Factorization of a rational matrix: The singular case
type: Item
instance of: scholarly article
title: Factorization of a rational matrix: The singular case
zbMATH Open document ID: 0553.47010
DOI: 10.1007/BF01195921
published in: Integral Equations and Operator Theory
publication date: 1984-01-01T00:00:00Z
review text: In this paper the author studies the problem of minimal factorization of an arbitrary rational matrix R(s), i.e. where R(s) is not necessarily square or invertible. The definition of minimality of the factorization: \(R(s)=R_ 1(s)\cdot R_ 2(s),\) where R(s), \(R_ 1(s)\) and \(R_ 2(s)\) have respective dimensions \(m\times n\), \(m\times r\) and \(r\times n\), supposes that: i) no pole-zero cancellations occur between the two factors ii) the dimension r common to both factors, equals the normal rank of R(s). Following this definition of minimality, it is shown that the problem can be solved via a generalized eigenvalue problem which will be singular when R(s) is singular. The concept of invariant subspace, which has been used in the solution of the minimal factorization problem for regular matrices, is now replaced by the recently introduced concept of reducing subspace, a logical extension of invariant and deflating subspaces to the singular pencil case. The theory naturally extends to the multiple factor case, which is also treated in detail in this paper. Finally, the author shortly discusses some related problems occurring in the area of systems and control theory.
Mathematics Subject Classification ID: 47A56
Mathematics Subject Classification ID: 47A68
Mathematics Subject Classification ID: 65F99
Mathematics Subject Classification ID: 93B40
Mathematics Subject Classification ID: 94C05
zbMATH DE Number: 3881335
zbMATH Keywords: minimal factorization of an arbitrary rational matrix
zbMATH Keywords: generalized eigenvalue problem
zbMATH Keywords: invariant subspace
zbMATH Keywords: reducing subspace
zbMATH Keywords: deflating subspaces
zbMATH Keywords: singular pencil
zbMATH Keywords: multiple factor case
author: Paul van Dooren
describes a project that uses: HQR3 EXCHNG
describes a project that uses: HQR3
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3881335