A formal theory of matrix primeness

Primeness of nD polynomial matrices is of fundamental importance in multidimensional systems theory. In this paper we define a quantity which describes the “amount of primeness” of a matrix and identify it as the concept of grade in commutative algebra. This enables us to produce a theory which unifies many existing results, such as the Bézout identities and complementation laws, while placing them on a firm algebraic footing. We also present applications to autonomous systems, behavioural minimality of regular systems, and transfer matrix factorization. This work has been sponsored by EPSRC Grant No. GR/K 18504.