Separation of matrix eigenvalues and structural decomposition of large-scale systems

The paper presents the separation of matrix eigenvalues relative to strips, sectors, trapezoids and circles in a complex plane without actually seeking the characteristic polynomial and the matrix eigenvalues themselves. The system matrix of interest is a real or complex matrix which may have a real or complex characteristic polynomial. Also, the paper develops a technique for block-diagonalisation and block-triangularisation of the system matrix according to the characteristics of the system eigenvalues. As each block-decomposed submatrix contains the matrix eigenvalues lying within a specific subregion of a complex plane, the existing design methods, such as the multi-stage design methods, can effectively be applied to the substructural models of large-scale systems for attaining a desired overall system behaviour. The fast matrix sign function, which has quick convergence property and convergence speed independent of the dimension of the system map, is used for the derivations.