ON CLASSES OF INVERSE Z-MATRICES

Recently, a classification of matrices of class Z was introduced by Fi edler and Markham. This classification contains the classes of M-matri ces and the classes of N-o- and F-o-matrices studied by Fan, G. Johnso n, and Smith. The problem of determining which nonsingular matrices ha ve inverses which are Z-matrices is called the inverse Z-matrix proble m. For special classes of Z-matrices, such as the M- and N-o-matrices, there exist at least partial results, i.e., special classes of matric es have been introduced for which the inverse of such a matrix is an M -matrix or an N-o-matrix. Here, we define a system of classes of matri ces for which the inverse of each matrix of each class belongs to one class of the classification of Z-matrices defined by Fiedler and Markh am. Moreover, certain properties of the matrices of each class are est ablished, e.g., inequalities for the sum of the entries of the inverse and the structure of certain Schur complements. We also give a necess ary and sufficient condition for regularity. The class of inverse N-o- matrices given here generalizes the class of inverse N-o-matrices disc ussed by Johnson. All results established here can be applied to a cla ss of distance matrices which corresponds to a nonarchimedean metric. This metric arises in p-adic number theory and in taxonomy.