Stability analysis of companion matrices 

Author 
w.auzinger@tuwien.ac.at; sroksolyana@yahoo.com
Institut fur Analysis und Scientific Computing, Technische Universitat Wien, Austria; Institute for Applied Mathematics and Fundamental Sciences, Lviv Polytechnic National University

Abstract 
Assume that a family of (nonnormal) matrices has stable spectra contained in the complex
unit circle. This does not necessarily imply that the 2norm of such matrices is small, and the
question is under what `natural' similarity transformation the transformed matrices will have
2norm smaller or equal to 1. Already for the $2\times2$case this is a nontrivial question, involving
the analysis of a function in two complex variables (the eigenvalues) and a positive scaling
parameter. We discuss and explain an approach to this problem which has been used before in
the analysis of companion matrices. For the 3D case we present a numerical example.

Keywords 
companion matrix; stable spectrum; similarity; contraction

DOI 
doi:10.15330/ms.46.2.115120

Reference 
1. W. Auzinger, A note on similarity to contraction for stable 2x2 companion matrices, Ukr. Mat. Zh., 68
(2016), ¹3, 400–407.
2. A. Eder, G. Kirlinger, A normal form for multistep companion matrices, Math. Models and Methods in Applied Sciences, 11 (2001), ¹1, 57–70. 3. E. Hairer, G.Wanner, Solving ordinary differential equations II. Stiff and DifferentialAlgebraic Problems, Springer Series in Computational Mathematics, V.14, 2nd rev. edn. SpringerVerlag Berlin, Heidelberg, 1996. 4. J.C. Strikwerda, B.A. Wade, A survey of the Kreiss matrix theorem for power bounded families of matrices and its extensions, in: Linear Operators, Banach Center Publ., 38 (1997), 339–360. 
Pages 
115120

Volume 
46

Issue 
2

Year 
2016

Journal 
Matematychni Studii

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