Complex-L matrix and its recognition

A matrix is called a complex-L matrix if its complex sign pattern implies that it is of full column rank. The definition is a generalization of L-matrices from the real field to the complex field. The recognition problem of complex-L matrices is studied in this paper; a combinatorial characterization of complex-L matrices is given, showing that the problem of recognizing complex-L matrices is in the class co-NP-complete. It is also shown that this recognition problem can be reduced to its subproblem for which each entry of the matrices is either real or pure imaginary.