Complex- 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.