Stability analysis of companion matrices |
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| 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
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| Abstract |
Assume that a family of (non-normal) matrices has stable spectra contained in the complex
unit circle. This does not necessarily imply that the 2-norm of such matrices is small, and the
question is under what `natural' similarity transformation the transformed matrices will have
2-norm 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.
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| Keywords |
companion matrix; stable spectrum; similarity; contraction
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| DOI |
doi:10.15330/ms.46.2.115-120
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| 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 Differential-Algebraic Problems, Springer Series in Computational Mathematics, V.14, 2nd rev. edn. Springer-Verlag 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 |
115-120
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| Volume |
46
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| Issue |
2
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| Year |
2016
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| Journal |
Matematychni Studii
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| Full text of paper | |
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