Exclusion and Inclusion Intervals for the Real Eigenvalues of Positive Matrices

Given a real matrix, we analyze an open interval, called row exclusion interval, such that the real eigenvalues do not belong to it. We characterize when the row exclusion interval is nonempty. In addition to the exclusion interval, inclusion intervals for the real eigenvalues, alternative to those provided by the Gerschgorin disks, are also considered for matrices whose off-diagonal entries present a restricted dispersion. The results are applied to obtain a sharp upper bound for the real eigenvalues different from 1 of a positive stochastic matrix and a sufficient condition for the stability of a negative matrix, among other applications. 1. Introduction. Several inclusion regions in the complex plane for the eigenvalues of a matrix have been considered: Gerschgorin disks (see [14]), Brauer ovals of Cassini (see [1] and [15]), Brualdi Lemniscata sets (see [3]) or the minimal Gerschgorin set (see [12]). These sets have been recently compared in [13], and in [4] appear other inclusion regions. In order to localize the real parts of the eigenvalues of a real matrix, alternative methods to Gerschgorin disks and Brauer ovals of Cassini have been presented in [10] and [11], respectively. A key tool for these alternative methods has been the use of a class of real matrices with positive determinant, called B-matrices.