Hypertournament matrices, score vectors and eigenvalues
Steve Kirkland · Linear and Multilinear Algebra · 1991
Consider a real square matrix A of order n which satisfies A+ At = J− I (where J is the all ones matrix) and its score vectors= Al (where 1 is the all ones vector). Here are our main results. If , then A has a real positive eigenvalue p with while the other eigenvalues satisfy A has eigenvalues p and λ such that and if and only if A has n − 2 eigenvalues with real part -1/2. If then for any eigenvalue . Further, if then a Perron-Frobenius result holds for A. A consequence of this is that if such an A is non-negative as well, and if n ≥ 9, then A is irreducible and primitive.