Nonnegative definite hermitian matrices with increasing principal minors

Abstract

A nonnegative definite hermitian m × m matrix A ≠ 0 has increasing principal minors if det A[I] ≤ det A[J] for I J, where det A[I] is the principal minor of A based on rows and columns in the set I {1, . . . , m}. For m > 1 we show A has increasing principal minors if and only if A-1 exists and its diagonal entries are less or equal to 1.