Z-matrices and inverse Z-matrices Original Research Article

We consider Z-matrices and inverse Z-matrices, i.e. those nonsingular matrices whose inverses are Z-matrices. Recently Fiedler and Markham introduced a classification of Z-matrices. This classification directly leads to a classification of inverse Z-matrices. Among all classes of Z-matrices and inverse Z-matrices, the classes of M-matrices, N0-matrices, F0-matrices, and inverse M-matrices, inverse N0-matrices and inverse F0-matrices, respectively, have been studied in detail. Here we discuss each single class of Z-matrices and inverse Z-matrices as well as considering the whole classes of Z-matrices and inverse Z-matrices. We establish some common properties of the classes, such as eigenvalue bounds and determinant inequalities, and we give a new characterization of some classes of Z-matrices and inverse Z-matrices. Moreover, we prove that other classes besides those of M-matrices, N0-matrices, and F0-matrices consist of matrices whose determinants have the same sign. Some of the results generalize known results for M-matrices, N0-matrices, and F0-matrices and for inverse M-matrices, inverse N0-matrices, and inverse F0-matrices. However, we also show that some properties of the specific classes mentioned above do not hold for all classes of Z-matrices and inverse Z-matrices.