Algebraic multiplicity of the eigenvalues of a tournament matrix

Abstract

Let T n denote the set of irreducible n × n tournament matrices. Here are our main results: (1) For all n ⩾ 3, every matrix in T n has at least three distinct eigenvalues; such a matrix has exactly three distinct eigenvalues if and only if it is a Hadamard tournament matrix. (2) For all n ⩾ 3 there is a matrix in T n having n distinct eigenvalues. (3) If α n denotes the maximum algebraic multiplicity of 0 as an eigenvalue of the matrices in T n , then ⌊ n 2⌋ − 2 ⩽ α n ⩽ n − 6 for all n ⩾ 8. Each algebraic multiplicity m with 1 ⩽ m ⩽ ⌊ n 2⌋ ; − 2 is achieved for the eigenvalue 0 by some matrix in T n for every n⩾6. (4) If π n is the minimum Perron value (i.e. spectral radius) of all matrices in T n , then 2 < π n < 2.5 for all n⩾8.