On classes of inverse Z-matrices

• Published in: Linear algebra and its applications 1995, Vol.223, p.521-552
• Main Authors: Nabben, Reinhard, Varga, Richard S.
• Format: Article
• Language: English
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/0024-3795(95)92718-8

Recently, a classification of matrices of class Z was introduced by Fiedler and Markham. This classification contains the classes of M-matrices and the classes of N 0- and F 0-matrices studied by Fan, G. Johnson, and Smith. The problem of determining which nonsingular matrices have inverses which are Z-matrices is called the inverse Z-matrix problem. For special classes of Z-matrices, such as the M- and N 0-matrices, there exist at least partial results, i.e., special classes of matrices have been introduced for which the inverse of such a matrix is an M-matrix or an N 0-matrix. Here, we define a system of classes of matrices for which the inverse of each matrix of each class belongs to one class of the classification of Z-matrices defined by Fiedler and Markham. Moreover, certain properties of the matrices of each class are established, e.g., inequalities for the sum of the entries of the inverse and the structure of certain Schur complements. We also give a necessary and sufficient condition for regularity. The class of inverse N 0-matrices given here generalizes the class of inverse N 0-matrices discussed by Johnson. All results established here can be applied to a class of distance matrices which corresponds to a nonarchimedean metric. This metric arises in p-adic number theory and in taxonomy.