Ela 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 + I = J, where I is the identity matrix, and J = Jn 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 rankp(A) ≥ (n − 1)/2 with equality if and only if n is odd and AA = O. This article shows that the rank of a tournament matrix A of even order n over a field of characteristic p satisfies rankp(A) ≥ n/2 with equality if and only if after simultaneous row and column permutations AA = [ ±Jm O O 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.