from sage.modules.free_module import VectorSpace
from sage.modules.with_basis.indexed_element import IndexedFreeModuleElement
-from mjo.eja.eja_operator import FiniteDimensionalEJAOperator
+from mjo.eja.eja_operator import EJAOperator
from mjo.eja.eja_utils import _scale
-class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
+class EJAElement(IndexedFreeModuleElement):
"""
An element of a Euclidean Jordan algebra.
"""
SETUP::
- sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
+ sage: from mjo.eja.eja_algebra import (AlbertEJA,
+ ....: JordanSpinEJA,
....: TrivialEJA,
....: RealSymmetricEJA,
....: ComplexHermitianEJA,
Ensure that the determinant is multiplicative on an associative
subalgebra as in Faraut and Korányi's Proposition II.2.2::
- sage: J = random_eja().random_element().subalgebra_generated_by()
+ sage: x0 = random_eja().random_element()
+ sage: J = x0.subalgebra_generated_by(orthonormalize=False)
sage: x,y = J.random_elements(2)
sage: (x*y).det() == x.det()*y.det()
True
sage: actual2 == expected
True
+ There's a formula for the determinant of the Albert algebra
+ (Yokota, Section 2.1)::
+
+ sage: def albert_det(x):
+ ....: X = x.to_matrix()
+ ....: res = X[0,0]*X[1,1]*X[2,2]
+ ....: res += 2*(X[1,2]*X[2,0]*X[0,1]).real()
+ ....: res -= X[0,0]*X[1,2]*X[2,1]
+ ....: res -= X[1,1]*X[2,0]*X[0,2]
+ ....: res -= X[2,2]*X[0,1]*X[1,0]
+ ....: return res.leading_coefficient()
+ sage: J = AlbertEJA(field=QQ, orthonormalize=False)
+ sage: xs = J.random_elements(10)
+ sage: all( albert_det(x) == x.det() for x in xs )
+ True
+
"""
P = self.parent()
r = P.rank()
of an element is the inverse of its left-multiplication operator
applied to the algebra's identity, when that inverse exists::
- sage: J = random_eja()
- sage: x = J.random_element()
- sage: (not x.operator().is_invertible()) or (
- ....: x.operator().inverse()(J.one()) == x.inverse() )
+ sage: J = random_eja() # long time
+ sage: x = J.random_element() # long time
+ sage: (not x.operator().is_invertible()) or ( # long time
+ ....: x.operator().inverse()(J.one()) # long time
+ ....: == # long time
+ ....: x.inverse() ) # long time
True
Check that the fast (cached) and slow algorithms give the same
True
"""
not_invertible_msg = "element is not invertible"
- if self.parent()._charpoly_coefficients.is_in_cache():
+
+ algebra = self.parent()
+ if algebra._charpoly_coefficients.is_in_cache():
# We can invert using our charpoly if it will be fast to
# compute. If the coefficients are cached, our rank had
# better be too!
if self.det().is_zero():
raise ZeroDivisionError(not_invertible_msg)
- r = self.parent().rank()
+ r = algebra.rank()
a = self.characteristic_polynomial().coefficients(sparse=False)
- return (-1)**(r+1)*sum(a[i+1]*self**i for i in range(r))/self.det()
+ return (-1)**(r+1)*algebra.sum(a[i+1]*self**i
+ for i in range(r))/self.det()
try:
inv = (~self.quadratic_representation())(self)
If computing my determinant will be fast, we do so and compare
with zero (Proposition II.2.4 in Faraut and
- Koranyi). Otherwise, Proposition II.3.2 in Faraut and Koranyi
+ Korányi). Otherwise, Proposition II.3.2 in Faraut and Korányi
reduces the problem to the invertibility of my quadratic
representation.
ALGORITHM:
- .........
+ First we handle the special cases where the algebra is
+ trivial, this element is zero, or the dimension of the algebra
+ is one and this element is not zero. With those out of the
+ way, we may assume that ``self`` is nonzero, the algebra is
+ nontrivial, and that the dimension of the algebra is at least
+ two.
+
+ Beginning with the algebra's unit element (power zero), we add
+ successive (basis representations of) powers of this element
+ to a matrix, row-reducing at each step. After row-reducing, we
+ check the rank of the matrix. If adding a row and row-reducing
+ does not increase the rank of the matrix at any point, the row
+ we've just added lives in the span of the previous ones; thus
+ the corresponding power of ``self`` lives in the span of its
+ lesser powers. When that happens, the degree of the minimal
+ polynomial is the rank of the matrix; if it never happens, the
+ degree must be the dimension of the entire space.
SETUP::
sage: x = random_eja().random_element()
sage: x.degree() == x.minimal_polynomial().degree()
True
-
"""
n = self.parent().dimension()
The minimal polynomial should always kill its element::
- sage: x = random_eja().random_element()
- sage: p = x.minimal_polynomial()
- sage: x.apply_univariate_polynomial(p)
+ sage: x = random_eja().random_element() # long time
+ sage: p = x.minimal_polynomial() # long time
+ sage: x.apply_univariate_polynomial(p) # long time
0
The minimal polynomial is invariant under a change of basis,
P = self.parent()
left_mult_by_self = lambda y: self*y
L = P.module_morphism(function=left_mult_by_self, codomain=P)
- return FiniteDimensionalEJAOperator(P, P, L.matrix() )
+ return EJAOperator(P, P, L.matrix() )
def quadratic_representation(self, other=None):
This subalgebra, being composed of only powers, is associative::
sage: x0 = random_eja().random_element()
- sage: A = x0.subalgebra_generated_by()
+ sage: A = x0.subalgebra_generated_by(orthonormalize=False)
sage: x,y,z = A.random_elements(3)
sage: (x*y)*z == x*(y*z)
True
the superalgebra::
sage: x = random_eja().random_element()
- sage: A = x.subalgebra_generated_by()
+ sage: A = x.subalgebra_generated_by(orthonormalize=False)
sage: A(x^2) == A(x)*A(x)
True
where there are non-nilpotent elements, or that we get the dumb
solution in the trivial algebra::
- sage: J = random_eja()
+ sage: J = random_eja(field=QQ, orthonormalize=False)
sage: x = J.random_element()
sage: while x.is_nilpotent() and not J.is_trivial():
....: x = J.random_element()
if self.is_nilpotent():
raise ValueError("this only works with non-nilpotent elements!")
- J = self.subalgebra_generated_by()
+ # The subalgebra is transient (we return an element of the
+ # superalgebra, i.e. this algebra) so why bother
+ # orthonormalizing?
+ J = self.subalgebra_generated_by(orthonormalize=False)
u = J(self)
# The image of the matrix of left-u^m-multiplication
# subspace... or do we? Can't we just solve, knowing that
# A(c) = u^(s+1) should have a solution in the big space,
# too?
- #
- # Beware, solve_right() means that we're using COLUMN vectors.
- # Our FiniteDimensionalAlgebraElement superclass uses rows.
u_next = u**(s+1)
A = u_next.operator().matrix()
c = J.from_vector(A.solve_right(u_next.to_vector()))
- # Now c is the idempotent we want, but it still lives in the subalgebra.
+ # Now c is the idempotent we want, but it still lives in
+ # the subalgebra.
return c.superalgebra_element()
sage: x.trace_inner_product(y) == y.trace_inner_product(x)
True
sage: # bilinear
- sage: a = J.base_ring().random_element();
+ sage: a = J.base_ring().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) )
return self.trace_inner_product(self).sqrt()
-class CartesianProductEJAElement(FiniteDimensionalEJAElement):
- def det(self):
+ def operator_trace_inner_product(self, other):
r"""
- Compute the determinant of this product-element using the
- determianants of its factors.
+ Return the operator inner product of myself and ``other``.
+
+ The "operator inner product," whose name is not standard, is
+ defined be the usual linear-algebraic trace of the
+ ``(x*y).operator()``.
+
+ Proposition III.1.5 in Faraut and Korányi shows that on any
+ Euclidean Jordan algebra, this is another associative inner
+ product under which the cone of squares is symmetric.
+
+ This works even if the basis hasn't been orthonormalized
+ because the eigenvalues of the corresponding matrix don't
+ change when the basis does (they're preserved by any
+ similarity transformation).
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
+ ....: RealSymmetricEJA,
+ ....: ComplexHermitianEJA,
+ ....: random_eja)
+
+ EXAMPLES:
+
+ Proposition III.4.2 of Faraut and Korányi shows that on a
+ simple algebra of rank `r` and dimension `n`, this inner
+ product is `n/r` times the canonical
+ :meth:`trace_inner_product`::
+
+ sage: J = JordanSpinEJA(4, field=QQ)
+ sage: x,y = J.random_elements(2)
+ sage: n = J.dimension()
+ sage: r = J.rank()
+ sage: actual = x.operator_trace_inner_product(y)
+ sage: expected = (n/r)*x.trace_inner_product(y)
+ sage: actual == expected
+ True
+
+ ::
+
+ sage: J = RealSymmetricEJA(3)
+ sage: x,y = J.random_elements(2)
+ sage: n = J.dimension()
+ sage: r = J.rank()
+ sage: actual = x.operator_trace_inner_product(y)
+ sage: expected = (n/r)*x.trace_inner_product(y)
+ sage: actual == expected
+ True
+
+ ::
+
+ sage: J = ComplexHermitianEJA(3, field=QQ, orthonormalize=False)
+ sage: x,y = J.random_elements(2)
+ sage: n = J.dimension()
+ sage: r = J.rank()
+ sage: actual = x.operator_trace_inner_product(y)
+ sage: expected = (n/r)*x.trace_inner_product(y)
+ sage: actual == expected
+ True
+
+ TESTS:
+
+ The operator inner product is commutative, bilinear, and
+ associative::
+
+ sage: J = random_eja()
+ sage: x,y,z = J.random_elements(3)
+ sage: # commutative
+ sage: actual = x.operator_trace_inner_product(y)
+ sage: expected = y.operator_trace_inner_product(x)
+ sage: actual == expected
+ True
+ sage: # bilinear
+ sage: a = J.base_ring().random_element()
+ sage: actual = (a*(x+z)).operator_trace_inner_product(y)
+ sage: expected = ( a*x.operator_trace_inner_product(y) +
+ ....: a*z.operator_trace_inner_product(y) )
+ sage: actual == expected
+ True
+ sage: actual = x.operator_trace_inner_product(a*(y+z))
+ sage: expected = ( a*x.operator_trace_inner_product(y) +
+ ....: a*x.operator_trace_inner_product(z) )
+ sage: actual == expected
+ True
+ sage: # associative
+ sage: actual = (x*y).operator_trace_inner_product(z)
+ sage: expected = y.operator_trace_inner_product(x*z)
+ sage: actual == expected
+ True
+
+ Despite the fact that the implementation uses a matrix representation,
+ the answer is independent of the basis used::
+
+ sage: J = RealSymmetricEJA(3, field=QQ, orthonormalize=False)
+ sage: V = RealSymmetricEJA(3)
+ sage: x,y = J.random_elements(2)
+ sage: w = V(x.to_matrix())
+ sage: z = V(y.to_matrix())
+ sage: expected = x.operator_trace_inner_product(y)
+ sage: actual = w.operator_trace_inner_product(z)
+ sage: actual == expected
+ True
- This result Follows from the spectral decomposition of (say)
- the pair `(x,y)` in terms of the Jordan frame `\left\{ (c_1,
- 0),(c_2, 0),...,(0,d_1),(0,d_2),... \right\}.
"""
- from sage.misc.misc_c import prod
- return prod( f.det() for f in self.cartesian_factors() )
+ if not other in self.parent():
+ raise TypeError("'other' must live in the same algebra")
+
+ return (self*other).operator().matrix().trace()
+
+
+ def operator_trace_norm(self):
+ """
+ The norm of this element with respect to
+ :meth:`operator_trace_inner_product`.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
+ ....: HadamardEJA)
+ EXAMPLES:
+
+ On a simple algebra, this will differ from :meth:`trace_norm`
+ by the scalar factor ``(n/r).sqrt()``, where `n` is the
+ dimension of the algebra and `r` its rank. This follows from
+ the corresponding result (Proposition III.4.2 of Faraut and
+ Korányi) for the trace inner product::
+
+ sage: J = HadamardEJA(2)
+ sage: x = sum(J.gens())
+ sage: x.operator_trace_norm()
+ 1.414213562373095?
+
+ ::
+
+ sage: J = JordanSpinEJA(4)
+ sage: x = sum(J.gens())
+ sage: x.operator_trace_norm()
+ 4
+
+ """
+ return self.operator_trace_inner_product(self).sqrt()
+
+
+class CartesianProductParentEJAElement(EJAElement):
+ r"""
+ An intermediate class for elements that have a Cartesian
+ product as their parent algebra.
+
+ This is needed because the ``to_matrix`` method (which gives you a
+ representation from the superalgebra) needs to do special stuff
+ for Cartesian products. Specifically, an EJA subalgebra of a
+ Cartesian product EJA will not itself be a Cartesian product (it
+ has its own basis) -- but we want ``to_matrix()`` to be able to
+ give us a Cartesian product representation.
+ """
def to_matrix(self):
# An override is necessary to call our custom _scale().
B = self.parent().matrix_basis()
# Aaaaand linear combinations don't work in Cartesian
# product spaces, even though they provide a method with
- # that name. This is hidden behind an "if" because the
+ # that name. This is hidden in a subclass because the
# _scale() function is slow.
pairs = zip(B, self.to_vector())
return W.sum( _scale(b, alpha) for (b,alpha) in pairs )
+
+class CartesianProductEJAElement(CartesianProductParentEJAElement):
+ def det(self):
+ r"""
+ Compute the determinant of this product-element using the
+ determianants of its factors.
+
+ This result Follows from the spectral decomposition of (say)
+ the pair `(x,y)` in terms of the Jordan frame `\left\{ (c_1,
+ 0),(c_2, 0),...,(0,d_1),(0,d_2),... \right\}.
+ """
+ from sage.misc.misc_c import prod
+ return prod( f.det() for f in self.cartesian_factors() )
projects / sage.d.git / blobdiff