Science:Math Exam Resources/Courses/MATH307/April 2012/Question 08 (b)

MATH307 April 2012

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Question 08 (b)
Consider the stochastic matrix $P={\begin{bmatrix}0&1/4&1/3\\1&1/2&1/3\\0&1/4&1/3\end{bmatrix}}$ (b) By considering P2, what can you say about the other eigenvalues of P? Give a reason.

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Hint

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. We will begin by calculating what $P^{2}$ is: ${\begin{aligned}P^{2}&={\begin{bmatrix}0&1/4&1/3\\1&1/2&1/3\\0&1/4&1/3\end{bmatrix}}{\begin{bmatrix}0&1/4&1/3\\1&1/2&1/3\\0&1/4&1/3\end{bmatrix}}\\&={\begin{bmatrix}0(0)+1/4(1)+1/3(0)&0(1/4)+1/4(1/2)+1/3(1/4)&0(1/3)+1/4(1/3)+1/3(1/3)\\1(0)+1/2(1)+1/3(0)&1(1/4)+1/2(1/2)+1/3(1/4)&1(1/3)+1/2(1/3)+1/3(1/3)\\0(0)+1/4(1)+1/3(0)&0(1/4)+1/4(1/2)+1/3(1/4)&0(1/3)+1/4(1/3)+1/3(1/3)\end{bmatrix}}\\&={\begin{bmatrix}1/4&5/24&7/36\\1/2&7/12&11/18\\1/4&5/24&7/36\end{bmatrix}}\end{aligned}}$ We know that one of the properties of a stochastic matrix is, if $P$ or some power $P^{k}$ has all positive entries (that is, no zero entries) then the other eigenvalues of $P$ besides $\lambda =1$ will have $|\lambda _{i}|<1$ (referenced from page 187 of the current 2014 Summer Math 307 Textbook). Since we just found that all the entries in $P^{2}$ are positive, we can see that the other eigenvalues of $P$ must follow: $\lambda _{i}\in (-1,1)$.