Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.02314 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
Table of Contents:
- 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, Csóka, Hubai, Lippner, and Lovász, and the proof relies on this surprising connection to graph theory.