逆三对角Z-矩阵
Inverse tridiagonal Z-matrices
摘要：
本文考虑逆矩阵为三对角Z-矩阵的矩阵。本文利用Kreinmacher和Krein对对称三对角矩阵的一个刻画，证明了一个矩阵是三对角Z-矩阵的逆矩阵当且仅当在正的行尺度下，它是一个所谓的弱型$\D$矩阵和一个参数满足一定二次条件的翻转弱型$\D$矩阵的Hadamard积.我们预测从这些参数的Z-矩阵的逆属于哪一类。特别地，我们给出了逆三对角M-矩阵的一个特征。此外，我们还刻画了满足一定行和准则的三对角M-矩阵的逆矩阵。这就产生了由$\D$型矩阵和翻转型$\D$型矩阵构成的cyclopses矩阵。我们建立了独眼巨人的一些性质，并给出了非奇异独眼巨人的逆元素的显式公式。我们还证明了独圆阵是唯一的广义超度量矩阵的逆是三对角的。
英文摘要：
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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