On Block Representations and Spectral Properties of Semimagic Square Matrices

TL;DR

This paper introduces a block-structured matrix representation for semimagic squares, enabling new insights into their symmetries, spectral properties, and low-rank structures, with applications to eigenvector decomposition and quadratic forms.

Contribution

It develops an equivalent block representation for semimagic squares, facilitating analysis of symmetries, spectral properties, and low-rank structures, advancing understanding of their algebraic behavior.

Findings

Block representation captures symmetry properties.

Spectral analysis reveals eigenvector structures.

Low-rank matrices constructed via tensor product blocks.

Abstract

Using the decomposition of semimagic squares into the associated and balanced symmetry types as a motivation, we introduce an equivalent representation in terms of block-structured matrices. This block representation provides a way of constructing such matrices with further symmetries and of studying their algebraic behaviour, significantly advancing and contributing to the understanding of these symmetry properties. In addition to studying classical attributes, such as dihedral equivalence and the spectral properties of these matrices, we show that the inherent structure of the block representation facilitates the definition of low-rank semimagic square matrices. This is achieved by means of tensor product blocks. Furthermore, we study the rank and eigenvector decomposition of these matrices, enabling the construction of a corresponding two-sided eigenvector matrix in rational terms of their entries. The paper concludes with the derivation of a correspondence between the tensor product block representations and quadratic form expressions of Gaussian type.