Derivatives and perturbations of eigenvectors

For a matrix ${\bf A}(z)$ whose entries are complex valued functions of a complex variable z, results are presented concerning derivatives of an eigenvector ${\bf x}(z)$ of ${\bf A}(z)$ associated with a simple eigenvalue $\lambda (z)$ when ${\bf x}(z)$ is restricted to satisfy a constraint of the form $\sigma ({\bf x}(z)) = 1$ where $\sigma $ is a rather arbitrary scaling function. The differentiation formulas lead to a new approach for analyzing the sensitivity of an eigenvector under small perturbations in the underlying matrix. Finally, an application is given which concerns the analysis of a finite Markov chain subject to perturbations in the transition probabilities.