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.