A transformation that preserves principal minors of skew-symmetric matrices

TL;DR

This paper introduces a novel method to construct pairs of skew-symmetric matrices that share identical principal minors of all orders, expanding understanding of matrix similarity and principal minor preservation.

Contribution

It provides a new construction approach for skew-symmetric matrices with equal principal minors, extending prior results on symmetric matrices.

Findings

New construction method for skew-symmetric matrices with equal principal minors

Extension of principal minor preservation results to skew-symmetric matrices

Deeper understanding of matrix similarity and principal minor invariants

Abstract

Our motivation comes from the work of Engel and Schneider (1980). Their main theorem implies that two symmetric matrices have equal corresponding principal minors of all orders if and only if they are diagonally similar. This study was continued by Hartfiel and Loewy (1984). They found sufficient conditions under which two $n\times n$ matrices\ $A$ and $B$ have equal corresponding principal minors of all orders if and only if $B$ or its transpose $B^{t}$ is diagonally similar to $A$. In this paper, we give a new way to construct a pair of skew-symmetric having equal corresponding principal minors of all orders.