On equitable partition of matrices and its applications

Abstract

ABSTRACT A partition of a square matrix A is said to be equitable if all the blocks of the partitioned matrix have constant row sums and each of the diagonal blocks are of square order. A quotient matrix Q of a square matrix A corresponding to an equitable partition is a matrix whose entries are the constant row sums of the corresponding blocks of A. A quotient matrix is a useful tool to find some eigenvalues of the matrix A. In this paper we determine some matrices whose eigenvalues are those eigenvalues of A which are not the eigenvalues of a quotient matrix of A. Using this result we find eigenvalue localization theorems for matrices having an equitable partition. In particular, we find eigenvalue localization theorems for stochastic matrices and give a suitable example to compare with the existing results.