EVEN AND ODD TOURNAMENT MATRICES WITH MINIMUM RANK OVER FINITE FIELDS

The (0, 1)-matrix A of order n is a tournament matrix provided A + A(T) + I = J, where I is the identity matrix, and J = J(n) is the all 1's matrix of order n. It was shown by de Caen and Michael that the rank of a tournament matrix A of order n over a field of characteristic p satisfies rank(p)(A) >= (n - 1)/2 with equality if and only if n is odd and AA(T) = O. This article shows that the rank of a tournament matrix A of even order n over a field of characteristic p satisfies rank(p)(A) >= n/2 with equality if and only if after simultaneous row and column permutations AA(T) = [+/- J(m) O], for a suitable integer m. The results and constructions for even order tournament matrices are related to and shed light on tournament matrices of odd order with minimum rank.