An Explicit Jordan Decomposition of Companion Matrices
DOI:
https://doi.org/10.5540/tema.2006.07.02.0209Abstract
We derive a closed form for the Jordan decomposition of companion matrices including properties of generalized eigenvectors. As a consequence, we provide a formula for the inverse of confluent Vandermonde matrices and results on sensitivity of multiple roots of polynomials.References
[1] F.S.V. Bazán, Ph.L. Toint, Error analysis of signal zeros from a related companion matrix eigenvalue problem, Applied Mathematics Letters, 14 (2001), 859-866.
F.S.V. Bazán. Error analysis of signal zeros: a projected companion matrix approach, Linear Algebra Appl., 369 (2003), 153-167.
L.H. Bezerra, F.S.V. Bazán, Eigenvalue locations of generalized predictor companion matrices, SIAM J. Matrix Anal. Appl. 19, No. 4 (1998), 886-897.
F. Chaitin-Chatelin, V. Frayssé, “Lectures on Finite Precision Computations”. SIAM, Philadelphia 1996.
M.I. Friswell, U. Prells, S.D. Garvey, Low-rank damping modifications and defective systems, Journal of Sound and Vibration, 279 (2005), 757-774.
K-C Toh, Lloyd N. Trefethen, Pseudozeros of polynomials and pseudospectra of companion matrices, Numer. Math. 68 (1994), 403-425.
W. Gautschi, Questions of numerical condition related to polynomials, in MAAA Studies in Mathematics, Vol. 24, Studies in Numerical Analysis, G. H. Golub, ed., USA, 1984, The Mathematical Association of America, pp. 140-177.
G.H. Golub, C.F. Van Loan, “Matrix Computations”, The Johns Hopkins University Press, Baltimore, 1996.
P. de Groen, B. de Moor, The fit of a sum of exponentials to noisy data, Comput. Appl. Math., 20 (1987), 175-187.
R. Horn, Ch.R. Johnson, “Matrix Analysis”, Cambridge University Press 1999.
H.M. M¨oller, J. Stetter, Multivariate polynomial equations with multiple zeros solved by matrix eigenproblems, Numer. Math., 70, 311-329.
J.H.Wilkinson, “The Algebraic Eigenvalue Problem”, Oxford University Press, Oxford, UK, 1965.
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