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Determine if $P=\left[\begin{array}{cc}{.2} & {1} \\ {.8} & {0}\end{array}\right]$ is a regular stochastic matrix.

$\mathrm { P }$ is a regular stochastic matrix.

Calculus 3

Chapter 4

Vector Spaces

Section 9

Applications to Markov Chains

Vectors

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

University of Nottingham

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Hello, Numerator. Welcome back. Okay, One question. Nine on page 2. 63 of Chapter 4 69 of Linear algebra Les Les Macdonald, fifth position. All right, and part a The question. Let me write this down. Let's see you do this. And let's say show number nine A. The question is, this is matrix. Mm. Let's see his metrics. M a scholar e can never say this word is stochastic stochastic matrix. But really, the question is, is it a regular sarcastic means critics, but first of all, as we stochastic, right, So let's check that out first. And I'm gonna say I'm gonna give you the answer right now. I'm going to say and as sort of like the timing here, Yes. Um because every column of the Matrix is a probability vector. In other words, each come the elements of each come at upto one on. Let's just show that so the matrix M is a two bit too. The first row is point to 10 It's because zero point consistent 0.2 circling to calm at 1.0 in the first row. This is the second room, and the second row is 0.8. So the first come at this 0.2 plus 2.8 is 100%. That's probabilistic, right? And come a 0.0. So the second Calumet's is 1.0 point. There was 1.0 is 100%. Right, So there you go. And let's show that. Yeah. Okay. There you go. So 0.2 point eight is 1.11 point 01 point. Oh, so, yes, it is sarcastic, but is it a regular stochastic? Ah ha. So wait, that's a little different By 3. 18 in the textbook regular Stochastic Majesties will give you a convergent Markov chain and end up with a steady state director when you apply it to any initial vector. Okay, so let's talk about that show. Okay? Number nine. A nice RV is matrix. Um, a regular stochastic matrix is matrix. I am a regular stochastic matrix. And the question and the answer in this case is No, because what makes it regular that all the entries air positive. Non zero non negative, Strictly positive. And this thing has a zero entry. So that's the problem. So, uh, no. Uh, since all entries must be positive. It was strictly greater than zero. Campy zero can't be negative. It's not like it's non negative. They can't include yourself. Must be positive. Strictly positive. Okay. And on top of that, we can also say right? Right. Um so the same should be true of all powers event. So the same should be true of all powers of them. And eventually you get a city state matrix as as the exponents school to infinity. So let's show what we got here. So for en in range, let's say I don't know 1, 200. So that gives me all the images from 1 to 99. Really? Let's show am to the end and see what it looks like. So the exponents, er and powers of 10th power of them, right? Yeah. What's familiar with that? So part is stochastic? Yes, because you have probability vectors for cones probably is a regular stochastic. No, because not all the interest in not non non not all of the interests of positive See that zero in the second row, second gang. And so so if we start off with all positive non zero entries, you should end up with all puzzled and zero entries, and Aziz luck would have it. All powers of them are still can't write our regular stochastic. But M itself is not. And there's no guarantee of that unless I'm itself is so. That's sort of a fluke. Let's make this a little bit easier to read the output. Um, let's put a quote comma. Come on. Quote about that so you can see the exponents. Kevin. So m to the one is not regular, but I'm to. The two is they're all stochastic, but empty one is not regular. The restaurants are regular stochastic matrices. So that part works out, I guess. Okay, so that's it for number nine. Guys are There's the work for number nine. The input for number nine. There's the upward for 98 and the output for nine B. That's sufficient right there. Okay. Bye Bye.

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