A transition effect matrix (TEM) is a quantum generalization of a classical stochastic matrix. By employing a TEM we obtain a quantum generalization of a classical Markov chain. We first discuss state and operator dynamics for a quantum Markov chain. We then consider various types of TEMs and vector states. In particular, we study invariant, equilibrium and singular vector states and investigate projective, bistochastic, invertible and unitary TEMs.