The sensitivity of eigenvalues under elementary matrix perturbations

AbstractLet M be an n × n real matrix, and let Exy be the elementary matrix with 1 in the (x, y) position and zero elsewhere. For z ϵ C we call the matrix M + zExy an elementary matrix perturbation of M. Let Λ be any eigenvalue of M. Then there exists an (x, y) pair, 1 ⩽ x, y ⩽ n, and an analytic function hxy(z) defined in a neighborhood N of the origin such that: (a) hxy(0) = Λ. (b) hxy(z) is an eigenvalue of the elementary matrix perturbation M + zk(Λ)Exy for any z ϵ N, where k(Λ) is the dimension of the largest block containing Λ in the Jordan canonical form of M. (c) For any z ϵ N, z ≠ 0, M + zExy has k(Λ) distinct eigenvalues, all different from Λ. If Λ(z) is any one of these, then |Λ − λ(z)| = O(|z|1k(Λ)). (d) For any z ϵ N,z ≠ 0, M + zExy has eigenvalue Λ with multiplicity s(Λ) − k(Λ), where s(Λ) is the (algebraic) multiplicity of Λ in M. (e) For all real positive or negative t ϵ N, but generally not for both, M + tExy has an eigenvalue with magnitude bigger than |Λ|. If k(Λ) ⩾ 3 this is true for all t ϵ N (both positive and negative). Part (e) has some interesting applications to matrices which have an eigenvalue of critical size with respect to some property of interest (for example, pulse processes on directed graphs)