TAYLOR EXPANSIONS OF EIGENVALUES OF PERTURBED MATRICES WITH APPLICATIONS TO SPECTRAL RADII OF NONNEGATIVE MATRICES

Let A and B be two n X n complex matrices, and let lambda be an eigenvalue of A. The purpose of this paper is to derive, under certain conditions, Taylor power series expansions of the form lambda + SIGMA(k=1)infinity lambda(k)epsilon(k) and SIGMA(k= 0)infinity upsilon(k)epsilon(k), respectively, for eigenvalues and corresponding eigenvectors of the perturbed matrices A + epsilon(B) for epsilon that has sufficiently small absolute value. Our results apply to die case where lambda is a simple eigenvalue of A, e.g., when A is nonnegative and irreducible and lambda is the spectral radius of A. In particular, if A + epsilon(B) is nonnegative for sufficiently small nonnegative epsilon and A is irreducible, we obtain power series expansions for the spectral radii of the perturbed matrices A + epsilon(B) 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.