Example 4.  Let    record the number of people in a certain city who use brands X, Y, and Z, respectively.  
Each month people decide to keep using the same brand or switch brands.  
The probability that a user of brand X will switch to brand Y or Z is 0.3 and 0.3, respectively.
The probability that a user of brand Y will switch to brand X or Z is 0.3 and 0.2, respectively.
The probability that a user of brand Z will switch to brand X or Y is 0.1 and 0.3, respectively.
The transition matrix for this process is or
        
Assume that the initial distribution .
4 (a).  Find the first few terms in the sequence .  
4 (b).  Verify that   is the dominant eigenvector of  A.  
4 (c).  Verify that a corresponding eigenvector is  .
4 (d).  Conclude that the limiting distribution is  .  

Solution 4.

4 (a).  Enter the matrix A and vector and use the subroutine Markov to find the first few terms in the sequence .  

4 (b).  Verify that   is an eigenvector of  A.  

4 (c).  Verify that   is an eigenvector of  A  and a corresponding eigenvector is  .

4 (d).  The iteration in part (a) appears to be converging to  .

We are done.

Aside.  We can graph the situation, this is just for fun !

(c) John H. Mathews 2004