  The pattern of a matrix M is a (0,1)-matrix which replaces all non-zero
entries of M with a 1. A directed graph is said to support M if its adjacency
matrix is the pattern of M. If M is an orthogonal matrix, then a digraph which
supports M must satisfy a condition known as quadrangularity. We look at
quadrangularity in tournaments and determine for which orders quadrangular
tournaments exist. We also look at a more restrictive necessary condition for a
digraph to support an orthogonal matrix, and give a construction for
tournaments which meet this condition.
