On minimial factorizations of rational matrix functions. (English) Zbl 0542.47010
The theory of factorization of rational \(n\times n\) matrix functions \(W(\lambda)=D+C(\lambda -A)^{-1}B,\) as developed by H. Bart, I. Gohberg and M. A. Kaashoek [Minimal factorization of matrix and operator functions (1979; Zbl 0424.47001)], is extended to the case where \(D=W(\infty)\) is not invertible. This extension requires a generalization of the notion of a supporting projection and includes a description of all minimal factorizations of which the factors are again square rational matrix functions. Special attention is paid to spectral factorizations, stable factorizations and to the case when \(\det W(\lambda)\not\equiv {\text{ß}}.\) As an application a new proof is given of the Gohberg- Lancaster-Rodman factorization theorem for monic matrix polynomials.
Reviewer: M.A.Kasshoek
MSC:
47A56 | Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones) |
15A23 | Factorization of matrices |
47A68 | Factorization theory (including Wiener-Hopf and spectral factorizations) of linear operators |
15A54 | Matrices over function rings in one or more variables |
Keywords:
factorization of rational \(n\times n\) matrix functions; supporting projection; minimal factorizations; spectral factorizations; stable factorizations; Gohberg-Lancaster-Rodman factorization theorem for monic matrix polynomialsCitations:
Zbl 0424.47001References:
[1] | Bart, H., I. Gohberg, M.A. Kaashoek: Minimal Factorization of Matrix and Operator Functions. Operator Theory: Advances and Applications, Birkhäuser (1979). |
[2] | Gohberg, I., P. Lancaster, L. Rodman: Spectral Analysis of matrix polynomials - 1. Canonical forms and divisors. Lin. Alg. and Appl. 20 (1978), 1–44. · Zbl 0375.15008 · doi:10.1016/0024-3795(78)90026-5 |
[3] | Gohberg, I., P. Lancaster, L. Rodman: Representations and divisibility of operator polynomials. Can. J. Math. 30 (1978), 1045–1069. · doi:10.4153/CJM-1978-088-2 |
[4] | Rosenbrock, H.: State-Space and Multivariable Theory. New-York, Wiley (1970). · Zbl 0246.93010 |
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