Tournament Matrices with Extremal Spectral Properties

For a tournament matrix M of order n, we de ne its walk space, WM , to be SpanfM 1 : j = 0; . . . ; n 1g where 1 is the all ones vector. We show that the dimension of WM equals the number of eigenvalues of M whose real parts are greater than 1=2. We then focus on tournament matrices whose walk space has particularly simple structure, and characterize them in terms of their spectra. Speci cally, we characterize those tournament matrices such that M 1 is an eigenvector of M for some j 0. We also characterize the tournament matrices M such that Jn 2M is a skew{ Hadamard matrix. Throughout, we illustrate our results with examples.