Perturbation analysis on matrix pencils for two specified eigenpairs of a semisimple eigenvalue with multiplicity two

Sk. Safique Ahmad and Prince Kanhya

Abstract

In this paper, we derive backward error formulas of two approximate eigenpairs of a semisimple eigenvalue with multiplicity two for structured and unstructured matrix pencils. We also construct the minimal structured perturbations with respect to the Frobenius norm such that these approximate eigenpairs become exact eigenpairs of an appropriately perturbed matrix pencil. The structures we consider include T-symmetric/T-skew-symmetric, Hermitian/skew-Hermitian, T-even/T-odd, and H-even/H-odd matrix pencils. Further, we establish various relationships between the backward error of a single approximate eigenpair and the backward error of two approximate eigenpairs of a semisimple eigenvalue with multiplicity two.

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Key words

multiple eigenvalue, semisimple eigenvalue, defective eigenvalue, structured generalized eigenvalue problem, eigenpair backward error

AMS subject classifications

65F15, 15A18, 65F35, 15A12

Links to the cited ETNA articles

 [2] Vol. 51 (2019), pp. 151-168 Sk. Safique Ahmad: Perturbation analysis for palindromic and anti-palindromic nonlinear eigenvalue problems [4] Vol. 38 (2011), pp. 275-302 Sk. Safique Ahmad and Volker Mehrmann: Perturbation analysis for complex symmetric, skew symmetric, even and odd matrix polynomials

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