Pole cancellation Original Research Article

The problem of eliminating the right half plane poles of an rmvf (rational matrix valued function) G(z) with minimal realization G(z) = D + C(zIn − A)−1B by multiplication on the right by a suitably chosen J-inner rmvf Θ(z) is considered from a number of different points of view, including the notion of minimal J conjugators that was introduced by Kimura, the null/pole structure of rmvf’s that is developed at length in the monograph of Ball–Gohberg–Rodman, and the theory of reproducing kernel Hilbert spaces. Connections between these different points of view are developed and correspondences between (1) the Jordan chains corresponding to the right half plane eigenvalues of A*, (2) the left null chains of Θ(z) in the sense of Ball–Gohberg–Rodman, and (3) the invariant subspaces of the generalized backwards shift operator applied to a suitably defined space of rmvf’s are established. Enroute, a theorem of Kimura that relates the existence of minimal pole conjugators to the existence of solutions of a related Riccati equation is refined and made more precise with the aid of the techniques referred to above.