Generic canonical form of pairs of matrices with zeros

TL;DR

This paper establishes a canonical form for pairs of matrices with prescribed zero entries under certain transformations, using combinatorial graph methods to identify generic cases and exceptions.

Contribution

It introduces a universal canonical form for matrix pairs with fixed zeros, applicable in a broad class of problems, and develops a combinatorial approach to characterize generic and special cases.

Findings

Almost all matrix pairs reduce to a common canonical form

The canonical form is determined by a polynomial condition

A combinatorial graph method is used to construct the form

Abstract

We consider a family of pairs of m-by-p and m-by-q matrices, in which some entries are required to be zero and the others are arbitrary, with respect to transformations (A,B)--> (SAR,SBL) with nonsingular S, R, L. We prove that almost all of these pairs reduce to the same pair (C, D) from this family, except for pairs whose arbitrary entries are zeros of a certain polynomial. The polynomial and the pair (C D) are constructed by a combinatorial method based on properties of a certain graph.