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.
343 def __init__(self
, A
, elt
=None):
347 The identity in `S^n` is converted to the identity in the EJA::
349 sage: J = RealSymmetricEJA(3)
350 sage: I = identity_matrix(QQ,3)
351 sage: J(I) == J.one()
354 This skew-symmetric matrix can't be represented in the EJA::
356 sage: J = RealSymmetricEJA(3)
357 sage: A = matrix(QQ,3, lambda i,j: i-j)
359 Traceback (most recent call last):
361 ArithmeticError: vector is not in free module
364 # Goal: if we're given a matrix, and if it lives in our
365 # parent algebra's "natural ambient space," convert it
366 # into an algebra element.
368 # The catch is, we make a recursive call after converting
369 # the given matrix into a vector that lives in the algebra.
370 # This we need to try the parent class initializer first,
371 # to avoid recursing forever if we're given something that
372 # already fits into the algebra, but also happens to live
373 # in the parent's "natural ambient space" (this happens with
376 FiniteDimensionalAlgebraElement
.__init
__(self
, A
, elt
)
378 natural_basis
= A
.natural_basis()
379 if elt
in natural_basis
[0].matrix_space():
380 # Thanks for nothing! Matrix spaces aren't vector
381 # spaces in Sage, so we have to figure out its
382 # natural-basis coordinates ourselves.
383 V
= VectorSpace(elt
.base_ring(), elt
.nrows()**2)
384 W
= V
.span( _mat2vec(s
) for s
in natural_basis
)
385 coords
= W
.coordinates(_mat2vec(elt
))
386 FiniteDimensionalAlgebraElement
.__init
__(self
, A
, coords
)
388 def __pow__(self
, n
):
390 Return ``self`` raised to the power ``n``.
392 Jordan algebras are always power-associative; see for
393 example Faraut and Koranyi, Proposition II.1.2 (ii).
397 We have to override this because our superclass uses row vectors
398 instead of column vectors! We, on the other hand, assume column
403 sage: set_random_seed()
404 sage: x = random_eja().random_element()
405 sage: x.operator_matrix()*x.vector() == (x^2).vector()
408 A few examples of power-associativity::
410 sage: set_random_seed()
411 sage: x = random_eja().random_element()
412 sage: x*(x*x)*(x*x) == x^5
414 sage: (x*x)*(x*x*x) == x^5
417 We also know that powers operator-commute (Koecher, Chapter
420 sage: set_random_seed()
421 sage: x = random_eja().random_element()
422 sage: m = ZZ.random_element(0,10)
423 sage: n = ZZ.random_element(0,10)
424 sage: Lxm = (x^m).operator_matrix()
425 sage: Lxn = (x^n).operator_matrix()
426 sage: Lxm*Lxn == Lxn*Lxm
436 return A( (self
.operator_matrix()**(n
-1))*self
.vector() )
439 def apply_univariate_polynomial(self
, p
):
441 Apply the univariate polynomial ``p`` to this element.
443 A priori, SageMath won't allow us to apply a univariate
444 polynomial to an element of an EJA, because we don't know
445 that EJAs are rings (they are usually not associative). Of
446 course, we know that EJAs are power-associative, so the
447 operation is ultimately kosher. This function sidesteps
448 the CAS to get the answer we want and expect.
452 sage: R = PolynomialRing(QQ, 't')
454 sage: p = t^4 - t^3 + 5*t - 2
455 sage: J = RealCartesianProductEJA(5)
456 sage: J.one().apply_univariate_polynomial(p) == 3*J.one()
461 We should always get back an element of the algebra::
463 sage: set_random_seed()
464 sage: p = PolynomialRing(QQ, 't').random_element()
465 sage: J = random_eja()
466 sage: x = J.random_element()
467 sage: x.apply_univariate_polynomial(p) in J
471 if len(p
.variables()) > 1:
472 raise ValueError("not a univariate polynomial")
475 # Convert the coeficcients to the parent's base ring,
476 # because a priori they might live in an (unnecessarily)
477 # larger ring for which P.sum() would fail below.
478 cs
= [ R(c
) for c
in p
.coefficients(sparse
=False) ]
479 return P
.sum( cs
[k
]*(self
**k
) for k
in range(len(cs
)) )
482 def characteristic_polynomial(self
):
484 Return the characteristic polynomial of this element.
488 The rank of `R^3` is three, and the minimal polynomial of
489 the identity element is `(t-1)` from which it follows that
490 the characteristic polynomial should be `(t-1)^3`::
492 sage: J = RealCartesianProductEJA(3)
493 sage: J.one().characteristic_polynomial()
494 t^3 - 3*t^2 + 3*t - 1
496 Likewise, the characteristic of the zero element in the
497 rank-three algebra `R^{n}` should be `t^{3}`::
499 sage: J = RealCartesianProductEJA(3)
500 sage: J.zero().characteristic_polynomial()
503 The characteristic polynomial of an element should evaluate
504 to zero on that element::
506 sage: set_random_seed()
507 sage: x = RealCartesianProductEJA(3).random_element()
508 sage: p = x.characteristic_polynomial()
509 sage: x.apply_univariate_polynomial(p)
513 p
= self
.parent().characteristic_polynomial()
514 return p(*self
.vector())
517 def inner_product(self
, other
):
519 Return the parent algebra's inner product of myself and ``other``.
523 The inner product in the Jordan spin algebra is the usual
524 inner product on `R^n` (this example only works because the
525 basis for the Jordan algebra is the standard basis in `R^n`)::
527 sage: J = JordanSpinEJA(3)
528 sage: x = vector(QQ,[1,2,3])
529 sage: y = vector(QQ,[4,5,6])
530 sage: x.inner_product(y)
532 sage: J(x).inner_product(J(y))
535 The inner product on `S^n` is `<X,Y> = trace(X*Y)`, where
536 multiplication is the usual matrix multiplication in `S^n`,
537 so the inner product of the identity matrix with itself
540 sage: J = RealSymmetricEJA(3)
541 sage: J.one().inner_product(J.one())
544 Likewise, the inner product on `C^n` is `<X,Y> =
545 Re(trace(X*Y))`, where we must necessarily take the real
546 part because the product of Hermitian matrices may not be
549 sage: J = ComplexHermitianEJA(3)
550 sage: J.one().inner_product(J.one())
553 Ditto for the quaternions::
555 sage: J = QuaternionHermitianEJA(3)
556 sage: J.one().inner_product(J.one())
561 Ensure that we can always compute an inner product, and that
562 it gives us back a real number::
564 sage: set_random_seed()
565 sage: J = random_eja()
566 sage: x = J.random_element()
567 sage: y = J.random_element()
568 sage: x.inner_product(y) in RR
574 raise TypeError("'other' must live in the same algebra")
576 return P
.inner_product(self
, other
)
579 def operator_commutes_with(self
, other
):
581 Return whether or not this element operator-commutes
586 The definition of a Jordan algebra says that any element
587 operator-commutes with its square::
589 sage: set_random_seed()
590 sage: x = random_eja().random_element()
591 sage: x.operator_commutes_with(x^2)
596 Test Lemma 1 from Chapter III of Koecher::
598 sage: set_random_seed()
599 sage: J = random_eja()
600 sage: u = J.random_element()
601 sage: v = J.random_element()
602 sage: lhs = u.operator_commutes_with(u*v)
603 sage: rhs = v.operator_commutes_with(u^2)
608 if not other
in self
.parent():
609 raise TypeError("'other' must live in the same algebra")
611 A
= self
.operator_matrix()
612 B
= other
.operator_matrix()
618 Return my determinant, the product of my eigenvalues.
622 sage: J = JordanSpinEJA(2)
623 sage: e0,e1 = J.gens()
624 sage: x = sum( J.gens() )
630 sage: J = JordanSpinEJA(3)
631 sage: e0,e1,e2 = J.gens()
632 sage: x = sum( J.gens() )
638 An element is invertible if and only if its determinant is
641 sage: set_random_seed()
642 sage: x = random_eja().random_element()
643 sage: x.is_invertible() == (x.det() != 0)
649 p
= P
._charpoly
_coeff
(0)
650 # The _charpoly_coeff function already adds the factor of
651 # -1 to ensure that _charpoly_coeff(0) is really what
652 # appears in front of t^{0} in the charpoly. However,
653 # we want (-1)^r times THAT for the determinant.
654 return ((-1)**r
)*p(*self
.vector())
659 Return the Jordan-multiplicative inverse of this element.
663 We appeal to the quadratic representation as in Koecher's
664 Theorem 12 in Chapter III, Section 5.
668 The inverse in the spin factor algebra is given in Alizadeh's
671 sage: set_random_seed()
672 sage: n = ZZ.random_element(1,10)
673 sage: J = JordanSpinEJA(n)
674 sage: x = J.random_element()
675 sage: while x.is_zero():
676 ....: x = J.random_element()
677 sage: x_vec = x.vector()
679 sage: x_bar = x_vec[1:]
680 sage: coeff = 1/(x0^2 - x_bar.inner_product(x_bar))
681 sage: inv_vec = x_vec.parent()([x0] + (-x_bar).list())
682 sage: x_inverse = coeff*inv_vec
683 sage: x.inverse() == J(x_inverse)
688 The identity element is its own inverse::
690 sage: set_random_seed()
691 sage: J = random_eja()
692 sage: J.one().inverse() == J.one()
695 If an element has an inverse, it acts like one::
697 sage: set_random_seed()
698 sage: J = random_eja()
699 sage: x = J.random_element()
700 sage: (not x.is_invertible()) or (x.inverse()*x == J.one())
703 The inverse of the inverse is what we started with::
705 sage: set_random_seed()
706 sage: J = random_eja()
707 sage: x = J.random_element()
708 sage: (not x.is_invertible()) or (x.inverse().inverse() == x)
711 The zero element is never invertible::
713 sage: set_random_seed()
714 sage: J = random_eja().zero().inverse()
715 Traceback (most recent call last):
717 ValueError: element is not invertible
720 if not self
.is_invertible():
721 raise ValueError("element is not invertible")
724 return P(self
.quadratic_representation().inverse()*self
.vector())
727 def is_invertible(self
):
729 Return whether or not this element is invertible.
731 We can't use the superclass method because it relies on
732 the algebra being associative.
736 The usual way to do this is to check if the determinant is
737 zero, but we need the characteristic polynomial for the
738 determinant. The minimal polynomial is a lot easier to get,
739 so we use Corollary 2 in Chapter V of Koecher to check
740 whether or not the paren't algebra's zero element is a root
741 of this element's minimal polynomial.
745 The identity element is always invertible::
747 sage: set_random_seed()
748 sage: J = random_eja()
749 sage: J.one().is_invertible()
752 The zero element is never invertible::
754 sage: set_random_seed()
755 sage: J = random_eja()
756 sage: J.zero().is_invertible()
760 zero
= self
.parent().zero()
761 p
= self
.minimal_polynomial()
762 return not (p(zero
) == zero
)
765 def is_nilpotent(self
):
767 Return whether or not some power of this element is zero.
769 The superclass method won't work unless we're in an
770 associative algebra, and we aren't. However, we generate
771 an assocoative subalgebra and we're nilpotent there if and
772 only if we're nilpotent here (probably).
776 The identity element is never nilpotent::
778 sage: set_random_seed()
779 sage: random_eja().one().is_nilpotent()
782 The additive identity is always nilpotent::
784 sage: set_random_seed()
785 sage: random_eja().zero().is_nilpotent()
789 # The element we're going to call "is_nilpotent()" on.
790 # Either myself, interpreted as an element of a finite-
791 # dimensional algebra, or an element of an associative
795 if self
.parent().is_associative():
796 elt
= FiniteDimensionalAlgebraElement(self
.parent(), self
)
798 V
= self
.span_of_powers()
799 assoc_subalg
= self
.subalgebra_generated_by()
800 # Mis-design warning: the basis used for span_of_powers()
801 # and subalgebra_generated_by() must be the same, and in
803 elt
= assoc_subalg(V
.coordinates(self
.vector()))
805 # Recursive call, but should work since elt lives in an
806 # associative algebra.
807 return elt
.is_nilpotent()
810 def is_regular(self
):
812 Return whether or not this is a regular element.
816 The identity element always has degree one, but any element
817 linearly-independent from it is regular::
819 sage: J = JordanSpinEJA(5)
820 sage: J.one().is_regular()
822 sage: e0, e1, e2, e3, e4 = J.gens() # e0 is the identity
823 sage: for x in J.gens():
824 ....: (J.one() + x).is_regular()
832 return self
.degree() == self
.parent().rank()
837 Compute the degree of this element the straightforward way
838 according to the definition; by appending powers to a list
839 and figuring out its dimension (that is, whether or not
840 they're linearly dependent).
844 sage: J = JordanSpinEJA(4)
845 sage: J.one().degree()
847 sage: e0,e1,e2,e3 = J.gens()
848 sage: (e0 - e1).degree()
851 In the spin factor algebra (of rank two), all elements that
852 aren't multiples of the identity are regular::
854 sage: set_random_seed()
855 sage: n = ZZ.random_element(1,10)
856 sage: J = JordanSpinEJA(n)
857 sage: x = J.random_element()
858 sage: x == x.coefficient(0)*J.one() or x.degree() == 2
862 return self
.span_of_powers().dimension()
865 def minimal_polynomial(self
):
867 Return the minimal polynomial of this element,
868 as a function of the variable `t`.
872 We restrict ourselves to the associative subalgebra
873 generated by this element, and then return the minimal
874 polynomial of this element's operator matrix (in that
875 subalgebra). This works by Baes Proposition 2.3.16.
879 The minimal polynomial of the identity and zero elements are
882 sage: set_random_seed()
883 sage: J = random_eja()
884 sage: J.one().minimal_polynomial()
886 sage: J.zero().minimal_polynomial()
889 The degree of an element is (by one definition) the degree
890 of its minimal polynomial::
892 sage: set_random_seed()
893 sage: x = random_eja().random_element()
894 sage: x.degree() == x.minimal_polynomial().degree()
897 The minimal polynomial and the characteristic polynomial coincide
898 and are known (see Alizadeh, Example 11.11) for all elements of
899 the spin factor algebra that aren't scalar multiples of the
902 sage: set_random_seed()
903 sage: n = ZZ.random_element(2,10)
904 sage: J = JordanSpinEJA(n)
905 sage: y = J.random_element()
906 sage: while y == y.coefficient(0)*J.one():
907 ....: y = J.random_element()
908 sage: y0 = y.vector()[0]
909 sage: y_bar = y.vector()[1:]
910 sage: actual = y.minimal_polynomial()
911 sage: t = PolynomialRing(J.base_ring(),'t').gen(0)
912 sage: expected = t^2 - 2*y0*t + (y0^2 - norm(y_bar)^2)
913 sage: bool(actual == expected)
916 The minimal polynomial should always kill its element::
918 sage: set_random_seed()
919 sage: x = random_eja().random_element()
920 sage: p = x.minimal_polynomial()
921 sage: x.apply_univariate_polynomial(p)
925 V
= self
.span_of_powers()
926 assoc_subalg
= self
.subalgebra_generated_by()
927 # Mis-design warning: the basis used for span_of_powers()
928 # and subalgebra_generated_by() must be the same, and in
930 elt
= assoc_subalg(V
.coordinates(self
.vector()))
932 # We get back a symbolic polynomial in 'x' but want a real
934 p_of_x
= elt
.operator_matrix().minimal_polynomial()
935 return p_of_x
.change_variable_name('t')
938 def natural_representation(self
):
940 Return a more-natural representation of this element.
942 Every finite-dimensional Euclidean Jordan Algebra is a
943 direct sum of five simple algebras, four of which comprise
944 Hermitian matrices. This method returns the original
945 "natural" representation of this element as a Hermitian
946 matrix, if it has one. If not, you get the usual representation.
950 sage: J = ComplexHermitianEJA(3)
953 sage: J.one().natural_representation()
963 sage: J = QuaternionHermitianEJA(3)
966 sage: J.one().natural_representation()
967 [1 0 0 0 0 0 0 0 0 0 0 0]
968 [0 1 0 0 0 0 0 0 0 0 0 0]
969 [0 0 1 0 0 0 0 0 0 0 0 0]
970 [0 0 0 1 0 0 0 0 0 0 0 0]
971 [0 0 0 0 1 0 0 0 0 0 0 0]
972 [0 0 0 0 0 1 0 0 0 0 0 0]
973 [0 0 0 0 0 0 1 0 0 0 0 0]
974 [0 0 0 0 0 0 0 1 0 0 0 0]
975 [0 0 0 0 0 0 0 0 1 0 0 0]
976 [0 0 0 0 0 0 0 0 0 1 0 0]
977 [0 0 0 0 0 0 0 0 0 0 1 0]
978 [0 0 0 0 0 0 0 0 0 0 0 1]
981 B
= self
.parent().natural_basis()
982 W
= B
[0].matrix_space()
983 return W
.linear_combination(zip(self
.vector(), B
))
986 def operator_matrix(self
):
988 Return the matrix that represents left- (or right-)
989 multiplication by this element in the parent algebra.
991 We have to override this because the superclass method
992 returns a matrix that acts on row vectors (that is, on
997 Test the first polarization identity from my notes, Koecher Chapter
998 III, or from Baes (2.3)::
1000 sage: set_random_seed()
1001 sage: J = random_eja()
1002 sage: x = J.random_element()
1003 sage: y = J.random_element()
1004 sage: Lx = x.operator_matrix()
1005 sage: Ly = y.operator_matrix()
1006 sage: Lxx = (x*x).operator_matrix()
1007 sage: Lxy = (x*y).operator_matrix()
1008 sage: bool(2*Lx*Lxy + Ly*Lxx == 2*Lxy*Lx + Lxx*Ly)
1011 Test the second polarization identity from my notes or from
1014 sage: set_random_seed()
1015 sage: J = random_eja()
1016 sage: x = J.random_element()
1017 sage: y = J.random_element()
1018 sage: z = J.random_element()
1019 sage: Lx = x.operator_matrix()
1020 sage: Ly = y.operator_matrix()
1021 sage: Lz = z.operator_matrix()
1022 sage: Lzy = (z*y).operator_matrix()
1023 sage: Lxy = (x*y).operator_matrix()
1024 sage: Lxz = (x*z).operator_matrix()
1025 sage: bool(Lx*Lzy + Lz*Lxy + Ly*Lxz == Lzy*Lx + Lxy*Lz + Lxz*Ly)
1028 Test the third polarization identity from my notes or from
1031 sage: set_random_seed()
1032 sage: J = random_eja()
1033 sage: u = J.random_element()
1034 sage: y = J.random_element()
1035 sage: z = J.random_element()
1036 sage: Lu = u.operator_matrix()
1037 sage: Ly = y.operator_matrix()
1038 sage: Lz = z.operator_matrix()
1039 sage: Lzy = (z*y).operator_matrix()
1040 sage: Luy = (u*y).operator_matrix()
1041 sage: Luz = (u*z).operator_matrix()
1042 sage: Luyz = (u*(y*z)).operator_matrix()
1043 sage: lhs = Lu*Lzy + Lz*Luy + Ly*Luz
1044 sage: rhs = Luyz + Ly*Lu*Lz + Lz*Lu*Ly
1045 sage: bool(lhs == rhs)
1049 fda_elt
= FiniteDimensionalAlgebraElement(self
.parent(), self
)
1050 return fda_elt
.matrix().transpose()
1053 def quadratic_representation(self
, other
=None):
1055 Return the quadratic representation of this element.
1059 The explicit form in the spin factor algebra is given by
1060 Alizadeh's Example 11.12::
1062 sage: set_random_seed()
1063 sage: n = ZZ.random_element(1,10)
1064 sage: J = JordanSpinEJA(n)
1065 sage: x = J.random_element()
1066 sage: x_vec = x.vector()
1068 sage: x_bar = x_vec[1:]
1069 sage: A = matrix(QQ, 1, [x_vec.inner_product(x_vec)])
1070 sage: B = 2*x0*x_bar.row()
1071 sage: C = 2*x0*x_bar.column()
1072 sage: D = identity_matrix(QQ, n-1)
1073 sage: D = (x0^2 - x_bar.inner_product(x_bar))*D
1074 sage: D = D + 2*x_bar.tensor_product(x_bar)
1075 sage: Q = block_matrix(2,2,[A,B,C,D])
1076 sage: Q == x.quadratic_representation()
1079 Test all of the properties from Theorem 11.2 in Alizadeh::
1081 sage: set_random_seed()
1082 sage: J = random_eja()
1083 sage: x = J.random_element()
1084 sage: y = J.random_element()
1085 sage: Lx = x.operator_matrix()
1086 sage: Lxx = (x*x).operator_matrix()
1087 sage: Qx = x.quadratic_representation()
1088 sage: Qy = y.quadratic_representation()
1089 sage: Qxy = x.quadratic_representation(y)
1090 sage: Qex = J.one().quadratic_representation(x)
1091 sage: n = ZZ.random_element(10)
1092 sage: Qxn = (x^n).quadratic_representation()
1096 sage: 2*Qxy == (x+y).quadratic_representation() - Qx - Qy
1101 sage: alpha = QQ.random_element()
1102 sage: (alpha*x).quadratic_representation() == (alpha^2)*Qx
1107 sage: not x.is_invertible() or (
1108 ....: Qx*x.inverse().vector() == x.vector() )
1111 sage: not x.is_invertible() or (
1114 ....: x.inverse().quadratic_representation() )
1117 sage: Qxy*(J.one().vector()) == (x*y).vector()
1122 sage: not x.is_invertible() or (
1123 ....: x.quadratic_representation(x.inverse())*Qx
1124 ....: == Qx*x.quadratic_representation(x.inverse()) )
1127 sage: not x.is_invertible() or (
1128 ....: x.quadratic_representation(x.inverse())*Qx
1130 ....: 2*x.operator_matrix()*Qex - Qx )
1133 sage: 2*x.operator_matrix()*Qex - Qx == Lxx
1138 sage: J(Qy*x.vector()).quadratic_representation() == Qy*Qx*Qy
1148 sage: not x.is_invertible() or (
1149 ....: Qx*x.inverse().operator_matrix() == Lx )
1154 sage: not x.operator_commutes_with(y) or (
1155 ....: J(Qx*y.vector())^n == J(Qxn*(y^n).vector()) )
1161 elif not other
in self
.parent():
1162 raise TypeError("'other' must live in the same algebra")
1164 L
= self
.operator_matrix()
1165 M
= other
.operator_matrix()
1166 return ( L
*M
+ M
*L
- (self
*other
).operator_matrix() )
1169 def span_of_powers(self
):
1171 Return the vector space spanned by successive powers of
1174 # The dimension of the subalgebra can't be greater than
1175 # the big algebra, so just put everything into a list
1176 # and let span() get rid of the excess.
1178 # We do the extra ambient_vector_space() in case we're messing
1179 # with polynomials and the direct parent is a module.
1180 V
= self
.vector().parent().ambient_vector_space()
1181 return V
.span( (self
**d
).vector() for d
in xrange(V
.dimension()) )
1184 def subalgebra_generated_by(self
):
1186 Return the associative subalgebra of the parent EJA generated
1191 sage: set_random_seed()
1192 sage: x = random_eja().random_element()
1193 sage: x.subalgebra_generated_by().is_associative()
1196 Squaring in the subalgebra should be the same thing as
1197 squaring in the superalgebra::
1199 sage: set_random_seed()
1200 sage: x = random_eja().random_element()
1201 sage: u = x.subalgebra_generated_by().random_element()
1202 sage: u.operator_matrix()*u.vector() == (u**2).vector()
1206 # First get the subspace spanned by the powers of myself...
1207 V
= self
.span_of_powers()
1208 F
= self
.base_ring()
1210 # Now figure out the entries of the right-multiplication
1211 # matrix for the successive basis elements b0, b1,... of
1214 for b_right
in V
.basis():
1215 eja_b_right
= self
.parent()(b_right
)
1217 # The first row of the right-multiplication matrix by
1218 # b1 is what we get if we apply that matrix to b1. The
1219 # second row of the right multiplication matrix by b1
1220 # is what we get when we apply that matrix to b2...
1222 # IMPORTANT: this assumes that all vectors are COLUMN
1223 # vectors, unlike our superclass (which uses row vectors).
1224 for b_left
in V
.basis():
1225 eja_b_left
= self
.parent()(b_left
)
1226 # Multiply in the original EJA, but then get the
1227 # coordinates from the subalgebra in terms of its
1229 this_row
= V
.coordinates((eja_b_left
*eja_b_right
).vector())
1230 b_right_rows
.append(this_row
)
1231 b_right_matrix
= matrix(F
, b_right_rows
)
1232 mats
.append(b_right_matrix
)
1234 # It's an algebra of polynomials in one element, and EJAs
1235 # are power-associative.
1237 # TODO: choose generator names intelligently.
1238 return FiniteDimensionalEuclideanJordanAlgebra(F
, mats
, assume_associative
=True, names
='f')
1241 def subalgebra_idempotent(self
):
1243 Find an idempotent in the associative subalgebra I generate
1244 using Proposition 2.3.5 in Baes.
1248 sage: set_random_seed()
1249 sage: J = random_eja()
1250 sage: x = J.random_element()
1251 sage: while x.is_nilpotent():
1252 ....: x = J.random_element()
1253 sage: c = x.subalgebra_idempotent()
1258 if self
.is_nilpotent():
1259 raise ValueError("this only works with non-nilpotent elements!")
1261 V
= self
.span_of_powers()
1262 J
= self
.subalgebra_generated_by()
1263 # Mis-design warning: the basis used for span_of_powers()
1264 # and subalgebra_generated_by() must be the same, and in
1266 u
= J(V
.coordinates(self
.vector()))
1268 # The image of the matrix of left-u^m-multiplication
1269 # will be minimal for some natural number s...
1271 minimal_dim
= V
.dimension()
1272 for i
in xrange(1, V
.dimension()):
1273 this_dim
= (u
**i
).operator_matrix().image().dimension()
1274 if this_dim
< minimal_dim
:
1275 minimal_dim
= this_dim
1278 # Now minimal_matrix should correspond to the smallest
1279 # non-zero subspace in Baes's (or really, Koecher's)
1282 # However, we need to restrict the matrix to work on the
1283 # subspace... or do we? Can't we just solve, knowing that
1284 # A(c) = u^(s+1) should have a solution in the big space,
1287 # Beware, solve_right() means that we're using COLUMN vectors.
1288 # Our FiniteDimensionalAlgebraElement superclass uses rows.
1290 A
= u_next
.operator_matrix()
1291 c_coordinates
= A
.solve_right(u_next
.vector())
1293 # Now c_coordinates is the idempotent we want, but it's in
1294 # the coordinate system of the subalgebra.
1296 # We need the basis for J, but as elements of the parent algebra.
1298 basis
= [self
.parent(v
) for v
in V
.basis()]
1299 return self
.parent().linear_combination(zip(c_coordinates
, basis
))
1304 Return my trace, the sum of my eigenvalues.
1308 sage: J = JordanSpinEJA(3)
1309 sage: x = sum(J.gens())
1315 sage: J = RealCartesianProductEJA(5)
1316 sage: J.one().trace()
1321 The trace of an element is a real number::
1323 sage: set_random_seed()
1324 sage: J = random_eja()
1325 sage: J.random_element().trace() in J.base_ring()
1331 p
= P
._charpoly
_coeff
(r
-1)
1332 # The _charpoly_coeff function already adds the factor of
1333 # -1 to ensure that _charpoly_coeff(r-1) is really what
1334 # appears in front of t^{r-1} in the charpoly. However,
1335 # we want the negative of THAT for the trace.
1336 return -p(*self
.vector())
1339 def trace_inner_product(self
, other
):
1341 Return the trace inner product of myself and ``other``.
1345 The trace inner product is commutative::
1347 sage: set_random_seed()
1348 sage: J = random_eja()
1349 sage: x = J.random_element(); y = J.random_element()
1350 sage: x.trace_inner_product(y) == y.trace_inner_product(x)
1353 The trace inner product is bilinear::
1355 sage: set_random_seed()
1356 sage: J = random_eja()
1357 sage: x = J.random_element()
1358 sage: y = J.random_element()
1359 sage: z = J.random_element()
1360 sage: a = QQ.random_element();
1361 sage: actual = (a*(x+z)).trace_inner_product(y)
1362 sage: expected = ( a*x.trace_inner_product(y) +
1363 ....: a*z.trace_inner_product(y) )
1364 sage: actual == expected
1366 sage: actual = x.trace_inner_product(a*(y+z))
1367 sage: expected = ( a*x.trace_inner_product(y) +
1368 ....: a*x.trace_inner_product(z) )
1369 sage: actual == expected
1372 The trace inner product satisfies the compatibility
1373 condition in the definition of a Euclidean Jordan algebra::
1375 sage: set_random_seed()
1376 sage: J = random_eja()
1377 sage: x = J.random_element()
1378 sage: y = J.random_element()
1379 sage: z = J.random_element()
1380 sage: (x*y).trace_inner_product(z) == y.trace_inner_product(x*z)
1384 if not other
in self
.parent():
1385 raise TypeError("'other' must live in the same algebra")
1387 return (self
*other
).trace()
1390 class RealCartesianProductEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1392 Return the Euclidean Jordan Algebra corresponding to the set
1393 `R^n` under the Hadamard product.
1395 Note: this is nothing more than the Cartesian product of ``n``
1396 copies of the spin algebra. Once Cartesian product algebras
1397 are implemented, this can go.
1401 This multiplication table can be verified by hand::
1403 sage: J = RealCartesianProductEJA(3)
1404 sage: e0,e1,e2 = J.gens()
1420 def __classcall_private__(cls
, n
, field
=QQ
):
1421 # The FiniteDimensionalAlgebra constructor takes a list of
1422 # matrices, the ith representing right multiplication by the ith
1423 # basis element in the vector space. So if e_1 = (1,0,0), then
1424 # right (Hadamard) multiplication of x by e_1 picks out the first
1425 # component of x; and likewise for the ith basis element e_i.
1426 Qs
= [ matrix(field
, n
, n
, lambda k
,j
: 1*(k
== j
== i
))
1427 for i
in xrange(n
) ]
1429 fdeja
= super(RealCartesianProductEJA
, cls
)
1430 return fdeja
.__classcall
_private
__(cls
, field
, Qs
, rank
=n
)
1432 def inner_product(self
, x
, y
):
1433 return _usual_ip(x
,y
)
1438 Return a "random" finite-dimensional Euclidean Jordan Algebra.
1442 For now, we choose a random natural number ``n`` (greater than zero)
1443 and then give you back one of the following:
1445 * The cartesian product of the rational numbers ``n`` times; this is
1446 ``QQ^n`` with the Hadamard product.
1448 * The Jordan spin algebra on ``QQ^n``.
1450 * The ``n``-by-``n`` rational symmetric matrices with the symmetric
1453 * The ``n``-by-``n`` complex-rational Hermitian matrices embedded
1454 in the space of ``2n``-by-``2n`` real symmetric matrices.
1456 * The ``n``-by-``n`` quaternion-rational Hermitian matrices embedded
1457 in the space of ``4n``-by-``4n`` real symmetric matrices.
1459 Later this might be extended to return Cartesian products of the
1465 Euclidean Jordan algebra of degree...
1469 # The max_n component lets us choose different upper bounds on the
1470 # value "n" that gets passed to the constructor. This is needed
1471 # because e.g. R^{10} is reasonable to test, while the Hermitian
1472 # 10-by-10 quaternion matrices are not.
1473 (constructor
, max_n
) = choice([(RealCartesianProductEJA
, 6),
1475 (RealSymmetricEJA
, 5),
1476 (ComplexHermitianEJA
, 4),
1477 (QuaternionHermitianEJA
, 3)])
1478 n
= ZZ
.random_element(1, max_n
)
1479 return constructor(n
, field
=QQ
)
1483 def _real_symmetric_basis(n
, field
=QQ
):
1485 Return a basis for the space of real symmetric n-by-n matrices.
1487 # The basis of symmetric matrices, as matrices, in their R^(n-by-n)
1491 for j
in xrange(i
+1):
1492 Eij
= matrix(field
, n
, lambda k
,l
: k
==i
and l
==j
)
1496 # Beware, orthogonal but not normalized!
1497 Sij
= Eij
+ Eij
.transpose()
1502 def _complex_hermitian_basis(n
, field
=QQ
):
1504 Returns a basis for the space of complex Hermitian n-by-n matrices.
1508 sage: set_random_seed()
1509 sage: n = ZZ.random_element(1,5)
1510 sage: all( M.is_symmetric() for M in _complex_hermitian_basis(n) )
1514 F
= QuadraticField(-1, 'I')
1517 # This is like the symmetric case, but we need to be careful:
1519 # * We want conjugate-symmetry, not just symmetry.
1520 # * The diagonal will (as a result) be real.
1524 for j
in xrange(i
+1):
1525 Eij
= matrix(field
, n
, lambda k
,l
: k
==i
and l
==j
)
1527 Sij
= _embed_complex_matrix(Eij
)
1530 # Beware, orthogonal but not normalized! The second one
1531 # has a minus because it's conjugated.
1532 Sij_real
= _embed_complex_matrix(Eij
+ Eij
.transpose())
1534 Sij_imag
= _embed_complex_matrix(I
*Eij
- I
*Eij
.transpose())
1539 def _quaternion_hermitian_basis(n
, field
=QQ
):
1541 Returns a basis for the space of quaternion Hermitian n-by-n matrices.
1545 sage: set_random_seed()
1546 sage: n = ZZ.random_element(1,5)
1547 sage: all( M.is_symmetric() for M in _quaternion_hermitian_basis(n) )
1551 Q
= QuaternionAlgebra(QQ
,-1,-1)
1554 # This is like the symmetric case, but we need to be careful:
1556 # * We want conjugate-symmetry, not just symmetry.
1557 # * The diagonal will (as a result) be real.
1561 for j
in xrange(i
+1):
1562 Eij
= matrix(Q
, n
, lambda k
,l
: k
==i
and l
==j
)
1564 Sij
= _embed_quaternion_matrix(Eij
)
1567 # Beware, orthogonal but not normalized! The second,
1568 # third, and fourth ones have a minus because they're
1570 Sij_real
= _embed_quaternion_matrix(Eij
+ Eij
.transpose())
1572 Sij_I
= _embed_quaternion_matrix(I
*Eij
- I
*Eij
.transpose())
1574 Sij_J
= _embed_quaternion_matrix(J
*Eij
- J
*Eij
.transpose())
1576 Sij_K
= _embed_quaternion_matrix(K
*Eij
- K
*Eij
.transpose())
1582 return vector(m
.base_ring(), m
.list())
1585 return matrix(v
.base_ring(), sqrt(v
.degree()), v
.list())
1587 def _multiplication_table_from_matrix_basis(basis
):
1589 At least three of the five simple Euclidean Jordan algebras have the
1590 symmetric multiplication (A,B) |-> (AB + BA)/2, where the
1591 multiplication on the right is matrix multiplication. Given a basis
1592 for the underlying matrix space, this function returns a
1593 multiplication table (obtained by looping through the basis
1594 elements) for an algebra of those matrices. A reordered copy
1595 of the basis is also returned to work around the fact that
1596 the ``span()`` in this function will change the order of the basis
1597 from what we think it is, to... something else.
1599 # In S^2, for example, we nominally have four coordinates even
1600 # though the space is of dimension three only. The vector space V
1601 # is supposed to hold the entire long vector, and the subspace W
1602 # of V will be spanned by the vectors that arise from symmetric
1603 # matrices. Thus for S^2, dim(V) == 4 and dim(W) == 3.
1604 field
= basis
[0].base_ring()
1605 dimension
= basis
[0].nrows()
1607 V
= VectorSpace(field
, dimension
**2)
1608 W
= V
.span( _mat2vec(s
) for s
in basis
)
1610 # Taking the span above reorders our basis (thanks, jerk!) so we
1611 # need to put our "matrix basis" in the same order as the
1612 # (reordered) vector basis.
1613 S
= tuple( _vec2mat(b
) for b
in W
.basis() )
1617 # Brute force the multiplication-by-s matrix by looping
1618 # through all elements of the basis and doing the computation
1619 # to find out what the corresponding row should be. BEWARE:
1620 # these multiplication tables won't be symmetric! It therefore
1621 # becomes REALLY IMPORTANT that the underlying algebra
1622 # constructor uses ROW vectors and not COLUMN vectors. That's
1623 # why we're computing rows here and not columns.
1626 this_row
= _mat2vec((s
*t
+ t
*s
)/2)
1627 Q_rows
.append(W
.coordinates(this_row
))
1628 Q
= matrix(field
, W
.dimension(), Q_rows
)
1634 def _embed_complex_matrix(M
):
1636 Embed the n-by-n complex matrix ``M`` into the space of real
1637 matrices of size 2n-by-2n via the map the sends each entry `z = a +
1638 bi` to the block matrix ``[[a,b],[-b,a]]``.
1642 sage: F = QuadraticField(-1,'i')
1643 sage: x1 = F(4 - 2*i)
1644 sage: x2 = F(1 + 2*i)
1647 sage: M = matrix(F,2,[[x1,x2],[x3,x4]])
1648 sage: _embed_complex_matrix(M)
1657 Embedding is a homomorphism (isomorphism, in fact)::
1659 sage: set_random_seed()
1660 sage: n = ZZ.random_element(5)
1661 sage: F = QuadraticField(-1, 'i')
1662 sage: X = random_matrix(F, n)
1663 sage: Y = random_matrix(F, n)
1664 sage: actual = _embed_complex_matrix(X) * _embed_complex_matrix(Y)
1665 sage: expected = _embed_complex_matrix(X*Y)
1666 sage: actual == expected
1672 raise ValueError("the matrix 'M' must be square")
1673 field
= M
.base_ring()
1678 blocks
.append(matrix(field
, 2, [[a
,b
],[-b
,a
]]))
1680 # We can drop the imaginaries here.
1681 return block_matrix(field
.base_ring(), n
, blocks
)
1684 def _unembed_complex_matrix(M
):
1686 The inverse of _embed_complex_matrix().
1690 sage: A = matrix(QQ,[ [ 1, 2, 3, 4],
1691 ....: [-2, 1, -4, 3],
1692 ....: [ 9, 10, 11, 12],
1693 ....: [-10, 9, -12, 11] ])
1694 sage: _unembed_complex_matrix(A)
1696 [ 10*i + 9 12*i + 11]
1700 Unembedding is the inverse of embedding::
1702 sage: set_random_seed()
1703 sage: F = QuadraticField(-1, 'i')
1704 sage: M = random_matrix(F, 3)
1705 sage: _unembed_complex_matrix(_embed_complex_matrix(M)) == M
1711 raise ValueError("the matrix 'M' must be square")
1712 if not n
.mod(2).is_zero():
1713 raise ValueError("the matrix 'M' must be a complex embedding")
1715 F
= QuadraticField(-1, 'i')
1718 # Go top-left to bottom-right (reading order), converting every
1719 # 2-by-2 block we see to a single complex element.
1721 for k
in xrange(n
/2):
1722 for j
in xrange(n
/2):
1723 submat
= M
[2*k
:2*k
+2,2*j
:2*j
+2]
1724 if submat
[0,0] != submat
[1,1]:
1725 raise ValueError('bad on-diagonal submatrix')
1726 if submat
[0,1] != -submat
[1,0]:
1727 raise ValueError('bad off-diagonal submatrix')
1728 z
= submat
[0,0] + submat
[0,1]*i
1731 return matrix(F
, n
/2, elements
)
1734 def _embed_quaternion_matrix(M
):
1736 Embed the n-by-n quaternion matrix ``M`` into the space of real
1737 matrices of size 4n-by-4n by first sending each quaternion entry
1738 `z = a + bi + cj + dk` to the block-complex matrix
1739 ``[[a + bi, c+di],[-c + di, a-bi]]`, and then embedding those into
1744 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1745 sage: i,j,k = Q.gens()
1746 sage: x = 1 + 2*i + 3*j + 4*k
1747 sage: M = matrix(Q, 1, [[x]])
1748 sage: _embed_quaternion_matrix(M)
1754 Embedding is a homomorphism (isomorphism, in fact)::
1756 sage: set_random_seed()
1757 sage: n = ZZ.random_element(5)
1758 sage: Q = QuaternionAlgebra(QQ,-1,-1)
1759 sage: X = random_matrix(Q, n)
1760 sage: Y = random_matrix(Q, n)
1761 sage: actual = _embed_quaternion_matrix(X)*_embed_quaternion_matrix(Y)
1762 sage: expected = _embed_quaternion_matrix(X*Y)
1763 sage: actual == expected
1767 quaternions
= M
.base_ring()
1770 raise ValueError("the matrix 'M' must be square")
1772 F
= QuadraticField(-1, 'i')
1777 t
= z
.coefficient_tuple()
1782 cplx_matrix
= matrix(F
, 2, [[ a
+ b
*i
, c
+ d
*i
],
1783 [-c
+ d
*i
, a
- b
*i
]])
1784 blocks
.append(_embed_complex_matrix(cplx_matrix
))
1786 # We should have real entries by now, so use the realest field
1787 # we've got for the return value.
1788 return block_matrix(quaternions
.base_ring(), n
, blocks
)
1791 def _unembed_quaternion_matrix(M
):
1793 The inverse of _embed_quaternion_matrix().
1797 sage: M = matrix(QQ, [[ 1, 2, 3, 4],
1798 ....: [-2, 1, -4, 3],
1799 ....: [-3, 4, 1, -2],
1800 ....: [-4, -3, 2, 1]])
1801 sage: _unembed_quaternion_matrix(M)
1802 [1 + 2*i + 3*j + 4*k]
1806 Unembedding is the inverse of embedding::
1808 sage: set_random_seed()
1809 sage: Q = QuaternionAlgebra(QQ, -1, -1)
1810 sage: M = random_matrix(Q, 3)
1811 sage: _unembed_quaternion_matrix(_embed_quaternion_matrix(M)) == M
1817 raise ValueError("the matrix 'M' must be square")
1818 if not n
.mod(4).is_zero():
1819 raise ValueError("the matrix 'M' must be a complex embedding")
1821 Q
= QuaternionAlgebra(QQ
,-1,-1)
1824 # Go top-left to bottom-right (reading order), converting every
1825 # 4-by-4 block we see to a 2-by-2 complex block, to a 1-by-1
1828 for l
in xrange(n
/4):
1829 for m
in xrange(n
/4):
1830 submat
= _unembed_complex_matrix(M
[4*l
:4*l
+4,4*m
:4*m
+4])
1831 if submat
[0,0] != submat
[1,1].conjugate():
1832 raise ValueError('bad on-diagonal submatrix')
1833 if submat
[0,1] != -submat
[1,0].conjugate():
1834 raise ValueError('bad off-diagonal submatrix')
1835 z
= submat
[0,0].real() + submat
[0,0].imag()*i
1836 z
+= submat
[0,1].real()*j
+ submat
[0,1].imag()*k
1839 return matrix(Q
, n
/4, elements
)
1842 # The usual inner product on R^n.
1844 return x
.vector().inner_product(y
.vector())
1846 # The inner product used for the real symmetric simple EJA.
1847 # We keep it as a separate function because e.g. the complex
1848 # algebra uses the same inner product, except divided by 2.
1849 def _matrix_ip(X
,Y
):
1850 X_mat
= X
.natural_representation()
1851 Y_mat
= Y
.natural_representation()
1852 return (X_mat
*Y_mat
).trace()
1855 class RealSymmetricEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1857 The rank-n simple EJA consisting of real symmetric n-by-n
1858 matrices, the usual symmetric Jordan product, and the trace inner
1859 product. It has dimension `(n^2 + n)/2` over the reals.
1863 sage: J = RealSymmetricEJA(2)
1864 sage: e0, e1, e2 = J.gens()
1874 The degree of this algebra is `(n^2 + n) / 2`::
1876 sage: set_random_seed()
1877 sage: n = ZZ.random_element(1,5)
1878 sage: J = RealSymmetricEJA(n)
1879 sage: J.degree() == (n^2 + n)/2
1882 The Jordan multiplication is what we think it is::
1884 sage: set_random_seed()
1885 sage: n = ZZ.random_element(1,5)
1886 sage: J = RealSymmetricEJA(n)
1887 sage: x = J.random_element()
1888 sage: y = J.random_element()
1889 sage: actual = (x*y).natural_representation()
1890 sage: X = x.natural_representation()
1891 sage: Y = y.natural_representation()
1892 sage: expected = (X*Y + Y*X)/2
1893 sage: actual == expected
1895 sage: J(expected) == x*y
1900 def __classcall_private__(cls
, n
, field
=QQ
):
1901 S
= _real_symmetric_basis(n
, field
=field
)
1902 (Qs
, T
) = _multiplication_table_from_matrix_basis(S
)
1904 fdeja
= super(RealSymmetricEJA
, cls
)
1905 return fdeja
.__classcall
_private
__(cls
,
1911 def inner_product(self
, x
, y
):
1912 return _matrix_ip(x
,y
)
1915 class ComplexHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1917 The rank-n simple EJA consisting of complex Hermitian n-by-n
1918 matrices over the real numbers, the usual symmetric Jordan product,
1919 and the real-part-of-trace inner product. It has dimension `n^2` over
1924 The degree of this algebra is `n^2`::
1926 sage: set_random_seed()
1927 sage: n = ZZ.random_element(1,5)
1928 sage: J = ComplexHermitianEJA(n)
1929 sage: J.degree() == n^2
1932 The Jordan multiplication is what we think it is::
1934 sage: set_random_seed()
1935 sage: n = ZZ.random_element(1,5)
1936 sage: J = ComplexHermitianEJA(n)
1937 sage: x = J.random_element()
1938 sage: y = J.random_element()
1939 sage: actual = (x*y).natural_representation()
1940 sage: X = x.natural_representation()
1941 sage: Y = y.natural_representation()
1942 sage: expected = (X*Y + Y*X)/2
1943 sage: actual == expected
1945 sage: J(expected) == x*y
1950 def __classcall_private__(cls
, n
, field
=QQ
):
1951 S
= _complex_hermitian_basis(n
)
1952 (Qs
, T
) = _multiplication_table_from_matrix_basis(S
)
1954 fdeja
= super(ComplexHermitianEJA
, cls
)
1955 return fdeja
.__classcall
_private
__(cls
,
1961 def inner_product(self
, x
, y
):
1962 # Since a+bi on the diagonal is represented as
1967 # we'll double-count the "a" entries if we take the trace of
1969 return _matrix_ip(x
,y
)/2
1972 class QuaternionHermitianEJA(FiniteDimensionalEuclideanJordanAlgebra
):
1974 The rank-n simple EJA consisting of self-adjoint n-by-n quaternion
1975 matrices, the usual symmetric Jordan product, and the
1976 real-part-of-trace inner product. It has dimension `2n^2 - n` over
1981 The degree of this algebra is `n^2`::
1983 sage: set_random_seed()
1984 sage: n = ZZ.random_element(1,5)
1985 sage: J = QuaternionHermitianEJA(n)
1986 sage: J.degree() == 2*(n^2) - n
1989 The Jordan multiplication is what we think it is::
1991 sage: set_random_seed()
1992 sage: n = ZZ.random_element(1,5)
1993 sage: J = QuaternionHermitianEJA(n)
1994 sage: x = J.random_element()
1995 sage: y = J.random_element()
1996 sage: actual = (x*y).natural_representation()
1997 sage: X = x.natural_representation()
1998 sage: Y = y.natural_representation()
1999 sage: expected = (X*Y + Y*X)/2
2000 sage: actual == expected
2002 sage: J(expected) == x*y
2007 def __classcall_private__(cls
, n
, field
=QQ
):
2008 S
= _quaternion_hermitian_basis(n
)
2009 (Qs
, T
) = _multiplication_table_from_matrix_basis(S
)
2011 fdeja
= super(QuaternionHermitianEJA
, cls
)
2012 return fdeja
.__classcall
_private
__(cls
,
2018 def inner_product(self
, x
, y
):
2019 # Since a+bi+cj+dk on the diagonal is represented as
2021 # a + bi +cj + dk = [ a b c d]
2026 # we'll quadruple-count the "a" entries if we take the trace of
2028 return _matrix_ip(x
,y
)/4
2031 class JordanSpinEJA(FiniteDimensionalEuclideanJordanAlgebra
):
2033 The rank-2 simple EJA consisting of real vectors ``x=(x0, x_bar)``
2034 with the usual inner product and jordan product ``x*y =
2035 (<x_bar,y_bar>, x0*y_bar + y0*x_bar)``. It has dimension `n` over
2040 This multiplication table can be verified by hand::
2042 sage: J = JordanSpinEJA(4)
2043 sage: e0,e1,e2,e3 = J.gens()
2061 def __classcall_private__(cls
, n
, field
=QQ
):
2063 id_matrix
= identity_matrix(field
, n
)
2065 ei
= id_matrix
.column(i
)
2066 Qi
= zero_matrix(field
, n
)
2068 Qi
.set_column(0, ei
)
2069 Qi
+= diagonal_matrix(n
, [ei
[0]]*n
)
2070 # The addition of the diagonal matrix adds an extra ei[0] in the
2071 # upper-left corner of the matrix.
2072 Qi
[0,0] = Qi
[0,0] * ~
field(2)
2075 # The rank of the spin algebra is two, unless we're in a
2076 # one-dimensional ambient space (because the rank is bounded by
2077 # the ambient dimension).
2078 fdeja
= super(JordanSpinEJA
, cls
)
2079 return fdeja
.__classcall
_private
__(cls
, field
, Qs
, rank
=min(n
,2))
2081 def inner_product(self
, x
, y
):
2082 return _usual_ip(x
,y
)