Science:Math Exam Resources/Courses/MATH152/April 2012/Question 07 (d)/Solution 1

The stationary distribution is the eigenvector corresponding to eigenvalue 1. The other two eigenvectors correspond to eigenvalues with absolute value strictly less than 1. This means that only the component of the initial distribution that is parallel to the stationary distribution will maintain its length over repeated applications of P. Since the 3x3 matrix P has three distinct eigenvalues, every vector of initial conditions can be composed to a sum of eigenvectors. Hence, all components of the initial distribution that are not parallel to the stationary distribution will decay with repeated applications of P.

Hence, all initial distributions eventually converge to the stationary distribution computed in part (c).