mjo/ldlt.py

   1 from sage.all import *
   2
   3 def is_positive_semidefinite_naive(A):
   4     r"""
   5     A naive positive-semidefinite check that tests the eigenvalues for
   6     nonnegativity.  We follow the sage convention that positive
   7     (semi)definite matrices must be symmetric or Hermitian.
   8
   9     SETUP::
  10
  11         sage: from mjo.ldlt import is_positive_semidefinite_naive
  12
  13     TESTS:
  14
  15     The trivial matrix is vaciously positive-semidefinite::
  16
  17         sage: A = matrix(QQ, 0)
  18         sage: A
  19         []
  20         sage: is_positive_semidefinite_naive(A)
  21         True
  22
  23     """
  24     if A.nrows() == 0:
  25         return True # vacuously
  26     return A.is_hermitian() and all( v >= 0 for v in A.eigenvalues() )