Z-MATRICES AND INVERSE Z-MATRICES

We consider Z-matrices and inverse Z-matrices, i.e. those nonsingular matrices whose inverses are Z-matrices. Recently Fiedler and Markham i ntroduced 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, N-0-mat rices, F-0-matrices, and inverse M-matrices, inverse N-0-matrices and inverse F-0-matrices, respectively, have been studied in detail. Here we discuss each single class of Z-matrices and inverse Z-matrices as w ell as considering the whole classes of Z-matrices and inverse Z-matri ces. We establish some common properties of the classes, such as eigen value bounds and determinant inequalities, and we give a new character ization of some classes of Z-matrices and inverse Z-matrices. Moreover , we prove that other classes besides those of M-matrices, N-0-matrice s, and F-0-matrices consist of matrices whose determinants have the sa me sign. Some of the results generalize known results for M-matrices, N-0-matrices, and F-0-matrices and for inverse M-matrices, inverse N-0 -matrices, and inverse F-0-matrices. However, we also show that some p roperties of the specific classes mentioned above do not hold for all classes of Z-matrices and inverse Z-matrices. (C) Elsevier Science Inc ., 1997.