X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_element.py;h=c98d9a2b64651562928a6489ef88e03fdd2b0dc9;hb=bc02bf48592e22d034310cfffef8fb2a062c0a43;hp=347aa3c211278a15a691944c9a559076c5817acf;hpb=028e405e8fd8672d0a68d21118305586a34927ad;p=sage.d.git diff --git a/mjo/eja/eja_element.py b/mjo/eja/eja_element.py index 347aa3c..c98d9a2 100644 --- a/mjo/eja/eja_element.py +++ b/mjo/eja/eja_element.py @@ -4,7 +4,7 @@ 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_utils import _mat2vec +from mjo.eja.eja_utils import _mat2vec, _scale class FiniteDimensionalEJAElement(IndexedFreeModuleElement): """ @@ -167,8 +167,8 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): sage: J = HadamardEJA(3) sage: p1 = J.one().characteristic_polynomial() sage: q1 = J.zero().characteristic_polynomial() - sage: e0,e1,e2 = J.gens() - sage: A = (e0 + 2*e1 + 3*e2).subalgebra_generated_by() # dim 3 + sage: b0,b1,b2 = J.gens() + sage: A = (b0 + 2*b1 + 3*b2).subalgebra_generated_by() # dim 3 sage: p2 = A.one().characteristic_polynomial() sage: q2 = A.zero().characteristic_polynomial() sage: p1 == p2 @@ -348,7 +348,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): EXAMPLES:: sage: J = JordanSpinEJA(2) - sage: e0,e1 = J.gens() sage: x = sum( J.gens() ) sage: x.det() 0 @@ -356,7 +355,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): :: sage: J = JordanSpinEJA(3) - sage: e0,e1,e2 = J.gens() sage: x = sum( J.gens() ) sage: x.det() -1 @@ -391,7 +389,7 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): sage: (x*y).det() == x.det()*y.det() True - The determinant in matrix algebras is just the usual determinant:: + The determinant in real matrix algebras is the usual determinant:: sage: set_random_seed() sage: X = matrix.random(QQ,3) @@ -406,21 +404,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): sage: actual2 == expected True - :: - - sage: set_random_seed() - sage: J1 = ComplexHermitianEJA(2) - sage: J2 = ComplexHermitianEJA(2,field=QQ,orthonormalize=False) - sage: X = matrix.random(GaussianIntegers(), 2) - sage: X = X + X.H - sage: expected = AA(X.det()) - sage: actual1 = J1(J1.real_embed(X)).det() - sage: actual2 = J2(J2.real_embed(X)).det() - sage: expected == actual1 - True - sage: expected == actual2 - True - """ P = self.parent() r = P.rank() @@ -664,7 +647,7 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): element should always be in terms of minimal idempotents:: sage: J = JordanSpinEJA(4) - sage: x = sum( i*J.gens()[i] for i in range(len(J.gens())) ) + sage: x = sum( i*J.monomial(i) for i in range(len(J.gens())) ) sage: x.is_regular() True sage: [ c.is_primitive_idempotent() @@ -793,7 +776,9 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): sage: J = JordanSpinEJA(5) sage: J.one().is_regular() False - sage: e0, e1, e2, e3, e4 = J.gens() # e0 is the identity + sage: b0, b1, b2, b3, b4 = J.gens() + sage: b0 == J.one() + True sage: for x in J.gens(): ....: (J.one() + x).is_regular() False @@ -843,8 +828,8 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): sage: J = JordanSpinEJA(4) sage: J.one().degree() 1 - sage: e0,e1,e2,e3 = J.gens() - sage: (e0 - e1).degree() + sage: b0,b1,b2,b3 = J.gens() + sage: (b0 - b1).degree() 2 In the spin factor algebra (of rank two), all elements that @@ -910,7 +895,7 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): M = matrix([(self.parent().one()).to_vector()]) old_rank = 1 - # Specifying the row-reduction algorithm can e.g. help over + # Specifying the row-reduction algorithm can e.g. help over # AA because it avoids the RecursionError that gets thrown # when we have to look too hard for a root. # @@ -1047,19 +1032,30 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): """ if self.is_zero(): - # We would generate a zero-dimensional subalgebra - # where the minimal polynomial would be constant. - # That might be correct, but only if *this* algebra - # is trivial too. - if not self.parent().is_trivial(): - # Pretty sure we know what the minimal polynomial of - # the zero operator is going to be. This ensures - # consistency of e.g. the polynomial variable returned - # in the "normal" case without us having to think about it. - return self.operator().minimal_polynomial() - + # Pretty sure we know what the minimal polynomial of + # the zero operator is going to be. This ensures + # consistency of e.g. the polynomial variable returned + # in the "normal" case without us having to think about it. + return self.operator().minimal_polynomial() + + # If we don't orthonormalize the subalgebra's basis, then the + # first two monomials in the subalgebra will be self^0 and + # self^1... assuming that self^1 is not a scalar multiple of + # self^0 (the unit element). We special case these to avoid + # having to solve a system to coerce self into the subalgebra. A = self.subalgebra_generated_by(orthonormalize=False) - return A(self).operator().minimal_polynomial() + + if A.dimension() == 1: + # Does a solve to find the scalar multiple alpha such that + # alpha*unit = self. We have to do this because the basis + # for the subalgebra will be [ self^0 ], and not [ self^1 ]! + unit = self.parent().one() + alpha = self.to_vector() / unit.to_vector() + return (unit.operator()*alpha).minimal_polynomial() + else: + # If the dimension of the subalgebra is >= 2, then we just + # use the second basis element. + return A.monomial(1).operator().minimal_polynomial() @@ -1077,44 +1073,64 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): SETUP:: sage: from mjo.eja.eja_algebra import (ComplexHermitianEJA, - ....: QuaternionHermitianEJA) + ....: HadamardEJA, + ....: QuaternionHermitianEJA, + ....: RealSymmetricEJA) EXAMPLES:: sage: J = ComplexHermitianEJA(3) sage: J.one() - e0 + e3 + e8 + b0 + b3 + b8 sage: J.one().to_matrix() - [1 0 0 0 0 0] - [0 1 0 0 0 0] - [0 0 1 0 0 0] - [0 0 0 1 0 0] - [0 0 0 0 1 0] - [0 0 0 0 0 1] + +---+---+---+ + | 1 | 0 | 0 | + +---+---+---+ + | 0 | 1 | 0 | + +---+---+---+ + | 0 | 0 | 1 | + +---+---+---+ :: sage: J = QuaternionHermitianEJA(2) sage: J.one() - e0 + e5 + b0 + b5 sage: J.one().to_matrix() - [1 0 0 0 0 0 0 0] - [0 1 0 0 0 0 0 0] - [0 0 1 0 0 0 0 0] - [0 0 0 1 0 0 0 0] - [0 0 0 0 1 0 0 0] - [0 0 0 0 0 1 0 0] - [0 0 0 0 0 0 1 0] - [0 0 0 0 0 0 0 1] + +---+---+ + | 1 | 0 | + +---+---+ + | 0 | 1 | + +---+---+ + + This also works in Cartesian product algebras:: + + 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() - # This is just a manual "from_vector()", but of course - # matrix spaces aren't vector spaces in sage, so they - # don't have a from_vector() method. - return W.linear_combination( zip(B, self.to_vector()) ) + if hasattr(W, 'cartesian_factors'): + # 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 + # _scale() function is slow. + pairs = zip(B, self.to_vector()) + return W.sum( _scale(b, alpha) for (b,alpha) in pairs ) + else: + # This is just a manual "from_vector()", but of course + # matrix spaces aren't vector spaces in sage, so they + # don't have a from_vector() method. + return W.linear_combination( zip(B, self.to_vector()) ) @@ -1322,11 +1338,11 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): sage: J = RealSymmetricEJA(3) sage: J.one() - e0 + e2 + e5 + b0 + b2 + b5 sage: J.one().spectral_decomposition() - [(1, e0 + e2 + e5)] + [(1, b0 + b2 + b5)] sage: J.zero().spectral_decomposition() - [(0, e0 + e2 + e5)] + [(0, b0 + b2 + b5)] TESTS:: @@ -1351,13 +1367,13 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): The spectral decomposition should work in subalgebras, too:: sage: J = RealSymmetricEJA(4) - sage: (e0, e1, e2, e3, e4, e5, e6, e7, e8, e9) = J.gens() - sage: A = 2*e5 - 2*e8 + sage: (b0, b1, b2, b3, b4, b5, b6, b7, b8, b9) = J.gens() + sage: A = 2*b5 - 2*b8 sage: (lambda1, c1) = A.spectral_decomposition()[1] sage: (J0, J5, J1) = J.peirce_decomposition(c1) sage: (f0, f1, f2) = J1.gens() sage: f0.spectral_decomposition() - [(0, f2), (1, f0)] + [(0, c2), (1, c0)] """ A = self.subalgebra_generated_by(orthonormalize=True) @@ -1380,7 +1396,20 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): SETUP:: - sage: from mjo.eja.eja_algebra import random_eja + sage: from mjo.eja.eja_algebra import (random_eja, + ....: HadamardEJA, + ....: RealSymmetricEJA) + + EXAMPLES: + + We can create subalgebras of Cartesian product EJAs that are not + themselves Cartesian product EJAs (they're just "regular" EJAs):: + + sage: J1 = HadamardEJA(3) + sage: J2 = RealSymmetricEJA(2) + sage: J = cartesian_product([J1,J2]) + sage: J.one().subalgebra_generated_by() + Euclidean Jordan algebra of dimension 1 over Algebraic Real Field TESTS: @@ -1413,12 +1442,12 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement): True """ - from mjo.eja.eja_subalgebra import FiniteDimensionalEJASubalgebra powers = tuple( self**k for k in range(self.degree()) ) - A = FiniteDimensionalEJASubalgebra(self.parent(), - powers, - associative=True, - **kwargs) + A = self.parent().subalgebra(powers, + associative=True, + check_field=False, + check_axioms=False, + **kwargs) A.one.set_cache(A(self.parent().one())) return A @@ -1619,39 +1648,3 @@ 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()) -