Commuting Magic Square Matrices

TL;DR

This paper reviews a method for constructing larger commuting magic square matrices through compounding, deriving their algebraic properties, and exploring their generalizations with numerical examples.

Contribution

It introduces a systematic approach to compound magic squares, analyzes their algebraic structures, and extends the method to broader classes of commuting magic squares.

Findings

Compounded magic squares commute and can be explicitly characterized.

Formulas for Jordan form and singular value decomposition are derived.

Regular and pandiagonal commuting magic squares can be constructed via compounding.

Abstract

We review a known method of compounding two magic square matrices of order m and n with the all-ones matrix to form two magic square matrices of order mn. We show that these compounded matrices commute. Simple formulas are derived for their Jordan form and singular value decomposition. We verify that regular (associative) and pandiagonal commuting magic squares can be constructed by compounding. In a special case the compounded matrices are similar. Generalization of compounding to a wider class of commuting magic squares is considered. Three numerical examples illustrate our theoretical results.