Exercise $4\mathrm{.}5$ a$.$ Use MATLAB to find a matrix Q and a diagonal matrix D such that P $=$ QDQ$-1$ b$.$ Now recall that pn $=$ QD$\text{'}$Q$-1.$ 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$?($on the appropriate sides$)$ to compute poo, the limit as n tends to infinity of P$.$ Store the answer in a variable called Pint. 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 vectory in R$4$ 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 Poo give a mathematical explanation. You're going to need the matrix P and initial distribution vector Xo from the exercise we did: $>>$ P $=[0\mathrm{.}81000\mathrm{.}08000\mathrm{.}16000\mathrm{.}1000$: $0\mathrm{.}09000\mathrm{.}84000\mathrm{.}05000\mathrm{.}0800$: $0\mathrm{.}06000\mathrm{.}04000\mathrm{.}74900\mathrm{.}0400$; $0\mathrm{.}04000\mathrm{.}04000\mathrm{.}05000\mathrm{.}7800]>>$ X$0=[0\mathrm{.}5106$: $0\mathrm{.}4720$: $0\mathrm{.}0075$: $0\mathrm{.}00991$