A class of P-matrices with applications to the localization of the eigenvalues of a real matrix

A matrix with positive row sums and all its off-diagonal elements bounded above by their corresponding row means is called a B-matrix. It is proved that the class of B-matrices is a subset of the class of P-matrices. Properties of B-matrices are used to localize the real eigenvalues of a real matrix and the real parts of all eigenvalues of a real matrix.