First Order Eigenvalue Perturbation Theory and the Newton Diagram

First order perturbation theory for eigenvalues of arbitrary matrices is systematically developed in all its generality with the aid of the Newton diagram, an elementary geometric construction first proposed by Isaac Newton. In its simplest form, a square matrix A with known Jordan canonical form is linearly perturbed to A(ε) = A+ ε B for an arbitrary perturbation matrix B, and one is interested in the leading term in the ε-expansion of the eigenvalues of A(ε). The perturbation of singular values and of generalized eigenvalues is also covered.