Factorization of a rational matrix: The singular case
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.
minimal factorization of an arbitrary rational matrix
scientific article; zbMATH DE number 3881335
scientific article