Let A k ; k 0, be a sequence of mm nonnegative matrices and let A(z) = P 1 k=0 A k z k be such that A(1) is an irreducible stochastic matrix. The unique power-bounded solution of the nonlinear matrix equation G = P 1 k=0 A k G k has been shown to play a key role in the analysis of Markov chains of M/G/1 type. Assuming that the matrix A(z) is rational, we show that the solution of this matrix eq...