Eigenvalue placement for regular matrix pencils with rank one perturbations

2016/04/12 by Hannes Gernandt, Gernandt, Hannes, Carsten Trunk +1 · 1 citation
Computer Science · Mathematics · #15A18 #15A22 #47A55 #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Matrix Theory and Algorithms #Numerical methods for differential equations #Rings and Algebras (math.RA) #math.FA #math.RA #msc:15A18 #msc:15A22 #msc:47A55

paper · pdf · doi:10.48550/arxiv.1604.06671

openalex publication_date 2016/04/12 · arxiv created 2017/01/05 · arxiv updated 2017/01/06 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28

Abstract

A regular matrix pencil sE-A and its rank one perturbations are considered. We determine the sets in the extended complex plane which are the eigenvalues of the perturbed pencil. We show that the largest Jordan chains at each eigenvalue of sE-A may disappear and the sum of the length of all destroyed Jordan chains is the number of eigenvalues (counted with multiplicities) which can be placed arbitrarily in the extended complex plane. We prove sharp upper and lower bounds of the change of the algebraic and geometric multiplicity of an eigenvalue under rank one perturbations. Finally we apply our results to a pole placement problem for a single-input differential algebraic equation with feedback.

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• Bounds for the change of the Weyr characteristic of matrix pencils after 1-rank perturbations

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