Derivatives and Perturbations of Eigenvectors

For a matrix A(z) whose entries are complex valued functions of a complex variable z, results are presented concerning derivatives of an eigenvector x(z) of A(z) associated with a simple eigenvalue λ (z) when x(z) is restricted to satisfy a constraint of the form σ ( x(z)) = 1 where σ 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.