The Main Diagonal of a Permutation Matrix

TL;DR

This paper presents a method to locate the main diagonal of permutation matrices and band-dominated matrices using counting techniques and Fredholm index, enabling matrix centering and factorization.

Contribution

It introduces a new counting-based approach for identifying the main diagonal in permutation and band-dominated matrices, extending previous methods.

Findings

Main diagonal located by counting 1's in specific rows

Matrix can be centered and factored into block-diagonal form

Main diagonal determined by Fredholm index at infinity

Abstract

By counting 1's in the "right half" of $2w$ consecutive rows, we locate the main diagonal of any doubly infinite permutation matrix with bandwidth $w$. Then the matrix can be correctly centered and factored into block-diagonal permutation matrices. Part II of the paper discusses the same questions for the much larger class of band-dominated matrices. The main diagonal is determined by the Fredholm index of a singly infinite submatrix. Thus the main diagonal is determined "at infinity" in general, but from only $2w$ rows for banded permutations.