Matrices with integer eigenvalues for all permutations of coefficients (thanks to Pythagoras!)

TL;DR

This paper demonstrates how Pythagorean triples can generate matrices with the property that all permutations of their coefficients produce matrices with integer eigenvalues, revealing an infinite family of such matrices.

Contribution

It introduces a novel method linking Pythagorean triples to matrices with permutation-invariant integer eigenvalues, expanding understanding of matrix eigenvalue properties.

Findings

Permutations of matrix coefficients from Pythagorean triples yield integer eigenvalues.

Each Pythagorean triple generates infinitely many related matrices with this property.

The approach provides a simple formula-based construction for such matrices.

Abstract

It is shown that Pythagorean triples can be used to generate matrices that have integer eigenvalues for all permutations of their coefficients, via simple formulas. For example, each and every permutation of the $2\times2$ matrix coefficients $\{12,6,7,1\}$, generated by the Pythagorean triple $(5,12,13)$, yields a matrix with integer eigenvalues. Further, each and every Pythagorean triple in fact generates a countable infinity of nontrivially related matrices having this property.