Inverse tridiagonal Z-Martices ∗

In this paper, we consider whose invereses are tridiagonal Z-matrices Based on a characterization of symmetric tridiagonal matirices by Gantmacher and Kein, 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 certiain quadratic conditions we predict from these parameters to which class of Z-matices the inverse belings to In particular, we give a characterization of inverse tridiagonal M-matrices Moreover ,we charactetrize inverese of diagonal M-matrices that saftisfy certain row sum ceriteria. 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 cyclopses. we also shoe that the cyclopses are the only generalized ultrametric matrices whose inverses are tridiagonal