A characterization for positive semi-definite matrix products

TL;DR

This paper investigates whether certain symbolic matrix products always have non-negative eigenvalues, providing a decidability result and a graph-theoretic characterization of such products.

Contribution

It introduces a decision procedure for symbolic matrix products to determine if they always have non-negative eigenvalues, linking linear algebra with graph theory.

Findings

The problem of checking non-negativity of eigenvalues in symbolic products is decidable.

A simple characterization of matrix products with only non-negative eigenvalues is provided.

The characterization connects matrix analysis to the positive graph conjecture in graph theory.

Abstract

A well-known fact in linear algebra is that $A^T A$ is always positive semi-definite for any real matrix $A$. We consider a generalization of this fact via the following decision problem. Given a symbolic product of length $k$, consisting of $\ell$ variables and their transposes, such as $ABB^TCA^T$, does there exist an $n\in\mathbb N$ and an assignment of matrices from $\mathbb R^{n\times n}$ such that the resulting matrix product has a negative eigenvalue? We show that this problem is decidable and provide a simple characterization of those symbolic products that have only non-negative real eigenvalues for any assignment of matrices. This characterization can also be understood as a matrix analogue of the positive graph conjecture by Camarena, Cs\'oka, Hubai, Lippner, and Lov\'asz, and the proof relies on this surprising connection to graph theory.