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

Consider a random walk that has transition matrix: Based on the results in (b), explain why the probability vector always approaches the equilibrium found in (c), regardless of the initial probability vector . 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
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Hint 

Remember that for any matrix , all eigenvectoreigenvalue combinations satisfy What happens when versus when as ? 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. 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). 