Low Rank Perturbations of Quaternion Matrices

Abstract

Dedicated to the memory of Leiba Rodman, whose work inspired us greatly.Abstract. Low rank perturbations of right eigenvalues of quaternion matrices are considered. For real and complex matrices it is well known that under a generic rank-k perturbation the k largest Jordan blocks of a given eigenvalue will disappear while additional smaller Jordan blocks will remain. In this paper, it is shown that the same is true for real eigenvalues of quaternion matrices, but for complex nonreal eigenvalues the situation is different: not only the largest k, but the largest 2k Jordan blocks of a given eigenvalue will disappear under generic quaternion perturbations of rank k. Special emphasis is also given to Hermitian and skew-Hermitian quaternion matrices and generic low rank perturbations that are structure-preserving.

Global Properties of Eigenvalues of Parametric Rank One Perturbations for Unstructured and Structured Matrices

“…Another type of structure was treated in [5], where nonnegative rank one perturbations of M-matrices are discussed. Finally, the quaternionic case was discussed in [31].…”