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

MATH152 April 2012

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Question 07 (b)
Consider a random walk that has transition matrix: $P={\begin{bmatrix}1/4&1/4&1/4\\3/4&3/4&1/4\\0&0&1/2\end{bmatrix}}$ Find all eigenvalues of the transition matrix P (no need to calculate the eigenvectors at this stage).

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?

If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
As it always has, the eigenvalues of a matrix P are the zeros of the characteristic polynomial $\operatorname {Det} (P-\lambda I)$

Hint 2

For this simple matrix however, the eigenvalue beg to be spotted without any calculation. How many eigenvalues does a 3x3 matrix have (at most)? P is a transition matrix. Which eigenvalue do all transition matrices have in common? Compare the first and the second row (or first and second column). Is P singular? What does that imply for the eigenvalues? The last row is very simple. What happens if you calculate Pe3?

Hint 3
This should already reveal all eigenvalues of P. If you miss one hint, calculate the missing eigenvalue from the fact that the sum of the diagonal entries of a matrix equals the sum of the eigenvalues.

Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution 1

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. The eigenvalues of P satisfy $\operatorname {Det} (P-\lambda I)=0.$ Performing the calculation of the determinant using the co-factor method gives ${\begin{aligned}\operatorname {Det} (P-\lambda I)&=\operatorname {Det} \left({\begin{bmatrix}1/4-\lambda &1/4&1/4\\3/4&3/4-\lambda &1/4\\0&0&1/2-\lambda \end{bmatrix}}\right)\\&=(-1)^{3+3}(1/2-\lambda )\left((1/4-\lambda )(3/4-\lambda )-3/16\right)\\&=(1/2-\lambda )(\lambda ^{2}-\lambda )\\&=\lambda (1/2-\lambda )(\lambda -1).\end{aligned}}$ Solving $\operatorname {Det} (P-\lambda I)=0$ gives the eigenvalues: ${\color {blue}\lambda =0,1/2,1}$.

Solution 2

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. If you want to be clever, you don't have to do any calculation to find the eigenvalues of P. P is a 3x3 matrix and hence has at most 3 eigenvalues. Since P is a transition matrix we know that $\displaystyle \lambda _{1}=1$ must be an eigenvalue. Since the first and second row (or first and second colum) of P are identical, P is singular. Singular matrices always have an eigenvalue $\displaystyle \lambda _{2}=0$. The only non-zero entry in the third row of P is the diagonal entry $1/2$. Hence $Pe_{3}=(1/2)e_{3}$ and therefore $\displaystyle \lambda _{3}=1/2$ is the last missing eigenvalue. We can double check our answer by verifying that the sum of eigenvalues equals the sum of diagonal entries: $\displaystyle \lambda _{1}+\lambda _{2}+\lambda _{3}=3/2=P_{11}+P_{22}+P_{33}$

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