#### Volume 54, pp. 128-149, 2021.

## Perturbation analysis of matrices over a quaternion division algebra

Sk. Safique Ahmad, Istkhar Ali, and Ivan Slapničar

### Abstract

In this paper, we present the concept of perturbation bounds for the right eigenvalues of a quaternionic matrix. In particular, a Bauer-Fike-type theorem for the right eigenvalues of a diagonalizable quaternionic matrix is derived. In addition, perturbations of a quaternionic matrix are discussed via a block-diagonal decomposition and the Jordan canonical form of a quaternionic matrix. The location of the standard right eigenvalues of a quaternionic matrix and a sufficient condition for the stability of a perturbed quaternionic matrix are given. As an application, perturbation bounds for the zeros of quaternionic polynomials are derived. Finally, we give numerical examples to illustrate our results.

Full Text (PDF) [333 KB], BibTeX

### Key words

quaternionic matrices, left eigenvalues, right eigenvalues, quaternionic polynomials, Bauer-Fike theorem, quaternionic companion matrices, quaternionic matrix norms

### AMS subject classifications

15A18, 15A66

### Links to the cited ETNA articles

[21] | Vol. 4 (1996), pp. 89-105 Volker Mehrmann and Hongguo Xu: An analysis of the pole placement problem. I. The single-input case |

[22] | Vol. 5 (1997), pp. 77-97 Volker Mehrmann and Hongguo Xu: An analysis of the pole placement problem II. The multi-input case |

[24] | Vol. 36 (2009-2010), pp. 9-16 Gerhard Opfer: Polynomials and Vandermonde matrices over the field of quaternions |

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