Linear Principal Minor Polynomials: Hyperbolic Determinantal Inequalities and Spectral Containment
Abstract
A linear principal minor polynomial or lpm polynomial is a linear combination of principal minors of a symmetric matrix. By restricting to the diagonal, lpm polynomials are in bijection to multiaffine polynomials. We show that this establishes a one-to-one correspondence between homogeneous multiaffine stable polynomials and PSD-stable lpm polynomials. This yields new construction techniques for hyperbolic polynomials and allows us to generalize the well-known Fisher--Hadamard and Koteljanskii inequalities from determinants to PSD-stable lpm polynomials. We investigate the relationship between the associated hyperbolicity cones and conjecture a relationship between the eigenvalues of a symmetric matrix and the values of certain lpm polynomials evaluated at that matrix. We refer to this relationship as spectral containment.
Keywords
Cite
@article{arxiv.2112.13321,
title = {Linear Principal Minor Polynomials: Hyperbolic Determinantal Inequalities and Spectral Containment},
author = {Grigoriy Blekherman and Mario Kummer and Raman Sanyal and Kevin Shu and Shengding Sun},
journal= {arXiv preprint arXiv:2112.13321},
year = {2024}
}