Factorization of a rational matrix: The singular case

Abstract

We study the problem of minimal factorization of an arbitrary rational matrix R(λ), i. e. where R(λ) is not necessarily square or invertible. Following the definition of minimality used here, we show that the problem can be solved via a generalized eigenvalue problem which will be singular when R(λ) 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 a reducing subspace, a recently introduced concept which is a logical extension of invariant and deflating subspaces to the singular pencil case.