Keldysh Chains and Factorizations of Matrix Polynomials

Abstract

The paper studies factorizations $a ( \lambda ) = b ( \lambda ) \cdot c ( \lambda )$ of $n \times n$ matrix polynomials, with emphasis on linear $b ( \lambda )$. It is assumed that $\det a ( \lambda )$ factorizes into scalar polynomials of degree 1. The tool is the concept of Keldysh chains (generalized Jordan chains). In Theorem 1 the linear independence of eigenvectors of a matrix is generalized to Keldysh chains.It is known that there is an $nm \times nm$ matrix polynomial $Kt ( m,a ( \lambda ) )$ such that the Keldysh chains of length $\leqq m$ with respect to an “eigenvalue” $\omega ( \det a ( \omega ) = 0 )$ may be identified with the nonzero vectors in the nullspace of $Kt ( m,a ( \omega ) )$. The main technique of the paper is to exploit the multiplicative property $Kt ( m,a ( \lambda ) ) = Kt ( m,b( \lambda ) ) \cdot Kt ( m,c ( \lambda ) )$ together with the mentioned generalization. Improved factorization results are obtained when det $a( \lambda )$ has more than $( n - 1 ) \cdot \deg a ( \lambd...