Symmetric and innerwise matrices for the root-clustering and root-distribution of a polynomial

Abstract

The general problem of root-clustering and root-distribution of a polynomial in a certain region Γ in the complex plane has been investigated in this paper. The region Γ is general and includes all the previously investigated regions. For the root-clustering problem, it is shown that by using a certain transformation, the necessary and sufficient condition can be represented either in terms of positive definite (p.d.) or positive innerwise (p.i.) matrices. The entries in these matrices are rational functions of the coefficients of the polynomial. The connection between p.d. and p.i. matrices is established in terms of matrix multiplication. The investigations of this paper are quite general in that all the previously known criteria for root-clustering can be obtained from the general formulation. Furthermore, this study shows that the “Inner” matrix representation offers unification, simplification and a systematic way for studying the mentioned problems.