+++ /dev/null
-"""
-There is an explicit isomorphism between all finite-dimensional vector
-spaces. In particular, there is an isomorphism between the m-by-n
-matrices and `$R^(m \times n)$`. Since most vector operations are not
-available on Sage matrices, we have to go back and forth between these
-two vector spaces often.
-"""
-
-from sage.all import *
-from sage.matrix.matrix_space import is_MatrixSpace
-
-def _mat2vec(m):
- return vector(m.base_ring(), m.list())
-
-def basis_repr(M):
- """
- 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:
-
- - ``M`` -- Either a ``MatrixSpace``, or a list of matrices that form
- a basis for a matrix space.
-
- OUTPUT:
-
- 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_repr
-
- 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_repr(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_repr(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_repr(M)
- sage: phi_inv(phi(X)) == X
- True
-
- """
- if is_MatrixSpace(M):
- basis_space = M
- basis = list(M.basis())
- else:
- basis_space = M[0].matrix_space()
- basis = M
-
- 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")
-
- 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_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 basis_space.linear_combination( zip(Y,basis) )
-
- return (phi, phi_inv)
-
-
-
-def basis_repr_of_operator(M, L):
- """
- Return the matrix of the operator `L` with respect to the basis
- `M` if `M` is a list of basis vectors for a matrix space; or with
- respect to the standard basis of `M` if `M` is a ``MatrixSpace``.
-
- This function is necessary because SageMath does not know that
- matrix spaces are vector spaces, and it moreover it doesn't know
- that (for example) the subspace of symmetric matrices is a matrix
- space in its own right.
-
- Use ``linear_transformation().matrix()`` instead if you have a
- true ``VectorSpace``.
-
- INPUT:
-
- - ``M`` -- Either a ``MatrixSpace``, or a list of matrices that form
- a basis for a matrix space.
-
- OUTPUT:
-
- If the matrix space associated with `M` has dimension `n`, then an
- `n`-by-`n` matrix over the same base ring is returned.
-
- SETUP::
-
- sage: from mjo.matrix_vector import (basis_repr,
- ....: basis_repr_of_operator)
-
- EXAMPLES:
-
- The matrix of the identity operator on the space of two-by-two
- symmetric matrices is the identity matrix, regardless of the basis::
-
- 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: identity = lambda X: X
- sage: basis_repr_of_operator(basis, identity)
- [1 0 0]
- [0 1 0]
- [0 0 1]
- sage: E11 = matrix(QQ,[[2,0],[0,0]])
- sage: E12 = matrix(QQ,[[0,2],[2,0]])
- sage: basis = [E11, E12, E22]
- sage: basis_repr_of_operator(basis, identity)
- [1 0 0]
- [0 1 0]
- [0 0 1]
-
- A more complicated example confirms that we get a matrix consistent
- with our ``matrix_to_vector`` function::
-
- sage: M = MatrixSpace(QQ,3,3)
- sage: Q = M([[0,1,0],[1,0,0],[0,0,1]])
- sage: def f(x):
- ....: return Q*x*Q.inverse()
- ....:
- sage: F = basis_repr_of_operator(M, f)
- sage: F
- [0 0 0 0 1 0 0 0 0]
- [0 0 0 1 0 0 0 0 0]
- [0 0 0 0 0 1 0 0 0]
- [0 1 0 0 0 0 0 0 0]
- [1 0 0 0 0 0 0 0 0]
- [0 0 1 0 0 0 0 0 0]
- [0 0 0 0 0 0 0 1 0]
- [0 0 0 0 0 0 1 0 0]
- [0 0 0 0 0 0 0 0 1]
- sage: phi, phi_inv = basis_repr(M)
- sage: X = M([[1,2,3],[4,5,6],[7,8,9]])
- sage: F*phi(X) == phi(f(X))
- True
-
- """
- if is_MatrixSpace(M):
- basis_space = M
- basis = list(M.basis())
- else:
- basis_space = M[0].matrix_space()
- basis = M
-
- (phi, phi_inv) = basis_repr(M)
-
- # Get a basis for the image space. Since phi is an isometry,
- # it takes one basis to another.
- image_basis = [ phi(b) for b in basis ]
-
- # Now construct the image space itself equipped with our custom basis.
- W = VectorSpace(basis_space.base_ring(), len(basis))
- W = W.span_of_basis(image_basis)
-
- return matrix.column( W.coordinates(phi(L(b))) for b in basis )
projects / sage.d.git / blobdiff