A Note on the Classification of Permutation Matrix

TL;DR

This paper explores the classification of permutation matrices through permutation similarity, focusing on canonical forms, cycle matrix decomposition, and cycle factorization to better understand their structural properties.

Contribution

It introduces a detailed approach to classify permutation matrices using canonical forms and cycle decompositions, enhancing understanding of their equivalence classes.

Findings

Canonical form characterization of permutation matrices

Cycle matrix decomposition method

Cycle factorization of permutation and monomial matrices

Abstract

This paper is concentrated on the classification of permutation matrix with the permutation similarity relation, mainly about the canonical form of a permutational similar equivalence class, the cycle matrix decomposition of a permutation matrix and the cycle factorization of a permutation matrix or monomial matrix.