Pseudo-tournament matrices and their eigenvalues (Q6486729)

A tournament matrix and its corresponding directed graph both arise as a record of the outcomes of a round Robin competition. An \(n\times n\) 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. The authors discuss the eigenproperties 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.