Real matrices with non-negative off-diagonal entries play a crucial role for modelling the positive linear dynamical systems. In the literature, these matrices are referred to as Metzler matrices or negated Z-matrices. Finding the closest stable Metzler matrix to an unstable one (and vice versa) is an important issue with many applications. The stability considered here is in the sense of Hurwitz, and the distance between matrices is measured in \(l_\infty ,\,l_1\), and in the max norm. We provide either explicit solutions or efficient algorithms for obtaining the closest (un)stable matrix. The procedure for finding the closest stable Metzler matrix is based on the recently introduced selective greedy spectral method for optimizing the Perron eigenvalue. Originally intended for non-negative matrices, here is generalized to Metzler matrices. The efficiency of the new algorithms is demonstrated in examples and numerical experiments for the dimension of up to 2000. Applications to dynamical systems, linear switching systems, and sign-matrices are considered.

中文翻译：

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稳定Metzler矩阵及其在动力系统中的应用

具有非负非对角线入口的实矩阵在建模正线性动力学系统中起着至关重要的作用。在文献中，这些矩阵称为Metzler矩阵或负Z矩阵。在许多应用中，寻找最接近不稳定的Metzler矩阵（反之亦然）是一个重要问题。此处考虑的稳定性在Hurwitz的意义上，矩阵之间的距离以\（l_ \ infty，\，l_1 \）度量。，并以最大标准。我们提供明确的解决方案或有效的算法来获取最接近的（不稳定）矩阵。查找最近的稳定Metzler矩阵的过程基于最近引入的用于优化Perron特征值的选择性贪婪谱方法。原本打算用于非负矩阵，但在这里泛化为Metzler矩阵。在高达2000的尺寸的示例和数值实验中证明了新算法的效率。考虑了在动态系统，线性开关系统和符号矩阵中的应用。