Taylor expansions of eigenvalues of perturbed matrices with applications to spectral radii of nonnegative matrices

Let A and B be two n×n complex matrices, and let λ be an eigenvalue of A. The purpose of this paper is to derive, under certain conditions, Taylor power series expansions of the form λ+σ^∞_k=1λ_kε^k and σ∞_k=0v{script}_kε^k, respectively, for eigenvalues and corresponding eigenvectors of the perturbed matrices A+εB for ε that has sufficiently small absolute value. Our results apply to the case where λ is a simple eigenvalue of A, e.g., when A is nonnegative and irreducible and λ is the spectral radius of A. In particular, if A+εB is nonnegative for sufficiently small nonnegative ε and A is irreducible, we obtain power series expansions for the spectral radii of the perturbed matrices A+εB and for corresponding eigenvectors. The coefficients of the expansions yield explicit expressions for the regular and mixed derivatives of the spectral radius and of a corresponding eigenvector of a nonnegative irreducible matrix when viewed as a function of the elements of the matrix. Our approach is constructive, and we present a recursive algorithm that will compute the coefficients of the above series.