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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
DOI: https://doi.org/10.1090/S0002-9939-1979-0532135-6
MathSciNet review: 532135
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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.


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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