From a3b94d0fdb734aeb91875d2a2ceaece99d129934 Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Sun, 13 Sep 2020 12:51:45 -0400 Subject: [PATCH] mjo/matrix_vector: replace isomorphism with basis_representation(). --- mjo/cone/doubly_nonnegative.py | 4 +- mjo/matrix_vector.py | 125 ++++++++++++++++++++++++++------- 2 files changed, 101 insertions(+), 28 deletions(-) diff --git a/mjo/cone/doubly_nonnegative.py b/mjo/cone/doubly_nonnegative.py index 1fb350c..373557c 100644 --- a/mjo/cone/doubly_nonnegative.py +++ b/mjo/cone/doubly_nonnegative.py @@ -16,7 +16,7 @@ from sage.all import * from mjo.cone.symmetric_psd import (factor_psd, is_symmetric_psd, random_symmetric_psd) -from mjo.matrix_vector import isomorphism +from mjo.matrix_vector import basis_representation def is_doubly_nonnegative(A): @@ -383,7 +383,7 @@ def is_extreme_doubly_nonnegative(A): # can't compute the dimension of a set of matrices anyway, so we # convert them all to vectors and just ask for the dimension of the # resulting vector space. - (phi, phi_inverse) = isomorphism(A.matrix_space()) + (phi, phi_inverse) = basis_representation(A.matrix_space()) vectors = map(phi,spanning_set) V = span(vectors, A.base_ring()) diff --git a/mjo/matrix_vector.py b/mjo/matrix_vector.py index 111712b..4b4818d 100644 --- a/mjo/matrix_vector.py +++ b/mjo/matrix_vector.py @@ -7,51 +7,123 @@ two vector spaces often. """ from sage.all import * +from sage.matrix.matrix_space import is_MatrixSpace -def isomorphism(matrix_space): +def _mat2vec(m): + return vector(m.base_ring(), m.list()) + +def basis_representation(M): """ - Create isomorphism (i.e. the function) that converts elements - of a matrix space into those of the corresponding finite-dimensional - vector space. + Return the forward (``MatrixSpace`` -> ``VectorSpace``) and + inverse isometries, as a pair, that take elements of the given + ``MatrixSpace`` `M` to their representations as "long vectors," + and vice-versa. + + The argument ``M`` can be either a ``MatrixSpace`` or a basis for + a space of matrices. This function is needed because SageMath does + not know that matrix spaces are vector spaces, and therefore + cannot perform common operations with them -- like computing the + basis representation of an element. + + Moreover, the ability to pass in a basis (rather than a + ``MatrixSpace``) is needed because SageMath has no way to express + that e.g. a (sub)space of symmetric matrices is itself a + ``MatrixSpace``. INPUT: - - matrix_space: A finite-dimensional ``MatrixSpace`` object. + - ``M`` -- Either a ``MatrixSpace``, or a list of matrices that form + a basis for a matrix space. OUTPUT: - - (phi, phi_inverse): If ``matrix_space`` has dimension m*n, then - ``phi`` will map m-by-n matrices to R^(m*n). - The inverse mapping ``phi_inverse`` will go - the other way. + A pair of isometries ``(phi, phi_inv)``. + + If the matrix space associated with `M` has dimension `n`, then + ``phi`` will map its elements to vectors of length `n` over the + same base ring. The inverse map ``phi_inv`` reverses that + operation. SETUP:: - sage: from mjo.matrix_vector import isomorphism + sage: from mjo.matrix_vector import basis_representation + + EXAMPLES: + + This function computes the correct coordinate representations (of + length 3) for a basis of the space of two-by-two symmetric + matrices, the the inverse does indeed invert the process:: + + sage: E11 = matrix(QQbar,[ [1,0], + ....: [0,0] ]) + sage: E12 = matrix(QQbar,[ [0, 1/sqrt(2)], + ....: [1/sqrt(2), 0] ]) + sage: E22 = matrix(QQbar,[ [0,0], + ....: [0,1] ]) + sage: basis = [E11, E12, E22] + sage: phi, phi_inv = basis_representation(basis) + sage: phi(E11); phi(E12); phi(E22) + (1, 0, 0) + (0, 1, 0) + (0, 0, 1) + sage: phi_inv(phi(E11)) == E11 + True + sage: phi_inv(phi(E12)) == E12 + True + sage: phi_inv(phi(E22)) == E22 + True + + MatrixSpace arguments work too:: + + sage: M = MatrixSpace(QQ,2) + sage: phi, phi_inv = basis_representation(M) + sage: X = matrix(QQ, [ [1,2], + ....: [3,4] ]) + sage: phi(X) + (1, 2, 3, 4) + sage: phi_inv(phi(X)) == X + True + + TESTS: - EXAMPLES:: + The inverse is generally an inverse:: - sage: M = MatrixSpace(QQ,4,4) - sage: (p, p_inv) = isomorphism(M) - sage: m = M(range(16)) - sage: p_inv(p(m)) == m + sage: set_random_seed() + sage: n = ZZ.random_element(10) + sage: M = MatrixSpace(QQ,n) + sage: X = M.random_element() + sage: (phi, phi_inv) = basis_representation(M) + sage: phi_inv(phi(X)) == X True """ - from sage.matrix.matrix_space import is_MatrixSpace - if not is_MatrixSpace(matrix_space): - raise TypeError('argument must be a matrix space') + if is_MatrixSpace(M): + basis_space = M + basis = list(M.basis()) + else: + basis_space = M[0].matrix_space() + basis = M - base_ring = matrix_space.base_ring() - vector_space = VectorSpace(base_ring, matrix_space.dimension()) + def phi(X): + """ + The isometry sending ``X`` to its representation as a long vector. + """ + if X not in basis_space: + raise ValueError("X does not live in the domain of phi") - def phi(m): - return vector_space(m.list()) + V = VectorSpace(basis_space.base_ring(), X.nrows()*X.ncols()) + W = V.span_of_basis( _mat2vec(s) for s in basis ) + return W.coordinate_vector(_mat2vec(X)) - def phi_inverse(v): - return matrix_space(v.list()) + def phi_inv(Y): + """ + The isometry sending the long vector `Y` to an element of either + `M` or the span of `M` (depending on whether or not ``M`` + is a ``MatrixSpace`` or a basis). + """ + return sum( Y[i]*basis[i] for i in range(len(Y)) ) - return (phi, phi_inverse) + return (phi, phi_inv) @@ -78,7 +150,8 @@ def matrix_of_transformation(T, V): SETUP:: - sage: from mjo.matrix_vector import isomorphism, matrix_of_transformation + sage: from mjo.matrix_vector import (basis_representation, + ....: matrix_of_transformation) EXAMPLES: -- 2.44.2