Eigenvalue perturbations for matrices with arbitrary Jordan form - Part I

In theoretical work it is often desirable to obtain an expression for the eigenvalue of a matrix when that that matrix is subject to a small perturbation. The Zeeman and Stark effects are well known examples. In these cases the matrices are Hermitian and therefore diagonalisable, and the perturbation theory is relatively straightforward. Perturbation theory for non-diagonalisable matrices (which can always be reduced to the canonical Jordan form) is a lot less well known. For this reason I would like to bring to attention some key results developed by Russian mathematicians and brought into the Anglophone scientific literature only in the last 15 years or so. I stress that this will be an introduction, and that I will sketch but not tackle in detail the full set of complications that can arise in perturbing matrices with completely arbitrary Jordan form. I will illustrate using a matrix that describes the asymmetric exclusion process.