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Inverse tridiagonal Z-matrices

J. J. McDonald·R. Nabben·M. Neumann·H. Schneider·M. Tsatsomeros
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Abstract

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.

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J. J. McDonald, R. Nabben, M. Neumann, H. Schneider, M. Tsatsomeros. 1998-03-25. Inverse tridiagonal Z-matrices. https://arxiv.org/abs/math/9803125

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