Orthogonal Matrices with Zero Diagonal. II

Abstract

C -matrices appear in the literature at various places; for a survey, see [ 11 ]. Important for the construction of Hadamard matrices are the symmetric C -matrices, of order v ≡ 2 (mod 4), and the skew C -matrices, of order v ≡ 0 (mod 4). In § 2 of the present paper it is shown that there are essentially no other C -matrices. A more general class of matrices with zero diagonal is investigated, which contains the C -matrices and the matrices of ( v, k, λ)-systems on k and k + 1 in the sense of Bridges and Ryser [ 6 ]. Skew C -matrices are interpreted in § 3 as the adjacency matrices of a special class of tournaments, which we call strong tournaments. They generalize the tournaments introduced by Szekeres [ 24 ] and by Reid and Brown [ 21 ].