Exercise 4.5 a. Use MATLAB to find a matrix Q and a diagonal matrix D such...

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Exercise 4.5 a. Use MATLAB to find a matrix Q and a diagonal matrix D such that P = QDQ-¹. b. Now recall that pn = QDQ-¹. Find the limit as n tends to infinity of D" by hand; we'll call the resulting matrix L. c. Using MATLAB, multiply L by Q and Q-1 (on the appropriate sides) to compute Poo, the limit as n tends to infinity of P. Store the answer in a variable called Pinf. d. Now use MATLAB to compute Poxo. How does your answer compare to the results you saw in the second part of the exercise from last lab? e. Make up any vector y in R4 whose entries add up to 1. Compute Pooy, and compare your result to Poxo. How does the initial distribution vector y of the electorate seem to affect the distribution in the long term? By looking at the matrix P... give a mathematical explanation. You're going to need the matrix P and initial distribution vector Xxo from the exercise we did: >> P = [0.8100 0.0800 0.1600 0.1000: 0.0900 0.8400 0.0500 0.0800: 0.0600 0.0400 0.7490 0.0400; 0.0400 0.0400 0.0500 0.7800] >> X0 = [0.5106: 0.4720: 0.0075; 0.0099] Exercise 4.5 a. Use MATLAB to find a matrix Q and a diagonal matrix D such that P = QDQ-¹. b. Now recall that pn = QDQ-¹. Find the limit as n tends to infinity of D" by hand; we'll call the resulting matrix L. c. Using MATLAB, multiply L by Q and Q-1 (on the appropriate sides) to compute Poo, the limit as n tends to infinity of P. Store the answer in a variable called Pinf. d. Now use MATLAB to compute Poxo. How does your answer compare to the results you saw in the second part of the exercise from last lab? e. Make up any vector y in R4 whose entries add up to 1. Compute Pooy, and compare your result to Poxo. How does the initial distribution vector y of the electorate seem to affect the distribution in the long term? By looking at the matrix P... give a mathematical explanation. You're going to need the matrix P and initial distribution vector Xxo from the exercise we did: >> P = [0.8100 0.0800 0.1600 0.1000: 0.0900 0.8400 0.0500 0.0800: 0.0600 0.0400 0.7490 0.0400; 0.0400 0.0400 0.0500 0.7800] >> X0 = [0.5106: 0.4720: 0.0075; 0.0099]