The Fiber of the Principal Minor Map

TL;DR

This paper characterizes when matrices are uniquely determined by their principal minors up to diagonal equivalence, providing necessary and sufficient conditions and exploring implications for symmetric, Hermitian, and real stable matrices.

Contribution

It offers a complete characterization of the fibers of the principal minor map, extending previous conditions and connecting matrix reducibility with determinantal representations.

Findings

A necessary and sufficient condition for the fiber to be a point up to diagonal equivalence.

Full characterization of the fiber for symmetric and Hermitian matrices.

Resolution of a question on real stable matrices by applying the developed techniques.

Abstract

This paper explores the fibers of the principal minor map over a general field. The principal minor map is the map that assigns to each $n\times n$ matrix the $2^n$-vector of its principal minors. In $1984$, Hartfiel and Loewy proposed a condition that was sufficient to ensure that the fiber of the principal minor map is a single point up to diagonal equivalence. Loewy later improved upon this condition in $1986$. In this paper, we provide a necessary and sufficient condition for the fiber to be a point up to diagonal equivalence. Additionally, we establish a connection between the reducibility of a matrix and the reducibility of its determinantal representation. Using this connection, we fully characterize the fiber of symmetric and Hermitian matrices in the space of $n\times n$ matrices over any field $\mathbb{F}$. We also use these techniques to answer a question of Borcea, Br\"and\'en, and Liggett concerning real stable matrices.