1130 sage: not x.is_invertible() or (
1133 ....: x.inverse().quadratic_representation() )
1136 sage: Qxy*(J.one().vector()) == (x*y).vector()
1141 sage: not x.is_invertible() or (
1142 ....: x.quadratic_representation(x.inverse())*Qx
1143 ....: == Qx*x.quadratic_representation(x.inverse()) )
1146 sage: not x.is_invertible() or (
1147 ....: x.quadratic_representation(x.inverse())*Qx
1149 ....: 2*x.operator_matrix()*Qex - Qx )
1152 sage: 2*x.operator_matrix()*Qex - Qx == Lxx
1157 sage: J(Qy*x.vector()).quadratic_representation() == Qy*Qx*Qy
1167 sage: not x.is_invertible() or (
1168 ....: Qx*x.inverse().operator_matrix() == Lx )
1173 sage: not x.operator_commutes_with(y) or (
1174 ....: J(Qx*y.vector())^n == J(Qxn*(y^n).vector()) )
1180 elif not other
in self
.parent():
1181 raise TypeError("'other' must live in the same algebra")
1183 L
= self
.operator_matrix()
1184 M
= other
.operator_matrix()
1185 return ( L
*M
+ M
*L
- (self
*other
).operator_matrix() )
1188 def span_of_powers(self
):
1190 Return the vector space spanned by successive powers of
1193 # The dimension of the subalgebra can't be greater than
1194 # the big algebra, so just put everything into a list
1195 # and let span() get rid of the excess.
1197 # We do the extra ambient_vector_space() in case we're messing
1198 # with polynomials and the direct parent is a module.
1199 V
= self
.vector().parent().ambient_vector_space()
1200 return V
.span( (self
**d
).vector() for d
in xrange(V
.dimension()) )
1203 def subalgebra_generated_by(self
):
1205 Return the associative subalgebra of the parent EJA generated
1210 sage: set_random_seed()
1211 sage: x = random_eja().random_element()
1212 sage: x.subalgebra_generated_by().is_associative()
1215 Squaring in the subalgebra should be the same thing as
1216 squaring in the superalgebra::
1218 sage: set_random_seed()
1219 sage: x = random_eja().random_element()
1220 sage: u = x.subalgebra_generated_by().random_element()
1221 sage: u.operator_matrix()*u.vector() == (u**2).vector()
1225 # First get the subspace spanned by the powers of myself...
1226 V
= self
.span_of_powers()
1227 F
= self
.base_ring()
1229 # Now figure out the entries of the right-multiplication
1230 # matrix for the successive basis elements b0, b1,... of
1233 for b_right
in V
.basis():
1234 eja_b_right
= self
.parent()(b_right
)
1236 # The first row of the right-multiplication matrix by
1237 # b1 is what we get if we apply that matrix to b1. The
1238 # second row of the right multiplication matrix by b1
1239 # is what we get when we apply that matrix to b2...
1241 # IMPORTANT: this assumes that all vectors are COLUMN
1242 # vectors, unlike our superclass (which uses row vectors).
1243 for b_left
in V
.basis():
1244 eja_b_left
= self
.parent()(b_left
)
1245 # Multiply in the original EJA, but then get the
1246 # coordinates from the subalgebra in terms of its
1248 this_row
= V
.coordinates((eja_b_left
*eja_b_right
).vector())
1249 b_right_rows
.append(this_row
)
1250 b_right_matrix
= matrix(F
, b_right_rows
)
1251 mats
.append(b_right_matrix
)
1253 # It's an algebra of polynomials in one element, and EJAs
1254 # are power-associative.
1256 # TODO: choose generator names intelligently.
1257 return FiniteDimensionalEuclideanJordanAlgebra(F
, mats
, assume_associative
=True, names
='f')
1260 def subalgebra_idempotent(self
):
1262 Find an idempotent in the associative subalgebra I generate
1263 using Proposition 2.3.5 in Baes.
1267 sage: set_random_seed()
1268 sage: J = random_eja()
1269 sage: x = J.random_element()
1270 sage: while x.is_nilpotent():
1271 ....: x = J.random_element()
1272 sage: c = x.subalgebra_idempotent()
1277 if self
.is_nilpotent():
1278 raise ValueError("this only works with non-nilpotent elements!")
1280 V
= self
.span_of_powers()
1281 J
= self
.subalgebra_generated_by()
1282 # Mis-design warning: the basis used for span_of_powers()
1283 # and subalgebra_generated_by() must be the same, and in
1285 u
= J(V
.coordinates(self
.vector()))
1287 # The image of the matrix of left-u^m-multiplication
1288 # will be minimal for some natural number s...
1290 minimal_dim
= V
.dimension()
1291 for i
in xrange(1, V
.dimension()):
1292 this_dim
= (u
**i
).operator_matrix().image().dimension()
1293 if this_dim
< minimal_dim
:
1294 minimal_dim
= this_dim
1297 # Now minimal_matrix should correspond to the smallest
1298 # non-zero subspace in Baes's (or really, Koecher's)
1301 # However, we need to restrict the matrix to work on the
1302 # subspace... or do we? Can't we just solve, knowing that
1303 # A(c) = u^(s+1) should have a solution in the big space,
1306 # Beware, solve_right() means that we're using COLUMN vectors.
1307 # Our FiniteDimensionalAlgebraElement superclass uses rows.
1309 A
= u_next
.operator_matrix()
1310 c_coordinates
= A
.solve_right(u_next
.vector())
1312 # Now c_coordinates is the idempotent we want, but it's in
1313 # the coordinate system of the subalgebra.
1315 # We need the basis for J, but as elements of the parent algebra.
1317 basis
= [self
.parent(v
) for v
in V
.basis()]
1318 return self
.parent().linear_combination(zip(c_coordinates
, basis
))
1323 Return my trace, the sum of my eigenvalues.
1327 sage: J = JordanSpinEJA(3)
1328 sage: x = sum(J.gens())
1334 sage: J = RealCartesianProductEJA(5)
1335 sage: J.one().trace()
1340 The trace of an element is a real number::
1342 sage: set_random_seed()
1343 sage: J = random_eja()
1344 sage: J.random_element().trace() in J.base_ring()
1350 p
= P
._charpoly
_coeff
(r
-1)
1351 # The _charpoly_coeff function already adds the factor of
1352 # -1 to ensure that _charpoly_coeff(r-1) is really what
1353 # appears in front of t^{r-1} in the charpoly. However,
1354 # we want the negative of THAT for the trace.
1355 return -p(*self
.vector())
1358 def trace_inner_product(self
, other
):
1360 Return the trace inner product of myself and ``other``.
1364 The trace inner product is commutative::
1366 sage: set_random_seed()
1367 sage: J = random_eja()
1368 sage: x = J.random_element(); y = J.random_element()
1369 sage: x.trace_inner_product(y) == y.trace_inner_product(x)
1372 The trace inner product is bilinear::
1374 sage: set_random_seed()
1375 sage: J = random_eja()
1376 sage: x = J.random_element()
1377 sage: y = J.random_element()
1378 sage: z = J.random_element()
1379 sage: a = QQ.random_element();
1380 sage: actual = (a*(x+z)).trace_inner_product(y)
1381 sage: expected = ( a*x.trace_inner_product(y) +
1382 ....: a*z.trace_inner_product(y) )
1383 sage: actual == expected
1385 sage: actual = x.trace_inner_product(a*(y+z))
1386 sage: expected = ( a*x.trace_inner_product(y) +
1387 ....: a*x.trace_inner_product(z) )
1388 sage: actual == expected
1391 The trace inner product satisfies the compatibility
1392 condition in the definition of a Euclidean Jordan algebra::
1394 sage: set_random_seed()
1395 sage: J = random_eja()
1396 sage: x = J.random_element()
1397 sage: y = J.random_element()
1398 sage: z = J.random_element()
1399 sage: (x*y).trace_inner_product(z) == y.trace_inner_product(x*z)
1403 if not other
in self
.parent():
1404 raise TypeError("'other' must live in the same algebra")
1406 return (self
*other
).trace()
1409 class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1411 Return the Euclidean Jordan Algebra corresponding to the set
1412 `R^n` under the Hadamard product.
1414 Note: this is nothing more than the Cartesian product of ``n``
1415 copies of the spin algebra. Once Cartesian product algebras
1416 are implemented, this can go.
1420 This multiplication table can be verified by hand::
1422 sage: J = RealCartesianProductEJA(3)
1423 sage: e0,e1,e2 = J.gens()
1439 def __classcall_private__(cls
, n
, field
=QQ
):
1440 # The FiniteDimensionalAlgebra constructor takes a list of
1441 # matrices, the ith representing right multiplication by the ith
1442 # basis element in the vector space. So if e_1 = (1,0,0), then
1443 # right (Hadamard) multiplication of x by e_1 picks out the first
1444 # component of x; and likewise for the ith basis element e_i.
1445 Qs
= [ matrix(field
, n
, n
, lambda k
,j
: 1*(k
== j
== i
))
1446 for i
in xrange(n
) ]
1448 fdeja
= super(RealCartesianProductEJA
, cls
)
1449 return fdeja
.__classcall
_private
__(cls
, field
, Qs
, rank
=n
)
1451 def inner_product(self
, x
, y
):
1452 return _usual_ip(x
,y
)
1457 Return a "random" finite-dimensional Euclidean Jordan Algebra.
1461 For now, we choose a random natural number ``n`` (greater than zero)
1462 and then give you back one of the following:
1464 * The cartesian product of the rational numbers ``n`` times; this is
1465 ``QQ^n`` with the Hadamard product.
1467 * The Jordan spin algebra on ``QQ^n``.
1469 * The ``n``-by-``n`` rational symmetric matrices with the symmetric
1472 * The ``n``-by-``n`` complex-rational Hermitian matrices embedded
1473 in the space of ``2n``-by-``2n`` real symmetric matrices.
1475 * The ``n``-by-``n`` quaternion-rational Hermitian matrices embedded
1476 in the space of ``4n``-by-``4n`` real symmetric matrices.
1478 Later this might be extended to return Cartesian products of the
1484 Euclidean Jordan algebra of degree...
1488 # The max_n component lets us choose different upper bounds on the
1489 # value "n" that gets passed to the constructor. This is needed
1490 # because e.g. R^{10} is reasonable to test, while the Hermitian
1491 # 10-by-10 quaternion matrices are not.
1492 (constructor
, max_n
) = choice([(RealCartesianProductEJA
, 6),
1494 (RealSymmetricEJA
, 5),
1495 (ComplexHermitianEJA
, 4),
1496 (QuaternionHermitianEJA
, 3)])
1497 n
= ZZ
.random_element(1, max_n
)
1498 return constructor(n
, field
=QQ
)
1502 def _real_symmetric_basis(n
, field
=QQ
):
1504 Return a basis for the space of real symmetric n-by-n matrices.
1506 # The basis of symmetric matrices, as matrices, in their R^(n-by-n)
1510 for j
in xrange(i
+1):
1511 Eij
= matrix(field
, n
, lambda k
,l
: k
==i
and l
==j
)
1515 # Beware, orthogonal but not normalized!
1516 Sij
= Eij
+ Eij
.transpose()
1521 def _complex_hermitian_basis(n
, field
=QQ
):
1523 Returns a basis for the space of complex Hermitian n-by-n matrices.
1527 sage: set_random_seed()
1528 sage: n = ZZ.random_element(1,5)
1529 sage: all( M.is_symmetric() for M in _complex_hermitian_basis(n) )
1533 F
= QuadraticField(-1, 'I')
1536 # This is like the symmetric case, but we need to be careful:
1538 # * We want conjugate-symmetry, not just symmetry.
1539 # * The diagonal will (as a result) be real.
1543 for j
in xrange(i
+1):
1544 Eij
= matrix(field
, n
, lambda k
,l
: k
==i
and l
==j
)
1546 Sij
= _embed_complex_matrix(Eij
)
1549 # Beware, orthogonal but not normalized! The second one
1550 # has a minus because it's conjugated.
1551 Sij_real
= _embed_complex_matrix(Eij
+ Eij
.transpose())
1553 Sij_imag
= _embed_complex_matrix(I
*Eij
- I
*Eij
.transpose())
1558 def _quaternion_hermitian_basis(n
, field
=QQ
):
1560 Returns a basis for the space of quaternion Hermitian n-by-n matrices.
1564 sage: set_random_seed()
1565 sage: n = ZZ.random_element(1,5)
1566 sage: all( M.is_symmetric() for M in _quaternion_hermitian_basis(n) )
1570 Q
= QuaternionAlgebra(QQ
,-1,-1)
1573 # This is like the symmetric case, but we need to be careful:
1575 # * We want conjugate-symmetry, not just symmetry.
1576 # * The diagonal will (as a result) be real.
1580 for j
in xrange(i
+1):
1581 Eij
= matrix(Q
, n
, lambda k
,l
: k
==i
and l
==j
)
1583 Sij
= _embed_quaternion_matrix(Eij
)
1586 # Beware, orthogonal but not normalized! The second,
1587 # third, and fourth ones have a minus because they're
1589 Sij_real
= _embed_quaternion_matrix(Eij
+ Eij
.transpose())
1591 Sij_I
= _embed_quaternion_matrix(I
*Eij
- I
*Eij
.transpose())
1593 Sij_J
= _embed_quaternion_matrix(J
*Eij
- J
*Eij
.transpose())
1595 Sij_K
= _embed_quaternion_matrix(K
*Eij
- K
*Eij
.transpose())
1601 return vector(m
.base_ring(), m
.list())
1604 return matrix(v
.base_ring(), sqrt(v
.degree()), v
.list())
1606 def _multiplication_table_from_matrix_basis(basis
):
1608 At least three of the five simple Euclidean Jordan algebras have the
1609 symmetric multiplication (A,B) |-> (AB + BA)/2, where the
1610 multiplication on the right is matrix multiplication. Given a basis
1611 for the underlying matrix space, this function returns a
1612 multiplication table (obtained by looping through the basis
1613 elements) for an algebra of those matrices. A reordered copy
1614 of the basis is also returned to work around the fact that
1615 the ``span()`` in this function will change the order of the basis
1616 from what we think it is, to... something else.
1618 # In S^2, for example, we nominally have four coordinates even
1619 # though the space is of dimension three only. The vector space V
1620 # is supposed to hold the entire long vector, and the subspace W
1621 # of V will be spanned by the vectors that arise from symmetric
1622 # matrices. Thus for S^2, dim(V) == 4 and dim(W) == 3.
1623 field
= basis
[0].base_ring()
1624 dimension
= basis
[0].nrows()
1626 V
= VectorSpace(field
, dimension
**2)
1627 W
= V
.span( _mat2vec(s
) for s
in basis
)
1629 # Taking the span above reorders our basis (thanks, jerk!) so we
1630 # need to put our "matrix basis" in the same order as the
1631 # (reordered) vector basis.
1632 S
= tuple( _vec2mat(b
) for b
in W
.basis() )
1636 # Brute force the multiplication-by-s matrix by looping
1637 # through all elements of the basis and doing the computation
1638 # to find out what the corresponding row should be. BEWARE:
1639 # these multiplication tables won't be symmetric! It therefore
1640 # becomes REALLY IMPORTANT that the underlying algebra
1641 # constructor uses ROW vectors and not COLUMN vectors. That's
1642 # why we're computing rows here and not columns.
1645 this_row
= _mat2vec((s
*t
+ t
*s
)/2)
1646 Q_rows
.append(W
.coordinates(this_row
))
1647 Q
= matrix(field
, W
.dimension(), Q_rows
)
1653 def _embed_complex_matrix(M
):
1655 Embed the n-by-n complex matrix ``M`` into the space of real
1656 matrices of size 2n-by-2n via the map the sends each entry `z = a +
1657 bi` to the block matrix ``[[a,b],[-b,a]]``.
1661 sage: F = QuadraticField(-1,'i')
1662 sage: x1 = F(4 - 2*i)
1663 sage: x2 = F(1 + 2*i)
1666 sage: M = matrix(F,2,[[x1,x2],[x3,x4]])
1667 sage: _embed_complex_matrix(M)
1676 Embedding is a homomorphism (isomorphism, in fact)::
1678 sage: set_random_seed()
1679 sage: n = ZZ.random_element(5)
1680 sage: F = QuadraticField(-1, 'i')
1681 sage: X = random_matrix(F, n)
1682 sage: Y = random_matrix(F, n)
1683 sage: actual = _embed_complex_matrix(X) * _embed_complex_matrix(Y)
1684 sage: expected = _embed_complex_matrix(X*Y)
1685 sage: actual == expected
1691 raise ValueError("the matrix 'M' must be square")
1692 field
= M
.base_ring()
1697 blocks
.append(matrix(field
, 2, [[a
,b
],[-b
,a
]]))
1699 # We can drop the imaginaries here.
1700 return block_matrix(field
.base_ring(), n
, blocks
)
1703 def _unembed_complex_matrix(M
):
1705 The inverse of _embed_complex_matrix().
1709 sage: A = matrix(QQ,[ [ 1, 2, 3, 4],
1710 ....: [-2, 1, -4, 3],
1711 ....: [ 9, 10, 11, 12],
1712 ....: [-10, 9, -12, 11] ])
1713 sage: _unembed_complex_matrix(A)
1715 [ 10*i + 9 12*i + 11]
1719 Unembedding is the inverse of embedding::
1721 sage: set_random_seed()
1722 sage: F = QuadraticField(-1, 'i')
1723 sage: M = random_matrix(F, 3)
1724 sage: _unembed_complex_matrix(_embed_complex_matrix(M)) == M
1730 raise ValueError("the matrix 'M' must be square")
1731 if not n
.mod(2).is_zero():
1732 raise ValueError("the matrix 'M' must be a complex embedding")
1734 F
= QuadraticField(-1, 'i')
1737 # Go top-left to bottom-right (reading order), converting every
1738 # 2-by-2 block we see to a single complex element.
1740 for k
in xrange(n
/2):
1741 for j
in xrange(n
/2):
1742 submat
= M
[2*k
:2*k
+2,2*j
:2*j
+2]
1743 if submat
[0,0] != submat
[1,1]:
1744 raise ValueError('bad on-diagonal submatrix')
1745 if submat
[0,1] != -submat
[1,0]:
1746 raise ValueError('bad off-diagonal submatrix')
1747 z
= submat
[0,0] + submat
[0,1]*i
1750 return matrix(F
, n
/2, elements
)
1753 def _embed_quaternion_matrix(M
):
1755 Embed the n-by-n quaternion matrix ``M`` into the space of real
1756 matrices of size 4n-by-4n by first sending each quaternion entry
1757 `z = a + bi + cj + dk` to the block-complex matrix
1758 ``[[a + bi, c+di],[-c + di, a-bi]]`, and then embedding those into
1763 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1764 sage: i,j,k = Q.gens()
1765 sage: x = 1 + 2*i + 3*j + 4*k
1766 sage: M = matrix(Q, 1, [[x]])
1767 sage: _embed_quaternion_matrix(M)
1773 Embedding is a homomorphism (isomorphism, in fact)::
1775 sage: set_random_seed()
1776 sage: n = ZZ.random_element(5)
1777 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1778 sage: X = random_matrix(Q, n)
1779 sage: Y = random_matrix(Q, n)
1780 sage: actual = _embed_quaternion_matrix(X)*_embed_quaternion_matrix(Y)
1781 sage: expected = _embed_quaternion_matrix(X*Y)
1782 sage: actual == expected
1786 quaternions
= M
.base_ring()
1789 raise ValueError("the matrix 'M' must be square")
1791 F
= QuadraticField(-1, 'i')
1796 t
= z
.coefficient_tuple()
1801 cplx_matrix
= matrix(F
, 2, [[ a
+ b
*i
, c
+ d
*i
],
1802 [-c
+ d
*i
, a
- b
*i
]])
1803 blocks
.append(_embed_complex_matrix(cplx_matrix
))
1805 # We should have real entries by now, so use the realest field
1806 # we've got for the return value.
1807 return block_matrix(quaternions
.base_ring(), n
, blocks
)
1810 def _unembed_quaternion_matrix(M
):
1812 The inverse of _embed_quaternion_matrix().
1816 sage: M = matrix(QQ, [[ 1, 2, 3, 4],
1817 ....: [-2, 1, -4, 3],
1818 ....: [-3, 4, 1, -2],
1819 ....: [-4, -3, 2, 1]])
1820 sage: _unembed_quaternion_matrix(M)
1821 [1 + 2*i + 3*j + 4*k]
1825 Unembedding is the inverse of embedding::
1827 sage: set_random_seed()
1828 sage: Q = QuaternionAlgebra(QQ, -1, -1)
1829 sage: M = random_matrix(Q, 3)
1830 sage: _unembed_quaternion_matrix(_embed_quaternion_matrix(M)) == M
1836 raise ValueError("the matrix 'M' must be square")
1837 if not n
.mod(4).is_zero():
1838 raise ValueError("the matrix 'M' must be a complex embedding")
1840 Q
= QuaternionAlgebra(QQ
,-1,-1)
1843 # Go top-left to bottom-right (reading order), converting every
1844 # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1
1847 for l
in xrange(n
/4):
1848 for m
in xrange(n
/4):
1849 submat
= _unembed_complex_matrix(M
[4*l
:4*l
+4,4*m
:4*m
+4])
1850 if submat
[0,0] != submat
[1,1].conjugate():
1851 raise ValueError('bad on-diagonal submatrix')
1852 if submat
[0,1] != -submat
[1,0].conjugate():
1853 raise ValueError('bad off-diagonal submatrix')
1854 z
= submat
[0,0].real() + submat
[0,0].imag()*i
1855 z
+= submat
[0,1].real()*j
+ submat
[0,1].imag()*k
1858 return matrix(Q
, n
/4, elements
)
1861 # The usual inner product on R^n.
1863 return x
.vector().inner_product(y
.vector())
1865 # The inner product used for the real symmetric simple EJA.
1866 # We keep it as a separate function because e.g. the complex
1867 # algebra uses the same inner product, except divided by 2.
1868 def _matrix_ip(X
,Y
):
1869 X_mat
= X
.natural_representation()
1870 Y_mat
= Y
.natural_representation()
1871 return (X_mat
*Y_mat
).trace()
1874 class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1876 The rank-n simple EJA consisting of real symmetric n-by-n
1877 matrices, the usual symmetric Jordan product, and the trace inner
1878 product. It has dimension `(n^2 + n)/2` over the reals.
1882 sage: J = RealSymmetricEJA(2)
1883 sage: e0, e1, e2 = J.gens()
1893 The degree of this algebra is `(n^2 + n) / 2`::
1895 sage: set_random_seed()
1896 sage: n = ZZ.random_element(1,5)
1897 sage: J = RealSymmetricEJA(n)
1898 sage: J.degree() == (n^2 + n)/2
1901 The Jordan multiplication is what we think it is::
1903 sage: set_random_seed()
1904 sage: n = ZZ.random_element(1,5)
1905 sage: J = RealSymmetricEJA(n)
1906 sage: x = J.random_element()
1907 sage: y = J.random_element()
1908 sage: actual = (x*y).natural_representation()
1909 sage: X = x.natural_representation()
1910 sage: Y = y.natural_representation()
1911 sage: expected = (X*Y + Y*X)/2
1912 sage: actual == expected
1914 sage: J(expected) == x*y
1919 def __classcall_private__(cls
, n
, field
=QQ
):
1920 S
= _real_symmetric_basis(n
, field
=field
)
1921 (Qs
, T
) = _multiplication_table_from_matrix_basis(S
)
1923 fdeja
= super(RealSymmetricEJA
, cls
)
1924 return fdeja
.__classcall
_private
__(cls
,
1930 def inner_product(self
, x
, y
):
1931 return _matrix_ip(x
,y
)
1934 class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1936 The rank-n simple EJA consisting of complex Hermitian n-by-n
1937 matrices over the real numbers, the usual symmetric Jordan product,
1938 and the real-part-of-trace inner product. It has dimension `n^2` over
1943 The degree of this algebra is `n^2`::
1945 sage: set_random_seed()
1946 sage: n = ZZ.random_element(1,5)
1947 sage: J = ComplexHermitianEJA(n)
1948 sage: J.degree() == n^2
1951 The Jordan multiplication is what we think it is::
1953 sage: set_random_seed()
1954 sage: n = ZZ.random_element(1,5)
1955 sage: J = ComplexHermitianEJA(n)
1956 sage: x = J.random_element()
1957 sage: y = J.random_element()
1958 sage: actual = (x*y).natural_representation()
1959 sage: X = x.natural_representation()
1960 sage: Y = y.natural_representation()
1961 sage: expected = (X*Y + Y*X)/2
1962 sage: actual == expected
1964 sage: J(expected) == x*y
1969 def __classcall_private__(cls
, n
, field
=QQ
):
1970 S
= _complex_hermitian_basis(n
)
1971 (Qs
, T
) = _multiplication_table_from_matrix_basis(S
)
1973 fdeja
= super(ComplexHermitianEJA
, cls
)
1974 return fdeja
.__classcall
_private
__(cls
,
1980 def inner_product(self
, x
, y
):
1981 # Since a+bi on the diagonal is represented as
1986 # we'll double-count the "a" entries if we take the trace of
1988 return _matrix_ip(x
,y
)/2
1991 class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1993 The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
1994 matrices, the usual symmetric Jordan product, and the
1995 real-part-of-trace inner product. It has dimension `2n^2 - n` over
2000 The degree of this algebra is `n^2`::
2002 sage: set_random_seed()
2003 sage: n = ZZ.random_element(1,5)
2004 sage: J = QuaternionHermitianEJA(n)
2005 sage: J.degree() == 2*(n^2) - n
2008 The Jordan multiplication is what we think it is::
2010 sage: set_random_seed()
2011 sage: n = ZZ.random_element(1,5)
2012 sage: J = QuaternionHermitianEJA(n)
2013 sage: x = J.random_element()
2014 sage: y = J.random_element()
2015 sage: actual = (x*y).natural_representation()
2016 sage: X = x.natural_representation()
2017 sage: Y = y.natural_representation()
2018 sage: expected = (X*Y + Y*X)/2
2019 sage: actual == expected
2021 sage: J(expected) == x*y
2026 def __classcall_private__(cls
, n
, field
=QQ
):
2027 S
= _quaternion_hermitian_basis(n
)
2028 (Qs
, T
) = _multiplication_table_from_matrix_basis(S
)
2030 fdeja
= super(QuaternionHermitianEJA
, cls
)
2031 return fdeja
.__classcall
_private
__(cls
,
2037 def inner_product(self
, x
, y
):
2038 # Since a+bi+cj+dk on the diagonal is represented as
2040 # a + bi +cj + dk = [ a b c d]
2045 # we'll quadruple-count the "a" entries if we take the trace of
2047 return _matrix_ip(x
,y
)/4
2050 class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra
):
2052 The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
2053 with the usual inner product and jordan product ``x*y =
2054 (<x_bar,y_bar>, x0*y_bar + y0*x_bar)``. It has dimension `n` over
2059 This multiplication table can be verified by hand::
2061 sage: J = JordanSpinEJA(4)
2062 sage: e0,e1,e2,e3 = J.gens()
2080 def __classcall_private__(cls
, n
, field
=QQ
):
2082 id_matrix
= identity_matrix(field
, n
)
2084 ei
= id_matrix
.column(i
)
2085 Qi
= zero_matrix(field
, n
)
2087 Qi
.set_column(0, ei
)
2088 Qi
+= diagonal_matrix(n
, [ei
[0]]*n
)
2089 # The addition of the diagonal matrix adds an extra ei[0] in the
2090 # upper-left corner of the matrix.
2091 Qi
[0,0] = Qi
[0,0] * ~
field(2)
2094 # The rank of the spin algebra is two, unless we're in a
2095 # one-dimensional ambient space (because the rank is bounded by
2096 # the ambient dimension).
2097 fdeja
= super(JordanSpinEJA
, cls
)
2098 return fdeja
.__classcall
_private
__(cls
, field
, Qs
, rank
=min(n
,2))
2100 def inner_product(self
, x
, y
):
2101 return _usual_ip(x
,y
)