A solution of a nonlinear system arising in spectral perturbation theory of nonnegative matrices
This paper develops an algorithm for computing the coefficients of the fractional power series expansions of the spectral radius \(\rho(P+\varepsilon E)\) and corresponding (normalized) eigenvector of \(P+\varepsilon E\), \(P\) and \(E\) are complex \(n\times n\) and \(P+\varepsilon E\) with sufficiently small \(\varepsilon>0\) is nonnegative and irreducible, \(P\) has a single Jordan block corresponding to \(\rho(P)\) and the left and right eigenvectors \(v\) and \(w\), respectively, of \(P\) corresponding to \(\rho(P)\) satisfy \(v^TEw\neq 0\). The novel idea is the recursive solution of a system of nonlinear equations, where in each step the coefficients of the expansion of \(\rho(P+\varepsilon E)\) are considered as parameters and a related linear system is solved parametrically, and the next coefficient of the development of \(\rho(P+\varepsilon E)\) is determined from feasibility considerations for a linear system. The method is also considered under relaxed conditions. The computed coefficients also yield a preferred basis of the generalized eigenspace corresponding to \(\rho(P)\).
scientific article; zbMATH DE number 1026429