Pseudo - tournament matrices , Brualdi - Li matrices and their new properties ∗

Pseudo-Tournament Matrices and Their Eigenvalues

Abstract. A tournament matrix and its corresponding directed graph both arise as a record of the outcomes of a round robin competition. An n × n complex matrix A is called h-pseudo-tournament if there exists a complex or real nonzero column vector h such that A+ A∗ = hh∗ − I . This class of matrices is a generalisation of well-studied tournament-like matrices such as h-hypertournament matrices,...

Ela on the Brualdi-li Matrix and Its Perron Eigenspace

The n × n Brualdi-Li matrix Bn has recently been shown to have maximal Perron value (spectral radius) ρ among all tournament matrices of even order n, thus settling the conjecture by the same name. This renews our interest in estimating ρ and 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 Pe...

Tournament Matrices with Extremal Spectral Properties

For a tournament matrix M of order n, we de ne its walk space, WM , to be SpanfM 1 : j = 0; . . . ; n 1g where 1 is the all ones vector. We show that the dimension of WM equals the number of eigenvalues of M whose real parts are greater than 1=2. We then focus on tournament matrices whose walk space has particularly simple structure, and characterize them in terms of their spectra. Speci cally,...

&mall Diameter Interchange Graphs of Classes of Matrices of Zeros and Ones

Let %I( R, S) denote the class of all m X n matrices of O’s and l’s having row sum vector R and column sum vector S. The interchange graph G( R, S) is the graph where the vertices are the matrices in %(R, S) and where two matrices are joined by an edge provided they differ by an interchange. We characterize those 81 (R, S) for which the graph C( R, S) has diameter at most 2 and those YI( R, S) ...

A Partition Problem for Sets of Permutation Matrices