On the Brualdi-Li matrix and its Perron Eigenspace

Abstract

The n × n Brualdi-Li matrix Bn has recently been shown to have maximal Perron value (spectral radius) � among all tournament matrices of even order n, thus settling the conjecture by the same name. This renews our interest in estimatingand motivates us to study the Perron eigenvector x of Bn, which is normalized to have 1-norm equal to one. It follows that x minimizes the 2-norm among all Perron vectors of n × n tournament matrices. There are also interesting relations among the entries of x and �, allowing us to rank the teams corresponding to a Brualdi-Li tournament according to the Kendall-Wei and Ramanajucharyula ranking schemes.