A note on the eigenvectors of perturbed matrices with applications to linear positive systems

A result is presented describing the eigenvectors of a perturbed matrix, for a class of structured perturbations. One motivation for doing so is that positive eigenvectors of nonnegative, irreducible matrices are known to induce norms - acting much like Lyapunov functions - for linear positive systems, which may help estimate or control transient dynamics. The results apply to both discrete- and continuous-time linear positive systems. The theory is illustrated with several examples.