Low-rank perturbation of symplectic matrices

Abstract

In this paper the generic behavior of the Jordan structure of symplectic matrices under low-rank structure-preserving transformations is considered. So far, only the case of rank-one perturbations was covered in the literature, and this result is now extended to the case of arbitrary rank $k$. As a key step toward the main result, it is shown that all symplectic matrices that differ from the identity by rank $k$ can be represented as a product of $k$ symplectic reflectors unless they are similar to a diagonalizable matrix having only the eigenvalues $\pm 1$.