Powers of doubly-affine integer square matrices with one non-zero eigenvalue

TL;DR

This paper explores how doubly-affine matrices like Latin and magic squares with one non-zero eigenvalue become constant matrices after powering, and how their powers can generate larger such matrices through a process involving the Cayley-Hamilton theorem.

Contribution

It introduces a method to generate infinite series of doubly-affine matrices of composite orders using powering and matrix multiplication, grounded in eigenvalue properties.

Findings

Matrices become constant after a few powers

Powering can generate larger doubly-affine matrices

Eigenvalue properties explain the matrices' behavior

Abstract

When doubly-affine matrices such as Latin and magic squares with a single non-zero eigenvalue are powered up they become constant matrices after a few steps. The process of compounding squares of orders m and n can then be used to generate an infinite series of such squares of orders mn. The Cayley-Hamilton theorem is used to understand this property, where their characteristic polynomials have just two terms.