X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fmatrix_vector.py;h=4b4818df133a54fc16c6cf7e22b9a0e8eda87796;hb=a3b94d0fdb734aeb91875d2a2ceaece99d129934;hp=49d112a9f832e8849428abab5955a9f779d884d2;hpb=5d22f22af29ba8f09a8c66244157e0e04c7bf43d;p=sage.d.git diff --git a/mjo/matrix_vector.py b/mjo/matrix_vector.py index 49d112a..4b4818d 100644 --- a/mjo/matrix_vector.py +++ b/mjo/matrix_vector.py @@ -7,47 +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 basis_representation EXAMPLES: - sage: M = MatrixSpace(QQ,4,4) - sage: (p, p_inv) = isomorphism(M) - sage: m = M(range(0,16)) - sage: p_inv(p(m)) == m + 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: + + The inverse is generally an inverse:: + + 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) @@ -72,6 +148,11 @@ def matrix_of_transformation(T, V): matrix that represents ``T`` with respect to the standard basis of ``V``. + SETUP:: + + sage: from mjo.matrix_vector import (basis_representation, + ....: matrix_of_transformation) + EXAMPLES: The matrix of a transformation on a simple vector space should be @@ -116,7 +197,7 @@ def matrix_of_transformation(T, V): """ n = V.dimension() - B = V.basis() + B = list(V.basis()) def inner_product(v, w): # An inner product function that works for both matrices and @@ -140,8 +221,8 @@ def matrix_of_transformation(T, V): return L(x) entries = [] - for j in range(0,n): - for i in range(0,n): + for j in range(n): + for i in range(n): entry = inner_product(apply(T,B[i]), B[j]) entries.append(entry)