Stability analysis of companion matrices
Institut fur Analysis und Scientific Computing, Technische Universitat Wien, Austria; Institute for Applied Mathematics and Fundamental Sciences, Lviv Polytechnic National University
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.
companion matrix; stable spectrum; similarity; contraction
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