On the nonnegative eigenvalue problem Original Research Article

{1, λ1, …, λn} is the spectrum of a stochastic matrix of order n + 1 if and only if there exists some real matrix B of order n with spectrum {λ1, …, λn} and some n-simplex S subset of imagen containing the origin such that BS subset of S. This result appears implicitly in Ciarletʹs work, and it provides us with a geometrical tool to obtain sufficient conditions for the nonnegative eigenvalue problem. We employ it here to generalize the sufficient conditions given by Kellogg.