The matrix P is called a homogeneous transition or stochastic matrix because all the transition probabilities p_ij are fixed and independent of time. The probability p_ij must satisfy the following conditions.
The Markov chain is now defined. A transition matrix P together with the initial probabilities a_j⁽⁰⁾ associated with the states E_j completely define a Markov chain. One usually thinks of a Markov chain as describing the transitional behaviour of a system over equal intervals. Situations exist where the length of the interval depends on the characteristics of the system and hence may not be equal. This case is referred to as imbedded Markov chains.