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, 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.