A Jordan Factorization Theorem for Polynomial Matrices

Abstract

It is shown that a complex polynomial matrix $M(\lambda )$ which has a proper rational inverse can be factored into $M(\lambda ) = \hat C(\lambda )(\lambda I - J)\hat B(\lambda )$ where J is a matrix in Jordan normal form and the columns of $\hat C(\lambda )$ consist of eigenvectors and generalized eigenvectors of a linear operator associated with $M(\lambda )$. For a proper rational matrix W with factorizations $W(\lambda ) = C{(\lambda I - J)^{ - 1}}B = M{(\lambda )^{ - 1}}P(\lambda ) = Q(\lambda )N{(\lambda )^{ - 1}}$ it will be proved that C consists of Jordan chains of M and B of Jordan chains of N.