On the minimal spectral radii of skew-reciprocal integer matrices

TL;DR

This paper investigates the smallest spectral radii of skew-reciprocal integer matrices, showing that, except for dimension six, these matrices generally have smaller spectral radii than their reciprocal counterparts, with implications for matrix theory.

Contribution

It provides the first comprehensive analysis of minimal spectral radii for skew-reciprocal integer matrices, highlighting differences from reciprocal matrices across dimensions.

Findings

Minimal spectral radii identified for skew-reciprocal matrices

Except for dimension six, skew-reciprocal matrices have smaller spectral radii than reciprocal matrices

Results contribute to understanding spectral properties of structured matrices

Abstract

We determine the minimal spectral radii among all skew-reciprocal integer matrices of a fixed even dimension that are primitive or nonnegative and irreducible. In particular, except for dimension six, we show that each such class of matrices realises smaller spectral radii than the corresponding reciprocal class.