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[M] Examine powers of a regular stochastic matrix.a. Compute $P^{k}$ for $k=2,3,4,5,$ when$$P=\left[\begin{array}{cccc}{.3355} & {.3682} & {.3067} & {.0389} \\ {.2663} & {.2723} & {.3277} & {.5451} \\ {.1935} & {.1502} & {.1589} & {.2395} \\ {.2047} & {.2093} & {.2067} & {.1765}\end{array}\right]$$Display calculations to four decimal places. What happens to the columns of $P^{k}$ as $k$ increases? Compute the steady-state vector for $P$ .b. Compute $Q^{k}$ for $k=10,20, \ldots, 80,$ when$$Q=\left[\begin{array}{ccc}{.97} & {.05} & {.10} \\ {0} & {.90} & {.05} \\ {.03} & {.05} & {.85}\end{array}\right]$$(Stability for $Q^{k}$ to four decimal places may require $k=116$ or more.) Compute the steady-state vector for $Q .$ Conjecture what might be true for any regular stochastic matrix.c. Use Theorem 18 to explain what you found in parts (a) and $(b) .$

a. $q = \left[ \begin{array} { l } { 0.2816 } \\ { 0.3355 } \\ { 0.1819 } \\ { 0.2009 } \end{array} \right]$b. The stready state vector $\mathrm { q }$ for $\theta$ is $q = \left[ \begin{array} { c } { 0.7353 } \\ { 0.882 } \\ { 0.1765 } \end{array} \right]$c. the $j ^ { t h }$ column of $P ^ { k }$ converges to q as $k \rightarrow \infty$ ; that is, $P ^ { k } \rightarrow \left[ \begin{array} { l l l } { q } & { q } & { \cdots } & { q } \end{array} \right]$

Calculus 3

Chapter 4

Vector Spaces

Section 9

Applications to Markov Chains

Vectors

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Hello, Numerator. Welcome back. Okay, so we're in Question 21 on page 2, 64 of Chapter four, Section nine and Linear Algebra Les les MacDonald, Fifth edition. Part A is, Let's see, uh, what happens to the powers of Peace of K, B, F, K, Peter, the K, the powers, Peter the K. Um, And let's see what happens if they converge in out if they converge quickly or slowly. That's sort of things. So let's define P. And it's a matrix and the list of four lists because it's a four by four matrix. Whoa! And they're giving us the road. The elements to four places. So 40.3355 come a 0.36 stopping from the book eight to comma 0.3067 com a point. 0389 Okay, good. Next row 0.2663 0.27 to 3. I'm sorry. There's a common miss singing here. Okay, karma 0.3 to 77 comma 0.54 51. If you're not used to the matrix, constructor and sage, just google it up sage matrix, and they will be very easy to see how this works. Okay, then point in 1935. Comma 15. 2 comma 15. 89. Comma 0.23 95 and the last row, The fourth row goes, Let's see 0.2 2047 come a 0.20 and 93 Come a 0.2067 Come a 0.17 65. All right, so let's evaluate that. Make sure you don't miss anything up. E didn't show it. Whoopsy I got what I want to show it. So semi colon show p equals peak. All right, That's right. Anyways all right. So let me just check my values here. 0.3, Let me go vertically. 3355 to 636319352047 And that I checked ahead of time. That's a probabilistic. That adds up to 100%. Second column 36 80 to 27 23 15 to 2093. Good Third column 30 67 32 77 15 89 2067 I'm likely to copy our hair guys. So just checking. And last column 0.389 point 54 51.23 95.17. 65. Perfect. Okay, so let's see what the powers of P look like already. So see if this goes to a limiter. A study state, So four K in range. All right, let's say I don't know. Let's go from 1 to 11. I remember Range 11 gives you 11 inches, just starting a 00 through 10. But if they say one common 11, it starts with one instead of instead, 1 to 10. That's what's gonna give me, right. I'm going to show the powers of K. I was powers. Repeat he tooth. Okay. There you go. Oops. I want that in quotes. And then I wanted evaluated here, this whole case sensitive past. And just for fun. I'm gonna put a blank show here to separate my outputs a little bit. All right, let's see what those personal Quick. All right. So, Peter, the one piece of the two does this computer is well is well. So what I got last time? Interesting. The Collins. Not only is it a convergent, all the columns are converging. The same thing is not interesting. All right. And so what's the steady state vector? If there is one. Let's do that again. But let's multiply it by some initial vector. How about let's say Okay, I'm gonna I'm gonna define it up here. What if I say x zero is and it's a vector, but matrix the Matrix Constructive So that I can tell it how much? Because nobody comes. I want. I want four rows and one columns of four time one. Otherwise, they get a role vector when a column back to here and any any probabilistic vector would be fine. Have a point. Three Come a 30.3 come a 0.3 comma 0.1. Let's show that Europe started going for two commands online. So the initial vector is so try and put it that correctly. Yep. Yep. I know. What I wanna do is want to show Let's make this a little more interesting about I make it. Um Oh, coach camera. No, I want to show the case. So come a quote. No quote karma show the K's equals. Okay, so let's see the side what I want. So I have Peter, the one they go people want. It's the same as right, but I start with a repeat of the tube. We do the three before. So for even a small values of K it it looks like it's converging already. It's not really changing much, so I should get a steady state vector very easily out of this. I'm gonna say similarly, Let's do this. Let's show whips out there. Let's show Peter the K Times x zero. Let's see where we get So p is a regular PSA test stochastic matrix Because all the column vectors are probabilistic. They had upto one and there are no not there. All the all the values of positive, non zero and non negative, right? And this seems to converge very quickly. The powers of P So what about the Do we get a steady state vector Very quickly, we dio and look at this. The steady state vector points to 816.3355 point 1819.2009 is what the columns converted to. They're all the same, and they all converge in that interesting Hm. Very, very interesting. All right, let's do the same thing for Q. All right, so we're gonna do the same. Basically the same thing for Q. Alright, so Cue were given. It's a three by three and it's not somebody decimal, so it should be pretty easy to figure this one out. Okay, so Q is a matrix, which is a list of lists. The lists are the rose. There's only three of them and the first row goes, let's see 97.5 point 10 all right. They were making sarcastic matrices with columns, air probabilistic sometimes in some applications, the rows of the probabilistic. But we're not doing that. So this rose not probabilistic. This road is not add up to one. That's okay. Don't get upset. Um, next row 0.0 comma, which is why no zero comma 90 comma 05 This is how was given in the books? I'm just copying what's camera? Separate all the elements. Okay. And then comma delimited, I suppose. Then 0.3 camera 0.5 come a 0.85 All right, let's show that makes you entered that correctly show local camera. Que is cute. Alright, let's see what he looks like, which come down, down, down, down down there. OK, so now let's see if it's probabilistic in the column. Sorry 0.97 point 0.3 at up to 11 or 100% 0.5 point. I know. 0.5. Add up. 200. 100% point one, Open of five 0.85 at up to 100%. All right, so this is probabilistic. I mean, this is stochastic, but it's not regular stochastic because it's a zero in there. Uh, that's different. Mhm. Okay. All right. So let's see if this thing converges. Now, there's a hint in the question that you're gonna need large values of K to make this one converge. Right. So let's make a loop some large values in it. So I'm gonna do something. Someone gonna do this right here, But it's gonna be powers of Q. Let's see and show that it's not really that converging that fast. Okay, so let's see the powers of Q. You know, the one could of to could have three could for really they're not very. I'm not getting closer and closer anything yet. Okay, so I'm going to use a lot bigger exponents for my powers. So how about instead of from 11 to 10. Technically, how about 10 to 100 by 10? So those little spit out. 10 2030 40 50 60 70 80 9100, But not 1 10. I don't see that comfort is better. It is converging with slowly. Right? So let's see. Let's see. Let's see, it's getting close. 2.73 point 73 runs in the first council. Well, look at this little three columns. Look the same already. Hmm. Interesting. Is this gonna happen? The same thing happening here because except for Peter, the one everything else is regular stochastic or And that's what that says Supply theorem 18 in part C. But you know there Which says if you have a regular stochastic matrix in the limit is K goes to infinity, you're gonna end up with a Markov chain that has, say, limiting steady state vector. So let's show with the city state Vector is is it like last time? It's gonna be point. Let's see. What did I get? I have my notes here. Is it gonna be? What is my study state vector 7352 08 a. Two and 17 64 It's already it's already looking like that. Okay, so let's show with the steady state victories. All right, So we're gonna do sort of like we did here. All right, I'm gonna top you this. But now I'm going to say times and we found another question. Doesn't matter what zero is. So for whatever eggs or you want to start with who is the same one? Let's see that. In fact, is it converging again to what the columns are converging to in the powers of Q. Let's see. So the powers of Q. It's one powers of Q. The columns and converging 2.7353 point 08 a 2.1764 Is that what we're getting in groups? What I dio I mess up something. Wine 16. That I do write something weird. Glean 16. Let's go back to line 16. Come on, where's my line? 16. In my notes. Oh, into something weird. What happened? I don't see the ER. Wait. Hang on. Show quote. Cue card. Quote comma K comma quote. Star X zero equals quote comma cute of the K times like zero. That's right, Yeah, What's the matter with that. Okay, hang on. Let's break this bill this up from scratch here. So this loop in this loop of the same so there's no problem with that, right? So in question 21 B, that's the same. Is that right? Okay, now what I want to do is print out eg zero times that. So let's do it. Times X zero, We have X zero in there somewhere. Still, right? Yeah. What is going on? Oh, I can't use the same zero because it's the wrong dimension. Okay, Okay. Okay, because now it's a three by three. Okay, I gotta change my eggs. Zero. Okay, let's see. Sorry, I gotta change my zero. So x zero here is a four by one. But I need a three by one. Right. Okay, so let's redefine this. Let's do it right up to a defined mix. Que let's make it 334 And it's a 3.1. All right, I see that. That doesn't Sorry about that, guys. Little mix up. Alright. So 21 b, we did 21 already. So it looks like the columns were converging. 2.7353 point 882.1764 and is that with the study State factor is 0.7353 ish 0.8 a three issue for running off to the nearest UH, 10,000 and 17.17 65 ish. Yep, but that's what we're getting into the columns here, too. Is that crazy? Alright, so that's it. I think we covered all of question 21. So there's a code for 21 except for the huge long matrix, which you can get from the textbook 21 A and 21 B well, 21 beef. It's nice. Here's the output for 21 A. There's p there's like zero. There's powers of P convergence pretty fast. And there's how was the P Times like zero, which conversions pretty fast to the same values as we have in each column of the convergent Matrix powers. Same thing happened with Q. Bert Q. Converge very slowly started to use very large powers. The same thing for the study state vector. And there you go. So alright, have fun with your homework, guys. Hope that was

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