ABSTRACT

A Markov matrix is embeddable if it can represent a homogeneous continuous-time Markov process. It is well known that if a Markov matrix has real and pairwise-different eigenvalues, then the embeddability can be determined by checking whether its principal logarithm is a rate matrix or not. The same holds for Markov matrices that are close enough to the identity matrix. In this paper we exhibit open sets of Markov matrices that are embeddable and whose principal logarithm is not a rate matrix, thus proving that the principal logarithm test above does not suffice generically.

中文翻译：

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其主要对数不是马尔可夫生成器的4×4可嵌入矩阵的开放集

摘要

如果可以表示齐次连续时间马尔可夫过程，则可以嵌入马尔可夫矩阵。众所周知，如果马尔可夫矩阵具有实数和成对的特征值，则可通过检查其主要对数是否为速率矩阵来确定可嵌入性。距离单位矩阵足够近的马尔可夫矩阵也是如此。在本文中，我们展示了可嵌入且主对数不是比率矩阵的可嵌入马尔可夫矩阵的开放集，从而证明上述主对数检验通常不能满足要求。