Almost Positive Matrices

Abstract

A symmetric matrix A = a _ij is said to be almost positive if the corresponding quadratic form satisfies the inequality

$$\sum {\sum {{a_{ij}}{z_i}{{\bar z}_j} \geqslant 0} }$$

whenever ∑ z _i = 0. It is clear that a positive matrix is an almost positive matrix and that the class of almost positive matrices forms a convex cone. It is easy to see that any matrix A of the form

$${a_{ij}} = {\alpha _i} + {\bar \alpha _j}$$

is an almost positive matrix, and therefore any matrix with

$${a_{ij}} = {b_{ij}} + {\alpha _i} + {\bar \alpha _j}$$

where B = b _ij is a positive matrix, is also almost positive. We first show that every almost positive matrix is of this form.