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.