On Inclusion and Exclusion Intervals for the Real Eigenvalues of Real Matrices

Abstract

A real square matrix with positive row sums and all its off-diagonal elements bounded below by the corresponding row means is called a $C$-matrix, which is introduced by Pena [Exclusion and inclusion intervals for the real eigenvalues of positive matrices, SIAM J. Matrix Anal. Appl., 26 (2005), pp. 908-917]. In this paper, a new class of nonsingular matrices—$MC$-matrices containing $C$-matrices—is first defined. By properties of a subclass of $MC$-matrices, we present new exclusion intervals of the real eigenvalues of a real matrix, which are further applied to localize the real eigenvalues different from 1 of a positive stochastic matrix. Secondly, an inclusion interval for the real parts of eigenvalues of a real matrix is established. Finally, for real matrices with nonnegative off-diagonal elements, lower and upper bounds of real eigenvalues are obtained. Furthermore, sufficient conditions are derived to indicate that the real inclusion intervals provided by Pena [On an alternative to Gerschgorin circles and ovals of Cassini, Numer. Math., 95 (2003), pp. 337-345] are subsets of those provided by the ovals of Cassini.