Hadamard-type inequalities for $k$-positive matrices

Abstract

We establish Hadamard-type inequalities for a class of symmetric matrices called $k$-positive matrices for which the $m$-th elementary symmetric functions of their eigenvalues are positive for all $m\le k$. These matrices arise naturally in the study of $k$-Hessian equations in Partial Differential Equations. For each $k$-positive matrix, we show that the sum of its principal minors of size $k$ is not larger than the $k$-th elementary symmetric function of their diagonal entries. The case $k=n$ corresponds to the classical Hadamard inequality for positive definite matrices. Some consequences are also obtained.

v2: The assumption in Lemma 2.2 of the published version in Linear Algebra Appl. was modified to make it invariant under conjugation with orthogonal matrices. All arguments and results remain unchanged