"Inners" approach to some problems of system theory

In this paper a formal definition of "inners" of a square matrix is introduced. Also defined are the concepts of positive, negative, null, and nonnull innerwise matrix. Applications of these definitions are shown 1) for the necessary and sufficient conditions for the roots of a real polynomial to be within the unit circle, 2) for the roots of a real polynomial to be distinct and on the real axis, and 3) to be distinct and on the imaginary axis in the complex plane. Conditions on relative stability of linear continuous systems as well as on root distribution within the unit circle are also presented in terms of inners. Furthermore, a theorem establishing the equivalence between positive innerwise matrix and a positive definite symmetric matrix is derived for the case when all the roots of a real polynomial are inside the unit circle. Extension of the inners approach to the general problems of clustering of roots as well as of the root distribution in the complex plane is briefly mentioned. Several examples are presented to elucidate the application of the inners concept to some problems that arise in system theory.