W. Auzinger:

"Normal forms for companion matrices and contractivity in inner product norms";

in: "ASC Report No. 24/2009", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2009, ISBN: 978-3-902627-02-5.

We study the problem of finding a inner product norm in which a given companion matrix with a [weakly] stable spectrum becomes contractive (or dissipative), via a preferably well-conditioned change of basis. To this end we use a basis transformation related to a rescaled LQ decomposition of the associated Vandermonde matrix which is robust to w.r.t. confluent or non-confluent spectra. For n=2 we give an explicit construction. The transformed, contractive matrix is non-normal in general, and it depends on the distribution of the spectrum in a nonlinear way. This analysis cannot be directly generalized to higher dimension, but it suggests an algebraic/numerical algorithm for a numerically given spectrum. This has been tested for small values of n and appears to be successful.

http://www.asc.tuwien.ac.at/preprint/2009/asc24x2009.pdf

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