Hadamard-type inequalities for $k$-positive matrices

TL;DR

This paper proves Hadamard-type inequalities for $k$-positive matrices, extending classical results to a broader class relevant in PDEs, with implications for matrix analysis and differential equations.

Contribution

It introduces new inequalities for $k$-positive matrices, linking principal minors and elementary symmetric functions, generalizing classical Hadamard inequalities.

Findings

Established inequalities for $k$-positive matrices.

Connected matrix inequalities with $k$-Hessian equations.

Derived consequences for matrix analysis and PDEs.

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 $m\leq k$. 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.