Skew prime polynomial matrices

A pair of polynomial matrices,P(s)andQ(s), is defined to be "externally skew prime" if and only if a solution,M(s), N(s), to the polynomial matrix equationP(s)M(s)+N(s)Q(s)=Iexists. It is shown thatP(s)andQ(s)are externally skew prime if and only ifQ(s)P(s)= P̄(s)R(s)withQ(s)andP̄(s)relatively left prime andP(s)andR(s)relatively right prime. This observation implies a new constructive procedure for determiningM(s)andN(s)whereP(s)andQ(s)are found to be externally skew prime andP(s)is nonsingular. A new procedure for obtaining solutions to the more general polynomial matrix equation,P(s)M(s)+N(s)Q(s)= V(s), based on the notion of skew-prime polynomial matrices is also presented. A characterization of all solutions whenV(s)= Iis also given, under appropriate assumptions, and then employed to determine a unique solution to this polynomial matrix equation.