The present paper examines matrix root properties and embedding conditions for discrete-time Markov chains with three states and a transition matrix having complex eigenvalues. Necessary as well as sufficient conditions for the existence of an m-th stochastic root of the transition matrix are investigated. Matrix roots are expressed in analytical form based on the spectral decomposition of the transition matrix, and properties of these matrix roots are proved.

中文翻译：

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具有复特征值的三态离散时间马尔可夫链的矩阵根和嵌入条件

本文研究了具有三个状态的离散时间马尔可夫链和具有复杂特征值的转移矩阵的矩阵根特性和嵌入条件。研究了转移矩阵的第m个随机根的存在的必要条件和充分条件。基于过渡矩阵的光谱分解，以分析形式表示矩阵根，并证明了这些矩阵根的性质。