A solution of a nonlinear system arising in spectral perturbation theory of nonnegative matrices

AbstractLet P and E be two n × n complex matrices such that for sufficiently small positive ε, P + εE is nonnegative and irreducible. It is known that the spectral radius of P + εE and corresponding (normalized) eigenvector have fractional power series expansions. The goal of the paper is to develop an algorithm for computing the coefficients 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 eigenvectors of P corresponding to its spectral radius, say v and w, satisfy vTEw ≠ 0. Our approach is to consider an associated countable system of nonlinear equations and solve this system recursively. At each step, we consider the coefficients of the expansion of the spectral radius of P + εE as parameters and solve a related linear system parametrically. The next coefficient of the expansion of the spectral radius is then determined from feasibility considerations for a linear system. This solution method is novel and seems useful for computing coefficients of corresponding expansions when the two (restrictive) assumptions are relaxed. Also, interestingly, the coefficients we compute yield a preferred basis of the generalized eigenspace corresponding to the spectral radius of the unperturbed matrix P