We extend here the Perron–Frobenius theory of nonnegative matrices to certain complex matrices. Following the generalization of the Perron–Frobenius theory to matrices that have some negative entries, given by Noutsos [14], we introduce here two types of extensions of the Perron–Frobenius theory to complex matrices. We present and prove here some sufficient conditions and some necessary and sufficient conditions for a complex matrix to have a Perron–Frobenius eigenpair. We apply this theory by introducing Perron–Frobenius splittings, as well as complex Perron–Frobenius splittings, for the solution of complex linear systems 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 also given and proved.