On Perron–Frobenius property of matrices having some negative entries☆

We extend the theory of nonnegative matrices to the matrices that have some negative entries. We present and prove some properties which give us information, when a matrix possesses a Perron–Frobenius eigenpair. We apply also this theory by proposing the Perron–Frobenius splitting for the solution of the linear system Ax = b by classical iterative methods. Perron– Frobenius splittings constitute an extension of the well known regular splittings, weak regular splittings and nonnegative splittings. Convergence and comparison properties are given and proved.