Symmetrizable Matrices I: Introduction

Abstract

We introduce some fundamental results concerning the structure of symmetrizable integer matrices. There is a beautiful combinatorial classification of these matrices as those that are sign symmetric and satisfy a certain cycle condition. The exact analogue of Cauchy interlacing will be seen to hold for symmetrizable matrices. We explore connections with equitable partitions of signed graphs. It will transpire that symmetrizable integer matrices are precisely those that occur as quotient matrices for these equitable partitions.