On the common (left,right) divisors of two polynomial matrices

The notions of common right and common left divisors of two polynomial matrices are well established. They can be used to characterize the unobservable and the uncontrollable modes of a system and to study polynomial matrix equations of the form a1V1 + a2V2 = V3. The new notion of common (right, left) divisors of two polynomial matrices is introduced here and it is used to characterize the modes of the system which are both uncontrollable and unobservable; it is also used to study polynomial matrix equations of the form Nx1 + x2D = V where N(qxp), D(rxm) and V(qxm) are given polynomial matrices; (1) has attracted recently considerable attention ([1] to [7]) mainly because of its importance in the output regulation problem with internal stability.