Pseudo-Tournament Matrices and Their Eigenvalues

A tournament matrix and its corresponding directed graph both arise as a record of the outcomes of a round robin competition. An n x 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, generalised tournament matrices, tournament matrices, and elliptic matrices. We discuss the eigen-properties of an h-pseudo-tournament matrix, and obtain new results when the matrix specialises to one of these tournament-like matrices. Further, several results derived in previous articles prove to be corollaries of those reached here.