mjo/matrix_vector.py

   1 """
   2 There is an explicit isomorphism between all finite-dimensional vector
   3 spaces. In particular, there is an isomorphism between the m-by-n
   4 matrices and `$R^(m \times n)$`. Since most vector operations are not
   5 available on Sage matrices, we have to go back and forth between these
   6 two vector spaces often.
   7 """
   8
   9 from sage.all import *
  10
  11 def isomorphism(matrix_space):
  12     """
  13     Create isomorphism (i.e. the function) that converts elements
  14     of a matrix space into those of the corresponding finite-dimensional
  15     vector space.
  16
  17     INPUT:
  18
  19     - matrix_space: A finite-dimensional ``MatrixSpace`` object.
  20
  21     OUTPUT:
  22
  23     - (phi, phi_inverse): If ``matrix_space`` has dimension m*n, then
  24                           ``phi`` will map m-by-n matrices to R^(m*n).
  25                           The inverse mapping ``phi_inverse`` will go
  26                           the other way.
  27
  28     EXAMPLES:
  29
  30         sage: M = MatrixSpace(QQ,4,4)
  31         sage: (p, p_inv) = isomorphism(M)
  32         sage: m = M(range(0,16))
  33         sage: p_inv(p(m)) == m
  34         True
  35
  36     """
  37     from sage.matrix.matrix_space import is_MatrixSpace
  38     if not is_MatrixSpace(matrix_space):
  39         raise TypeError('argument must be a matrix space')
  40
  41     base_ring = matrix_space.base_ring()
  42     vector_space = VectorSpace(base_ring, matrix_space.dimension())
  43
  44     def phi(m):
  45         return vector_space(m.list())
  46
  47     def phi_inverse(v):
  48         return matrix_space(v.list())
  49
  50     return (phi, phi_inverse)