A general theory for matrix root-clustering in subregions of the complex plane

We consider the general problem of root-clustering of a matrix in the complex plane: Let A \in C^{n \times n} and S \subset C . Find the largest class of S and an algebraic criterion which is necessary and sufficient for \lambda_{i}[A] \in S, i=1,2,..., n . We introduce two types of regions which constitute the largest class of S known to date. The criterion is presented both for open regions and closed ones. The results are used to define a design methodology for control systems. Moreover, all classical results are shown to be special cases of the present theory.