\(Z\)-matrices and inverse \(Z\)-matrices (Q1355225)

The whole class of \(n\times n\) \(Z\)-matrices (i.e. such matrices whose off-diagonal entries are nonpositive) and inverse \(Z\)-matrices are considered. They comprise \(M\)-matrices, \(N_0\)-matrices, \(F_0\)-matrices, and their inverses. Common properties of these classes, such as eigenvalue bounds, determinant inequalities, and new characterizations of some classes are discussed. It is shown that other classes besides those of \(M\)-, \(N_0\)-, and \(F_0\)-matrices have determinants with the same sign. Some characteristics are generalized for these classes, while some characteristics do not hold for all of them.