The Ranks of Extremal Positive Semidefinite Matrices with Given Sparsity Pattern

Abstract

Let P be a symmetric set of ordered pairs of integers from 1 to n, and define $M^ + (P)$ to be the closed cone of all positive semidefinite Hermitian matrices whose $(i,j)$ entry is zero whenever $i \ne j$ and $(i,j)$ is not in P. The extreme points of $M^ + (P)$ are considered. In some special cases, the maximum rank that such an extreme point can have is calculated.