New results on the separation of matrix eigenvalues and the clustering of matrix elements

The problem of separation of matrix eigenvalues is considered in this paper. Necessary and sufficient conditions are given for a matrix to have all eigenvalues contained in an open set defined on the complex plane by a separable polynomial γ(z₁, z₂). The problem of clustering of matrix elements is also considered. In this case, a set of matrices is identified, having all eigenvalues in a given subset of the complex plane. The presented results can be useful in both robust stability analysis and the design of control systems