مرکز منطقه ای اطلاع رساني علوم و فناوري

Abstract :

A pair of polynomial matrices, $P(s)$ and $Q(s)$ , is defined to be "externally skew prime" if and only if a solution, $M(s), N(s)$ , to the polynomial matrix equation $P(s)M(s)+N(s)Q(s)=I$ exists. It is shown that $P(s)$ and $Q(s)$ are externally skew prime if and only if $Q(s)P(s)= \\bar{P}(s)R(s)$ with $Q(s)$ and $\\bar{P}(s)$ relatively left prime and $P(s)$ and $R(s)$ relatively right prime. This observation implies a new constructive procedure for determining $M(s)$ and $N(s)$ where $P(s)$ and $Q(s)$ are found to be externally skew prime and $P(s)$ 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 when $V(s)= I$ is also given, under appropriate assumptions, and then employed to determine a unique solution to this polynomial matrix equation.