MAXIMIZING THE SPECTRAL-RADIUS OF FIXED TRACE DIAGONAL PERTURBATIONS OF NONNEGATIVE MATRICES

Let A be an n-by-n irreducible, entrywise nonnegative matrix. For a gi ven t > 0, we consider the problem of maximizing the Perron root of a nonnegative, diagonal, trace t perturbation of A. Because of the conve xity of the Perron root as a function of diagonal entries, the maximum occurs for some tE(ii). Such an index i, which is called a winner, ma y depend on t. We show how to determine the (nonempty) set of indices i that are winners for all sufficiently small t and the possibly diffe rent (nonempty) set of indices that are winners for all sufficiently l arge t. We also show how to determine if there are indices that are wi nners for all t.