Matrix Roots and Embedding Conditions for Three-State Discrete-Time Markov Chains with Complex Eigenvalues

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.