Spectral Estimates for Partitioned Matrices Based on the Pupkov Condition

The spectrum of a matrix contains information essential for understanding the dynamics of the real system behind the matrix. Knowledge of eigenvalues and generalized eigenvalues is of great importance in stability analysis. In this paper, we present a new technique for obtaining spectrum localizations of any given square complex matrix with respect to any chosen partition of the index set. The construction of the localization set is tightly related to special H-matrices introduced in this paper by the name Partitioned Pupkov matrices. The new matrix class is applied in order to obtain an estimate of the spectral radius of graphs and matrices. Results on spectral radius for a weighted digraph with matrix weights are widely applicable in modeling multidimensional dynamics, analysis of system stability and connectivity strengths of corresponding networks. We present further possibilities for application to localization of generalized eigenvalues of matrix pencils. We obtain a norm estimate of the inverse for matrices in the new class. The relationships between the new criteria and well-known matrix classes are analyzed, and numerical examples illustrate how the proposed estimates and localization sets compare to existing bounds.