Extreme vectors of doubly nonnegative matrices

Abstract

An n × n real symmetric matrix is doubly nonnegative if it is positive semi-definite and entrywise nonnegative. It is easy to check that the collection of all n × n doubly nonnegative matrices forms a closed convex cone. A vector lying on an extreme ray of this cone is referred to as an extreme DN matrix. In this note we obtain characterizations of extreme DN matrices and show that there exist n×n extreme DN matrices with rank k if and only if k ≠ 2 and k≤(max(1, n − 3) if n is even, max(1, n − 2) if n is odd.Using these results, we obtain an algorithm for checking whether a given DN matrix is extreme. Some other results concerning extreme DN matrices are also proved. © 1996 Rocky Mountain Mathematics Consortium.