  This paper is devoted to the study of the separability problem in the field
of Quantum information theory. We deal mainly with the bipartite finite
dimensional case and with two types of matrices, one of them being the PPT
matrices. We proved that many results holds for both types. If these matrices
have specific Hermitian Schmidt decompositions then the matrices are separable
in a very strong sense. We proved that both types have what we call split
decompositions.
  We defined the notion of weak irreducible matrix, based on the concept of
irreducible state defined recently. These split decomposition theorems together
with the notion of weak irreducible matrix, imply that these matrices are weak
irreducible or a sum of weak irreducible matrices of the same type. The
separability problem for these types of matrices can be reduced to the set of
weak irreducible matrices of the same type. We also provided a complete
description of weak irreducible matrices of both types.
  Using the fact that every positive semidefinite Hermitian matrix with tensor
rank 2 is separable, we found sharp inequalites providing separability for both
types.
