Separation theorems

Abstract

Consider a matrix A of order m×n defined on a field F. Let x and y be vectors in the column spans of A and A' respectively. The vectors x and y are said to be separable if A admits a partition into disjoint matrices of the same order (A=A₁⊕A₂) such that x belongs to the column span of A₂ and y to that of A'. Additional conditions imposed on A₁ and A₂ reflect stronger shades of separability or of inseparability. For complex matrices, star separability is one such instance. Necessary and sufficient conditions are obtained for separability and star separability of the pair (x, y). An EP matrix and its transpose (conjugate transpose in the complex case) have the same column span. It is shown that in the class of EP matrices, the separability of the pair (x, x) for every x in the column span of the matrix characterizes the skew symmetric matrices, and in the class of complex EP matrices, inseparability of similar pairs characterizes the almost definite matrices of Duffin and Morley.