Some characterizations of Nekrasov and $S$-Nekrasov matrices

It is known that the Nekrasov and $S$-Nekrasov matrices form subclasses of (nonsingular) $H$-matrices. The paper presents some necessary and sufficient conditions for a square matrix with complex entries to be a Nekrasov and an $S$-Nekrasov matrix. In particular, characterizations of the Nekrasov and $S$-Nekrasov matrices in terms of the diagonal column scaling matrices transforming them into strictly diagonally dominant matrices are obtained.