{1, [lambda] 1 [z.upto] [lambda] n } 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 {[lambda] 1 [z.upto] [lambda] n } and somen -simplex S [sub ] R n containing the origin such that BS [sub ] S. This result appears implicitly in Ciarlet's 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.