Use of Eigen values and eigen vectors to calculate higher transition probabilities

Suppose you have a Markov chain with  states, where  is finite, and it is also given that the transition probability matrix is , then one can very easily calculate the  step transition probability matrix, , such that we utilize the equation, , , where  and  for . Now our main intention of this discussion is to utilize the concept of eigen values and eigen vectors to calculate .
One must remember that for a square matrix, , the characteristics roots of the equation: , where  are called the eigen values, and they are given by . While , , is the right/left eigen vector, corresponding to , , such that the following equation (for), s, i.e., , , i.e.,

 holds for , such that we will finally obtain , where the  column corresponds to the eigen vector corresponding to the  characteristics root, .