This paper proposes a new condition for the root clustering of a real matrix A in a complex region D defined by a polynomial matrix inequality (PMI region). For a general case, a sufficient condition is given so that the eigenvalues of A lie in D . It was shown that this condition is necessary and sufficient for some particular PMI regions including linear matrix inequality (LMI) regions, quadratic matrix inequality (QMI) regions, polynomial regions and many others.

This paper also provides an extension to the classical theorem of Cyparissos Stephanos, which formulates the relationship between the eigenvalues of two matrices and those of composite matrices of their Kronecker products. This extension turned out to be crucial for proving the main results obtained.

Based on the analysis of the proposed condition, a guardian map function can be used for tackling the problem of robust D-stability of single-parameter uncertain linear systems, for which the exact and possibly disconnected domain of D-stability was determined. All the obtained results were illustrated by certain examples.