Nonnegative definite hermitian matrices with increasing principal minors

TL;DR

This paper characterizes nonnegative definite Hermitian matrices with increasing principal minors, showing they are precisely those invertible matrices with diagonal entries less than or equal to one.

Contribution

It provides a necessary and sufficient condition for matrices to have increasing principal minors based on the properties of their inverse.

Findings

Matrices with increasing principal minors are invertible.

Diagonal entries of the inverse are at most 1.

Characterization applies for matrices of size greater than 1.

Abstract

A nonzero nonnegative definite hermitian m by m matrix A has increasing principal minors if the value of each principle minor of A is not less than the value each of its subminors. 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.