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arXiv · 2112.01462

Hadamard-type inequalities for $k$-positive matrices

Nam Q. Le
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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.

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Nam Q. Le. 2021-12-02. Hadamard-type inequalities for $k$-positive matrices. https://doi.org/10.1016/j.laa.2021.11.018

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