Tournament matrices and their generalizations, I.
John S. Maybee, Norman J. Pullman · Linear and Multilinear Algebra · 1990
If M is any complex matrix with rank (M + M * + I) = 1, we show that any eigenvalue of M that is not geometrically simple has 1/2 for its real part. This generalizes a recent finding of de Caen and Hoffman: the rank of any n × n tournament matrix is at least n − 1. We extend several spectral properties of tournament matrices to this and related types of matrices. For example, we characterize the singular real matrices M with 0 diagonal for which rank (M + MT + I) = 1 and we characterize the vectors that can be in the kernels of such matrices. We show that singular, irreducible n × n tournament matrices exist if and only n∉ {2,3,4,5} and exhibit many infinite families of such matrices. Connections with signed digraphs are explored and several open problems are presented.