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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.