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

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.