from sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra import FiniteDimensionalAlgebra
from sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra_element import FiniteDimensionalAlgebraElement
+from sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra_morphism import FiniteDimensionalAlgebraMorphism
+
+
+class FiniteDimensionalEuclideanJordanAlgebraMorphism(FiniteDimensionalAlgebraMorphism):
+ """
+ A very thin wrapper around FiniteDimensionalAlgebraMorphism that
+ does only two things:
+
+ 1. Avoids the ``unitary`` and ``check`` arguments to the constructor
+ that will always be ``False``. This is necessary because these
+ are homomorphisms with respect to ADDITION, but the SageMath
+ machinery wants to check that they're homomorphisms with respect
+ to (Jordan) MULTIPLICATION. That obviously doesn't work.
+
+ 2. Inputs and outputs the underlying matrix with respect to COLUMN
+ vectors, unlike the parent class.
+
+ If this seems a bit heavyweight, it is. I would have been happy to
+ use a the ring morphism that underlies the finite-dimensional
+ algebra morphism, but they don't seem to be callable on elements of
+ our EJA.
+ """
+ def __init__(self, parent, f):
+ FiniteDimensionalAlgebraMorphism.__init__(self,
+ parent,
+ f.transpose(),
+ unitary=False,
+ check=False)
+
+
+ def _repr_(self):
+ """
+ We override only the representation that is shown to the user,
+ because we want the matrix to be with respect to COLUMN vectors.
+
+ TESTS:
+
+ Ensure that we see the transpose of the underlying matrix object:
+
+ sage: J = RealSymmetricEJA(3)
+ sage: x = J.linear_combination(zip(range(len(J.gens())), J.gens()))
+ sage: L = x.operator()
+ sage: L
+ Morphism from Euclidean Jordan algebra of degree 6 over Rational
+ Field to Euclidean Jordan algebra of degree 6 over Rational Field
+ given by matrix
+ [ 0 1 2 0 0 0]
+ [1/2 3/2 2 1/2 1 0]
+ [ 1 2 5/2 0 1/2 1]
+ [ 0 1 0 3 4 0]
+ [ 0 1 1/2 2 4 2]
+ [ 0 0 2 0 4 5]
+ sage: L._matrix
+ [ 0 1/2 1 0 0 0]
+ [ 1 3/2 2 1 1 0]
+ [ 2 2 5/2 0 1/2 2]
+ [ 0 1/2 0 3 2 0]
+ [ 0 1 1/2 4 4 4]
+ [ 0 0 1 0 2 5]
+
+ """
+ return "Morphism from {} to {} given by matrix\n{}".format(
+ self.domain(), self.codomain(), self.matrix())
+
+ def matrix(self):
+ """
+ Return the matrix of this morphism with respect to a left-action
+ on column vectors.
+ """
+ return FiniteDimensionalAlgebraMorphism.matrix(self).transpose()
+
class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
@staticmethod
return fmt.format(self.degree(), self.base_ring())
+ def _a_regular_element(self):
+ """
+ Guess a regular element. Needed to compute the basis for our
+ characteristic polynomial coefficients.
+ """
+ gs = self.gens()
+ z = self.sum( (i+1)*gs[i] for i in range(len(gs)) )
+ if not z.is_regular():
+ raise ValueError("don't know a regular element")
+ return z
+
+
+ @cached_method
+ def _charpoly_basis_space(self):
+ """
+ Return the vector space spanned by the basis used in our
+ characteristic polynomial coefficients. This is used not only to
+ compute those coefficients, but also any time we need to
+ evaluate the coefficients (like when we compute the trace or
+ determinant).
+ """
+ z = self._a_regular_element()
+ V = z.vector().parent().ambient_vector_space()
+ V1 = V.span_of_basis( (z**k).vector() for k in range(self.rank()) )
+ b = (V1.basis() + V1.complement().basis())
+ return V.span_of_basis(b)
+
@cached_method
def _charpoly_coeff(self, i):
store the trace/determinant (a_{r-1} and a_{0} respectively)
separate from the entire characteristic polynomial.
"""
- (A_of_x, x) = self._charpoly_matrix()
+ (A_of_x, x, xr, detA) = self._charpoly_matrix_system()
R = A_of_x.base_ring()
- A_cols = A_of_x.columns()
- A_cols[i] = (x**self.rank()).vector()
- numerator = column_matrix(A_of_x.base_ring(), A_cols).det()
- denominator = A_of_x.det()
+ if i >= self.rank():
+ # Guaranteed by theory
+ return R.zero()
+
+ # Danger: the in-place modification is done for performance
+ # reasons (reconstructing a matrix with huge polynomial
+ # entries is slow), but I don't know how cached_method works,
+ # so it's highly possible that we're modifying some global
+ # list variable by reference, here. In other words, you
+ # probably shouldn't call this method twice on the same
+ # algebra, at the same time, in two threads
+ Ai_orig = A_of_x.column(i)
+ A_of_x.set_column(i,xr)
+ numerator = A_of_x.det()
+ A_of_x.set_column(i,Ai_orig)
# We're relying on the theory here to ensure that each a_i is
# indeed back in R, and the added negative signs are to make
# the whole charpoly expression sum to zero.
- return R(-numerator/denominator)
+ return R(-numerator/detA)
@cached_method
- def _charpoly_matrix(self):
+ def _charpoly_matrix_system(self):
"""
Compute the matrix whose entries A_ij are polynomials in
- X1,...,XN. This same matrix is used in more than one method and
- it's not so fast to construct.
+ X1,...,XN, the vector ``x`` of variables X1,...,XN, the vector
+ corresponding to `x^r` and the determinent of the matrix A =
+ [A_ij]. In other words, all of the fixed (cachable) data needed
+ to compute the coefficients of the characteristic polynomial.
"""
r = self.rank()
n = self.dimension()
idmat = identity_matrix(J.base_ring(), n)
+ W = self._charpoly_basis_space()
+ W = W.change_ring(R.fraction_field())
+
+ # Starting with the standard coordinates x = (X1,X2,...,Xn)
+ # and then converting the entries to W-coordinates allows us
+ # to pass in the standard coordinates to the charpoly and get
+ # back the right answer. Specifically, with x = (X1,X2,...,Xn),
+ # we have
+ #
+ # W.coordinates(x^2) eval'd at (standard z-coords)
+ # =
+ # W-coords of (z^2)
+ # =
+ # W-coords of (standard coords of x^2 eval'd at std-coords of z)
+ #
+ # We want the middle equivalent thing in our matrix, but use
+ # the first equivalent thing instead so that we can pass in
+ # standard coordinates.
x = J(vector(R, R.gens()))
- l1 = [column_matrix((x**k).vector()) for k in range(r)]
+ l1 = [column_matrix(W.coordinates((x**k).vector())) for k in range(r)]
l2 = [idmat.column(k-1).column() for k in range(r+1, n+1)]
A_of_x = block_matrix(R, 1, n, (l1 + l2))
- return (A_of_x, x)
+ xr = W.coordinates((x**r).vector())
+ return (A_of_x, x, xr, A_of_x.det())
@cached_method
def characteristic_polynomial(self):
"""
+
+ .. WARNING::
+
+ This implementation doesn't guarantee that the polynomial
+ denominator in the coefficients is not identically zero, so
+ theoretically it could crash. The way that this is handled
+ in e.g. Faraut and Koranyi is to use a basis that guarantees
+ the denominator is non-zero. But, doing so requires knowledge
+ of at least one regular element, and we don't even know how
+ to do that. The trade-off is that, if we use the standard basis,
+ the resulting polynomial will accept the "usual" coordinates. In
+ other words, we don't have to do a change of basis before e.g.
+ computing the trace or determinant.
+
EXAMPLES:
The characteristic polynomial in the spin algebra is given in
An element of a Euclidean Jordan algebra.
"""
+ def __dir__(self):
+ """
+ Oh man, I should not be doing this. This hides the "disabled"
+ methods ``left_matrix`` and ``matrix`` from introspection;
+ in particular it removes them from tab-completion.
+ """
+ return filter(lambda s: s not in ['left_matrix', 'matrix'],
+ dir(self.__class__) )
+
+
def __init__(self, A, elt=None):
"""
EXAMPLES:
sage: J = JordanSpinEJA(2)
sage: e0,e1 = J.gens()
- sage: x = e0 + e1
+ sage: x = sum( J.gens() )
sage: x.det()
0
+
+ ::
+
sage: J = JordanSpinEJA(3)
sage: e0,e1,e2 = J.gens()
- sage: x = e0 + e1 + e2
+ sage: x = sum( J.gens() )
sage: x.det()
-1
+ TESTS:
+
+ An element is invertible if and only if its determinant is
+ non-zero::
+
+ sage: set_random_seed()
+ sage: x = random_eja().random_element()
+ sage: x.is_invertible() == (x.det() != 0)
+ True
+
"""
- cs = self.characteristic_polynomial().coefficients(sparse=False)
- r = len(cs) - 1
- if r >= 0:
- return cs[0] * (-1)**r
- else:
- raise ValueError('charpoly had no coefficients')
+ P = self.parent()
+ r = P.rank()
+ p = P._charpoly_coeff(0)
+ # The _charpoly_coeff function already adds the factor of
+ # -1 to ensure that _charpoly_coeff(0) is really what
+ # appears in front of t^{0} in the charpoly. However,
+ # we want (-1)^r times THAT for the determinant.
+ return ((-1)**r)*p(*self.vector())
def inverse(self):
"""
Return the Jordan-multiplicative inverse of this element.
- We can't use the superclass method because it relies on the
- algebra being associative.
+ ALGORITHM:
+
+ We appeal to the quadratic representation as in Koecher's
+ Theorem 12 in Chapter III, Section 5.
EXAMPLES:
sage: J.one().inverse() == J.one()
True
- If an element has an inverse, it acts like one. TODO: this
- can be a lot less ugly once ``is_invertible`` doesn't crash
- on irregular elements::
+ If an element has an inverse, it acts like one::
sage: set_random_seed()
sage: J = random_eja()
sage: x = J.random_element()
- sage: try:
- ....: x.inverse()*x == J.one()
- ....: except:
- ....: True
+ sage: (not x.is_invertible()) or (x.inverse()*x == J.one())
True
- """
- if self.parent().is_associative():
- elt = FiniteDimensionalAlgebraElement(self.parent(), self)
- return elt.inverse()
+ The inverse of the inverse is what we started with::
- # TODO: we can do better once the call to is_invertible()
- # doesn't crash on irregular elements.
- #if not self.is_invertible():
- # raise ValueError('element is not invertible')
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: x = J.random_element()
+ sage: (not x.is_invertible()) or (x.inverse().inverse() == x)
+ True
- # We do this a little different than the usual recursive
- # call to a finite-dimensional algebra element, because we
- # wind up with an inverse that lives in the subalgebra and
- # we need information about the parent to convert it back.
- V = self.span_of_powers()
- assoc_subalg = self.subalgebra_generated_by()
- # Mis-design warning: the basis used for span_of_powers()
- # and subalgebra_generated_by() must be the same, and in
- # the same order!
- elt = assoc_subalg(V.coordinates(self.vector()))
+ The zero element is never invertible::
+
+ sage: set_random_seed()
+ sage: J = random_eja().zero().inverse()
+ Traceback (most recent call last):
+ ...
+ ValueError: element is not invertible
- # This will be in the subalgebra's coordinates...
- fda_elt = FiniteDimensionalAlgebraElement(assoc_subalg, elt)
- subalg_inverse = fda_elt.inverse()
+ """
+ if not self.is_invertible():
+ raise ValueError("element is not invertible")
- # So we have to convert back...
- basis = [ self.parent(v) for v in V.basis() ]
- pairs = zip(subalg_inverse.vector(), basis)
- return self.parent().linear_combination(pairs)
+ P = self.parent()
+ return P(self.quadratic_representation().inverse()*self.vector())
def is_invertible(self):
return self.span_of_powers().dimension()
+ def left_matrix(self):
+ """
+ Our parent class defines ``left_matrix`` and ``matrix``
+ methods whose names are misleading. We don't want them.
+ """
+ raise NotImplementedError("use operator_matrix() instead")
+
+ matrix = left_matrix
+
+
def minimal_polynomial(self):
"""
Return the minimal polynomial of this element,
Return the matrix that represents left- (or right-)
multiplication by this element in the parent algebra.
- We have to override this because the superclass method
- returns a matrix that acts on row vectors (that is, on
- the right).
+ We implement this ourselves to work around the fact that
+ our parent class represents everything with row vectors.
EXAMPLES:
sage: J = random_eja()
sage: x = J.random_element()
sage: y = J.random_element()
+ sage: Lx = x.operator_matrix()
+ sage: Lxx = (x*x).operator_matrix()
+ sage: Qx = x.quadratic_representation()
+ sage: Qy = y.quadratic_representation()
+ sage: Qxy = x.quadratic_representation(y)
+ sage: Qex = J.one().quadratic_representation(x)
+ sage: n = ZZ.random_element(10)
+ sage: Qxn = (x^n).quadratic_representation()
Property 1:
- sage: actual = x.quadratic_representation(y)
- sage: expected = ( (x+y).quadratic_representation()
- ....: -x.quadratic_representation()
- ....: -y.quadratic_representation() ) / 2
- sage: actual == expected
+ sage: 2*Qxy == (x+y).quadratic_representation() - Qx - Qy
True
Property 2:
sage: alpha = QQ.random_element()
- sage: actual = (alpha*x).quadratic_representation()
- sage: expected = (alpha^2)*x.quadratic_representation()
- sage: actual == expected
+ sage: (alpha*x).quadratic_representation() == (alpha^2)*Qx
+ True
+
+ Property 3:
+
+ sage: not x.is_invertible() or (
+ ....: Qx*x.inverse().vector() == x.vector() )
+ True
+
+ sage: not x.is_invertible() or (
+ ....: Qx.inverse()
+ ....: ==
+ ....: x.inverse().quadratic_representation() )
+ True
+
+ sage: Qxy*(J.one().vector()) == (x*y).vector()
+ True
+
+ Property 4:
+
+ sage: not x.is_invertible() or (
+ ....: x.quadratic_representation(x.inverse())*Qx
+ ....: == Qx*x.quadratic_representation(x.inverse()) )
+ True
+
+ sage: not x.is_invertible() or (
+ ....: x.quadratic_representation(x.inverse())*Qx
+ ....: ==
+ ....: 2*x.operator_matrix()*Qex - Qx )
+ True
+
+ sage: 2*x.operator_matrix()*Qex - Qx == Lxx
True
Property 5:
- sage: Qy = y.quadratic_representation()
- sage: actual = J(Qy*x.vector()).quadratic_representation()
- sage: expected = Qy*x.quadratic_representation()*Qy
- sage: actual == expected
+ sage: J(Qy*x.vector()).quadratic_representation() == Qy*Qx*Qy
True
Property 6:
- sage: k = ZZ.random_element(1,10)
- sage: actual = (x^k).quadratic_representation()
- sage: expected = (x.quadratic_representation())^k
- sage: actual == expected
+ sage: Qxn == (Qx)^n
+ True
+
+ Property 7:
+
+ sage: not x.is_invertible() or (
+ ....: Qx*x.inverse().operator_matrix() == Lx )
+ True
+
+ Property 8:
+
+ sage: not x.operator_commutes_with(y) or (
+ ....: J(Qx*y.vector())^n == J(Qxn*(y^n).vector()) )
True
"""
TESTS::
sage: set_random_seed()
- sage: J = RealCartesianProductEJA(5)
- sage: c = J.random_element().subalgebra_idempotent()
- sage: c^2 == c
- True
- sage: J = JordanSpinEJA(5)
- sage: c = J.random_element().subalgebra_idempotent()
+ sage: J = random_eja()
+ sage: x = J.random_element()
+ sage: while x.is_nilpotent():
+ ....: x = J.random_element()
+ sage: c = x.subalgebra_idempotent()
sage: c^2 == c
True
def trace_inner_product(self, other):
"""
Return the trace inner product of myself and ``other``.
+
+ TESTS:
+
+ The trace inner product is commutative::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: x = J.random_element(); y = J.random_element()
+ sage: x.trace_inner_product(y) == y.trace_inner_product(x)
+ True
+
+ The trace inner product is bilinear::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: x = J.random_element()
+ sage: y = J.random_element()
+ sage: z = J.random_element()
+ sage: a = QQ.random_element();
+ sage: actual = (a*(x+z)).trace_inner_product(y)
+ sage: expected = ( a*x.trace_inner_product(y) +
+ ....: a*z.trace_inner_product(y) )
+ sage: actual == expected
+ True
+ sage: actual = x.trace_inner_product(a*(y+z))
+ sage: expected = ( a*x.trace_inner_product(y) +
+ ....: a*x.trace_inner_product(z) )
+ sage: actual == expected
+ True
+
+ The trace inner product satisfies the compatibility
+ condition in the definition of a Euclidean Jordan algebra::
+
+ sage: set_random_seed()
+ sage: J = random_eja()
+ sage: x = J.random_element()
+ sage: y = J.random_element()
+ sage: z = J.random_element()
+ sage: (x*y).trace_inner_product(z) == y.trace_inner_product(x*z)
+ True
+
"""
if not other in self.parent():
raise TypeError("'other' must live in the same algebra")
projects / sage.d.git / blobdiff