Structured perturbation for eigenvalues of symplectic matrices: a multiplicative approach

Given an eigenvalue of a symplectic matrix, we analyze its change under small structure-preserving perturbations, i.e., perturbations which maintain the symplectic nature of the matrix. Modelling such perturbations multiplicatively allows us to make use of the first order multiplicative perturbation theory developed in [2] via Newton diagram techniques. This leads to both leading exponents and explicit formulas for the leading coefficients in the corresponding eigenvalue asymptotic expansions. These formulas only involve the perturbation matrices and appropriately normalized eigenvectors. Thus, a very detailed description is needed of the connections which symplectic structure induces between left and right eigenvectors. This information can be recovered by making use of symplectic structured canonical forms. We prove that in most cases, no generalized eigenvectors are required, provided some mild genericity conditions are met.