X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_element.py;h=42e5782c4539c97fcd71805b50c81cd4a96877a5;hb=f292189be8e5a79f9ae2b80ddaff76460e0d14c2;hp=f0f6da756e841e65b7d9272f5feb09404c9f1a09;hpb=35ecc332201ce37f6ad1f6ac05b696d8c73c9cb3;p=sage.d.git diff --git a/mjo/eja/eja_element.py b/mjo/eja/eja_element.py index f0f6da7..42e5782 100644 --- a/mjo/eja/eja_element.py +++ b/mjo/eja/eja_element.py @@ -1,4 +1,5 @@ from sage.matrix.constructor import matrix +from sage.misc.cachefunc import cached_method from sage.modules.free_module import VectorSpace from sage.modules.with_basis.indexed_element import IndexedFreeModuleElement @@ -438,6 +439,7 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): return ((-1)**r)*p(*self.to_vector()) + @cached_method def inverse(self): """ Return the Jordan-multiplicative inverse of this element. @@ -482,7 +484,7 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): sage: JordanSpinEJA(3).zero().inverse() Traceback (most recent call last): ... - ValueError: element is not invertible + ZeroDivisionError: element is not invertible TESTS: @@ -534,40 +536,38 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): sage: slow == fast # long time True """ - if not self.is_invertible(): - raise ValueError("element is not invertible") - + not_invertible_msg = "element is not invertible" if self.parent()._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() 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 (~self.quadratic_representation())(self) + try: + inv = (~self.quadratic_representation())(self) + self.is_invertible.set_cache(True) + return inv + except ZeroDivisionError: + self.is_invertible.set_cache(False) + raise ZeroDivisionError(not_invertible_msg) + @cached_method def is_invertible(self): """ Return whether or not this element is invertible. ALGORITHM: - The usual way to do this is to check if the determinant is - zero, but we need the characteristic polynomial for the - determinant. The minimal polynomial is a lot easier to get, - so we use Corollary 2 in Chapter V of Koecher to check - whether or not the parent algebra's zero element is a root - of this element's minimal polynomial. - - That is... unless the coefficients of our algebra's - "characteristic polynomial of" function are already cached! - In that case, we just use the determinant (which will be fast - as a result). - - Beware that we can't use the superclass method, because it - relies on the algebra being associative. + 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 + reduces the problem to the invertibility of my quadratic + representation. SETUP:: @@ -600,7 +600,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): sage: fast = x.is_invertible() # long time sage: slow == fast # long time True - """ if self.is_zero(): if self.parent().is_trivial(): @@ -609,15 +608,17 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): return False if self.parent()._charpoly_coefficients.is_in_cache(): - # The determinant will be quicker than computing the minimal - # polynomial from scratch, most likely. + # The determinant will be quicker than inverting the + # quadratic representation, most likely. return (not self.det().is_zero()) - # In fact, we only need to know if the constant term is non-zero, - # so we can pass in the field's zero element instead. - zero = self.base_ring().zero() - p = self.minimal_polynomial() - return not (p(zero) == zero) + # The easiest way to determine if I'm invertible is to try. + try: + inv = (~self.quadratic_representation())(self) + self.inverse.set_cache(inv) + return True + except ZeroDivisionError: + return False def is_primitive_idempotent(self): @@ -1057,7 +1058,7 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): # in the "normal" case without us having to think about it. return self.operator().minimal_polynomial() - A = self.subalgebra_generated_by(orthonormalize_basis=False) + A = self.subalgebra_generated_by(orthonormalize=False) return A(self).operator().minimal_polynomial() @@ -1116,6 +1117,7 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): return W.linear_combination( zip(B, self.to_vector()) ) + def norm(self): """ The norm of this element with respect to :meth:`inner_product`. @@ -1358,7 +1360,7 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): [(0, f2), (1, f0)] """ - A = self.subalgebra_generated_by(orthonormalize_basis=True) + A = self.subalgebra_generated_by(orthonormalize=True) result = [] for (evalue, proj) in A(self).operator().spectral_decomposition(): result.append( (evalue, proj(A.one()).superalgebra_element()) ) @@ -1411,8 +1413,10 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): True """ - from mjo.eja.eja_element_subalgebra import FiniteDimensionalEJAElementSubalgebra - return FiniteDimensionalEJAElementSubalgebra(self, **kwargs) + powers = tuple( self**k for k in range(self.degree()) ) + A = self.parent().subalgebra(powers, associative=True, **kwargs) + A.one.set_cache(A(self.parent().one())) + return A def subalgebra_idempotent(self): @@ -1519,6 +1523,15 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): sage: J.random_element().trace() in RLF True + The trace is linear:: + + sage: set_random_seed() + sage: J = random_eja() + sage: x,y = J.random_elements(2) + sage: alpha = J.base_ring().random_element() + sage: (alpha*x + y).trace() == alpha*x.trace() + y.trace() + True + """ P = self.parent() r = P.rank() @@ -1602,3 +1615,38 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): """ return self.trace_inner_product(self).sqrt() + + + +class CartesianProductEJAElement(FiniteDimensionalEJAElement): + + def to_matrix(self): + r""" + SETUP:: + + sage: from mjo.eja.eja_algebra import (HadamardEJA, + ....: RealSymmetricEJA) + + EXAMPLES:: + + sage: J1 = HadamardEJA(1) + sage: J2 = RealSymmetricEJA(2) + sage: J = cartesian_product([J1,J2]) + sage: x = sum(J.gens()) + sage: x.to_matrix()[0] + [1] + sage: x.to_matrix()[1] + [ 1 0.7071067811865475?] + [0.7071067811865475? 1] + + """ + B = self.parent().matrix_basis() + W = self.parent().matrix_space() + + # Aaaaand linear combinations don't work in Cartesian + # product spaces, even though they provide a method + # with that name. + pairs = zip(B, self.to_vector()) + return sum( ( W(tuple(alpha*b_i for b_i in b)) + for (b,alpha) in pairs ), + W.zero())