Inverse Tridiagonal Z – Matrices

In this paper, we consider matrices whose inverses are tridiagonal Z–matrices. Based on a characterization of symmetric tridiagonal matrices by Gantmacher and Krein, we show that a matrix is the inverse of a tridiagonal Z–matrix if and only if, up to a positive scaling of the rows, it is the Hadamard product of a so called weak type D matrix and a flipped weak type D matrix whose parameters satisfy certain quadratic conditions. We predict from these parameters to which class of Z–matrices the inverse belongs to. In particular, we give a characterization of inverse tridiagonal M–matrices. Moreover, we characterize inverses of tridiagonal M–matrices that satisfy certain row sum criteria. This leads to the cyclopses that are matrices constructed from type D and flipped type D matrices. We establish some properties of the cyclopses and provide explicit formulae for the entries of the inverse of a nonsingular cyclops. We also show that the cyclopses are the only generalized ultrametric matrices whose inverses are tridiagonal.