The partial realization theory of finite size two‐dimensional arrays

Abstract

AbstractThe authors have already proposed the realization problem for (infinite) figures (arrays) as an application of realization theory for nonlinear systems, and the array realization theorem was obtained by commutative linear representation system. A discussion was also made of the relation between the infinite array and the finite‐dimensional commutative linear representation system. This paper discusses the partial realization problem for finite arrays. The notion of natural partial realization system is introduced for a finite array and the following properties are shown. For any arbitrary given finite array, there exists a minimum‐dimensional commutative linear representation system that partially realizes it. The necessary and sufficient condition for the natural partial realization system to exist for a finite array is derived in terms of the condition for the rank of the bounded Hankel matrix of the finite array. The necessary and sufficient condition for the minimum‐dimensional partial realization system to be unique except for the isomorphism is that there exists a natural partial realization system for the finite array. The proposed theory for partial realization of a finite array will be applicable to the theory for the communication of pattern information.