Spectrum of MATLABs magic squares

TL;DR

This paper analyzes the eigenvalues of MATLAB-generated magic squares across different types, providing approximations, error bounds, and connections to circulant matrices, enhancing understanding of their spectral properties.

Contribution

It introduces methods to approximate eigenvalues of all types of MATLAB magic squares and establishes error bounds, linking their spectra to g-circulant matrices.

Findings

Eigenvalues of odd and singly even magic squares approximated with error bounds.

Eigenpairs of doubly even magic squares obtained explicitly.

Connections established between magic square spectra and g-circulant matrices.

Abstract

This article looks at the eigenvalues of magic squares generated by the MATLAB's magic($n$) function. The magic($n$) function constructs doubly even ($n = 4k$) magic squares, singly even ($n = 4k+2$) magic squares and odd ($n = 2k+1$) magic squares using different algorithms. The doubly even magic squares are constructed by a criss-cross method that involves reflecting the entries of a simple square about the center. The odd magic squares are constructed using the Siamese method. The singly even magic squares are constructed using a lower-order odd magic square (Strachey method). We obtain approximations of eigenvalues of odd and singly even magic squares and prove error bounds on the approximation. For the sake of completeness, we also obtain the eigenpairs of doubly even magic squares generated by MATLAB. The approximation of the spectra involves some interesting connections with the spectrum of g-circulant matrices and the use of Bauer-Fike theorem.