Rosenbrock's theorem on system matrices over elementary divisor domains

Rosenbrock"s theorem on polynomial system matrices is a classical result in linear systems theory that relates the SmithMcMillan form of a rational matrix Gwith the Smith form of an irreducible polynomial system matrix Pgiving rise to Gand the Smith form of a submatrix of P. This theorem has been essential in the development of algorithms for computing the poles and zeros of a rational matrix via linearizations and generalized eigenvalue algorithms. In this paper, we extend Rosenbrock"s theorem to system matrices Pwith entries in an arbitrary elementary divisor domain Rand matrices Gwith entries in the field of fractions of R. These are the most general rings where the involved Smith-McMillan and Smith forms both exist and, so, where the problem makes sense. Moreover, we analyze in detail what happens when the system matrix is not irreducible. Finally, we explore how Rosenbrock"s theorem can be extended when the system matrix Pitself has entries in the field of fractions of the elementary divisor domain.

smith form; smith-mcmillan form; elementary divisor domain; field of fractions; polynomial matrices; rational matrices; polynomial system matrices; schur complement

Rosenbrock's theorem on system matrices over elementary divisor domains Articles