mjo/cone/symmetric_pd.py

   1 r"""
   2 The symmetric positive definite cone `$S^{n}_{++}$` is the cone
   3 consisting of all symmetric positive-definite matrices (as a subset of
   4 $\mathbb{R}^{n \times n}$`. It is the interior of the symmetric positive
   5 SEMI-definite cone.
   6 """
   7
   8 from sage.all import *
   9 from mjo.cone.symmetric_psd import random_symmetric_psd
  10
  11 def random_symmetric_pd(V):
  12     r"""
  13     Generate a random symmetric positive-definite matrix over the
  14     vector space ``V``. That is, the returned matrix will be a linear
  15     transformation on ``V``, with the same base ring as ``V``.
  16
  17     (We take a very loose interpretation of "random," here.)
  18
  19     INPUT:
  20
  21     - ``V`` -- The vector space on which the returned matrix will act.
  22
  23     OUTPUT:
  24
  25     A random symmetric positive-definite matrix, i.e. a linear
  26     transformation from ``V`` to itself.
  27
  28     ALGORITHM:
  29
  30     We request a full-rank positive-semidefinite matrix from the
  31     :func:`mjo.cone.symmetric_psd.random_symmetric_psd` function.
  32
  33     SETUP::
  34
  35         sage: from mjo.cone.symmetric_pd import random_symmetric_pd
  36
  37     EXAMPLES:
  38
  39     Well, it doesn't crash at least::
  40
  41         sage: set_random_seed()
  42         sage: V = VectorSpace(QQ, 2)
  43         sage: A = random_symmetric_pd(V)
  44         sage: A.matrix_space()
  45         Full MatrixSpace of 2 by 2 dense matrices over Rational Field
  46
  47     """
  48     # We accept_zero because the trivial matrix is (trivially)
  49     # symmetric and positive-definite, but it's also "zero." The rank
  50     # condition ensures that we don't get the zero matrix in a
  51     # nontrivial space.
  52     return random_symmetric_psd(V, rank=V.dimension())