First Order Structure-Preserving Perturbation Theory for Eigenvalues of Symplectic Matrices
Fredy Sosa, Julio Moro, Christian Mehl · SIAM Journal on Matrix Analysis and Applications · 2020
A first order perturbation theory for eigenvalues of real or complex $J$-symplectic matrices under structure-preserving perturbations is developed. The main tools in the analysis are structured canonical forms, together with Lidskii-like formulas for eigenvalues of multiplicative perturbations. Explicit formulas, depending only on appropriately normalized left and right eigenvectors, are obtained for the leading terms of asymptotic expansions describing the perturbed eigenvalues. Special attention is given to eigenvalues on the unit circle, especially to the exceptional eigenvalues $\pm 1$, whose behavior under structure-preserving perturbations is known to differ significantly from the behavior under general perturbations. Several numerical examples are used to illustrate the asymptotic expansions.