A SOLUTION OF A NONLINEAR-SYSTEM ARISING IN SPECTRAL PERTURBATION-THEORY OF NONNEGATIVE MATRICES

Let P and E be two n x n complex matrices such that for sufficiently s mall positive epsilon, P + epsilon E is nonnegative and irreducible. I t is known that the spectral radius of P + epsilon E and corresponding (normalized) eigenvector have fractional power series expansions. The goal of the paper is to develop an algorithm for computing the coeffi cients of these expansions under two (restrictive) assumptions, namely that P has a single Jordan block corresponding to its spectral radius and that the (unique up to scalar multiples) left and right eigenvect ors of P corresponding to its spectral radius, say v and w, satisfy v( T)Ew not equal 0. Our approach is to consider an associated countable system of nonlinear equations and solve this system recursively. At ea ch step, we consider the coefficients of the expansion of the spectral radius of P + epsilon E as parameters and solve a related linear syst em parametrically. The next coefficient of the expansion of the spectr al radius is then determined from feasibility considerations for a lin ear system. This solution method is novel and seems useful for computi ng coefficients of corresponding expansions when the two (restrictive) assumptions are relaxed, Also, interestingly, the coefficients we com pute yield a preferred basis of the generalized eigenspace correspondi ng to the spectral radius of the unperturbed matrix P. (C) 1997 Elsevi er Science Inc., 1997.