A Jordan factorization theorem for polynomial matrices

Wimmer, H. K.

doi:10.1090/S0002-9939-1979-0532135-6

A Jordan factorization theorem for polynomial matrices
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by H. K. Wimmer PDF

Proc. Amer. Math. Soc. 75 (1979), 201-206 Request permission

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

Hellmut Baumgärtel, Endlichdimensionale analytische Störungstheorie, Mathematische Lehrbücher und Monographien, II. Abteilung. Mathematische Monographien, Band 28, Akademie-Verlag, Berlin, 1972 (German). MR 0634965

Finite dimensional linear systems

W. A. Coppel, Matrices of rational functions, Bull. Austral. Math. Soc. 11 (1974), 89–113. MR 401805, DOI 10.1017/S0004972700043677
Paul A. Fuhrmann, Algebraic system theory: an analyst’s point of view, J. Franklin Inst. 301 (1976), no. 6, 521–540. MR 414159, DOI 10.1016/0016-0032(76)90076-4

Matrizenrechnung

Peter Lancaster and Harald K. Wimmer, Zur Theorie der $\lambda$-Matrizen, Math. Nachr. 68 (1975), 325–330 (German). MR 573015, DOI 10.1002/mana.19750680124
Harald K. Wimmer, Jordan-Ketten und Realisierungen rationaler Matrizen, Linear Algebra Appl. 20 (1978), no. 2, 101–110 (German, with English summary). MR 466172, DOI 10.1016/0024-3795(78)90044-7