BOUNDS FOR THE NONTRIVIAL EIGENVALUES OF STOCHASTIC MATRICES: A LOCAL APPROACH

摘要

Let S = {l,2,...,r}. Let P = (P_(ij)_(i,j∈S) be a stochastic matrix and put P_U = (P_(ij))_(i∈U,j∈S') where 0 ≠ U is contains in S. Our problem is to determine bounds for the nontrivial eigenvalues of P in terms of the matrices P_u, 0 ≠ U is contained in S. A main theorem is given and, by means of it, we get bounds for the eigenvalues λ with |λ| ≠ 1 of cyclic stochastic matrices and for the spectral radius of nonnegative matrices.