Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A Jordan factorization theorem for polynomial matrices


Author: H. K. Wimmer
Journal: Proc. Amer. Math. Soc. 75 (1979), 201-206
MSC: Primary 15A54; Secondary 15A23
MathSciNet review: 532135
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that a complex polynomial matrix $ M(\lambda )$ which has a proper rational inverse can be factored into $ M(\lambda ) = \hat C(\lambda )(\lambda I - J)\hat B(\lambda )$ where J is a matrix in Jordan normal form and the columns of $ \hat C(\lambda )$ consist of eigenvectors and generalized eigenvectors of a linear operator associated with $ M(\lambda )$. For a proper rational matrix W with factorizations $ W(\lambda ) = C{(\lambda I - J)^{ - 1}}B = M{(\lambda )^{ - 1}}P(\lambda ) = Q(\lambda )N{(\lambda )^{ - 1}}$ it will be proved that C consists of Jordan chains of M and B of Jordan chains of N.


References [Enhancements On Off] (What's this?)

  • [1] Hellmut Baumgärtel, Endlichdimensionale analytische Störungstheorie, Akademie-Verlag, Berlin, 1972 (German). Mathematische Lehrbücher und Monographien, II. Abteilung. Mathematische Monographien, Band 28. MR 0634965
  • [2] R. W. Brockett, Finite dimensional linear systems, Wiley, New York, 1970.
  • [3] W. A. Coppel, Matrices of rational functions, Bull. Austral. Math. Soc. 11 (1974), 89–113. MR 0401805
  • [4] Paul A. Fuhrmann, Algebraic system theory: an analyst’s point of view, J. Franklin Inst. 301 (1976), no. 6, 521–540. MR 0414159
  • [5] F. R. Gantmacher, Matrizenrechnung, Teil II, 3. Aufl., VEB Deutscher Verlag der Wissenschaften, Berlin, 1971.
  • [6] Peter Lancaster and Harald K. Wimmer, Zur Theorie der 𝜆-Matrizen, Math. Nachr. 68 (1975), 325–330 (German). MR 0573015
  • [7] Harald K. Wimmer, Jordan-Ketten und Realisierungen rationaler Matrizen, Linear Algebra and Appl. 20 (1978), no. 2, 101–110 (German, with English summary). MR 0466172

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 15A54, 15A23

Retrieve articles in all journals with MSC: 15A54, 15A23


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1979-0532135-6
Keywords: Polynomial matrices, Jordan chains, Jordan normal form, realizations, shift operator
Article copyright: © Copyright 1979 American Mathematical Society