An Elementary Derivation of the Projection Method for Nonlinear Eigenvalue Problems Based on Complex Contour Integration

The Sakurai-Sugiura (SS) projection method for the generalized eigenvalue problem has been extended to the nonlinear eigenvalue problem A(z)w = 0, where A(z) is an analytic matrix valued function, by several authors. To the best of the authors' knowledge, existing derivations of these methods rely on canonical forms of an analytic matrix function such as the Smith form or the theorem of Keldysh. While these theorems are powerful tools, they require advanced knowledge of both analysis and linear algebra and are rarely mentioned even in advanced textbooks of linear algebra. In this paper, we present an elementary derivation of the SS-type algorithm for the nonlinear eigenvalue problem, assuming that the wanted eigenvalues are all simple. Our derivation uses only the analyticity of the eigenvalues and eigenvectors of a parametrized matrix A(z), which is a standard result in matrix perturbation theory. Thus we expect that our approach will provide an easily accessible path to the theory of nonlinear SS-type methods.