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Perturbation analysis of matrices over a quaternion division algebra

    Sk. Safique Ahmad, Istkhar Ali, Ivan Slapničar

ETNA - Electronic Transactions on Numerical Analysis, pp. 128-149, 2021/01/21

doi: 10.1553/etna_vol54s128

doi: 10.1553/etna_vol54s128


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doi:10.1553/etna_vol54s128



doi:10.1553/etna_vol54s128

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.

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