Stability analysis of companion matrices

Author
W. Auzinger, R. Stolyarchuk
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 (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.
Keywords
companion matrix; stable spectrum; similarity; contraction
DOI
doi:10.15330/ms.46.2.115-120
Reference
1. W. Auzinger, A note on similarity to contraction for stable 2x2 companion matrices, Ukr. Mat. Zh., 68 (2016), 3, 400407.

2. A. Eder, G. Kirlinger, A normal form for multistep companion matrices, Math. Models and Methods in Applied Sciences, 11 (2001), 1, 5770.

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), 339360.

Pages
115-120
Volume
46
Issue
2
Year
2016
Journal
Matematychni Studii
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