Extreme points in convex sets of symmetric matrices

Abstract:

This paper deals with the following problem: What are the extreme points of a convex set $K$ of $n \times n$ matrices, which is the intersection of the set ${S_n}$ of symmetric matrices of nonnegative type, with another convex subset of symmetric matrices $H?$? In the case where the facial structure of $H$ is known, we expose a general method to determine the extreme points of $K$ (Theorem 1). Then, we apply this method to the set of correlation matrices, characterizing its extreme points in Theorem 2, which is our main theorem. A corollary describes thoroughly the extreme points of rank 2.