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A Markov matrix M has each m+Jj 2 and the sum of each column equal For given matrix Mwe can inspect its entries to determine if they are nonnegative and MATLAB can ...

Question

A Markov matrix M has each m+Jj 2 and the sum of each column equal For given matrix Mwe can inspect its entries to determine if they are nonnegative and MATLAB can be used to compute the sum of the columns Type help sum for description of the command sum(M):Ml =V2 =[3 221 46 39 24 V3 = V3 -19 -11 [Use MATLAB to determine which of Ml through M4 is Markov matrix (indicate yes OI no.)MlM2M34For those that are Markov matrices use gig to determine mar- {V1 where; / 1S an eigenvalue. Record your resul

A Markov matrix M has each m+Jj 2 and the sum of each column equal For given matrix Mwe can inspect its entries to determine if they are nonnegative and MATLAB can be used to compute the sum of the columns Type help sum for description of the command sum(M): Ml = V2 = [3 221 46 39 24 V3 = V3 -19 -11 [Use MATLAB to determine which of Ml through M4 is Markov matrix (indicate yes OI no.) Ml M2 M3 4 For those that are Markov matrices use gig to determine mar- {V1 where; / 1S an eigenvalue. Record your results here. Complete the following using the terminology of the introduction to these exercises Any process using Markov matfix Check your conjecture On a 4 x 4 Markov matrix you construct_ Provide your matrix and response. For each of the Markov matrices of Mlthrough M4 find the eigenvector corresponding TO T ffom part b. How are these eigenvectors related to your conjecture in part c? Be specific_



Answers

The transition matrix in Example 5 has the property that both its rows and its columns add up to $1 .$ In general, a matrix $A$ is said to be doubly stochastic if both $A$ and $A^{T}$ are stochastic. Let $A$ be an $n \times n$ doubly stochastic matrix whose eigenvalues satisfy $$ \lambda_{1}=1 \quad \text { and } \quad\left|\lambda_{j}\right|<1 \text { for } j=2,3, \ldots, n $$ Show that if $\mathbf{e}$ is the vector in $\mathbb{R}^{n}$ whose entries are all equal to $1,$ then the Markov chain will converge to the steady-state vector $\mathbf{x}=\frac{1}{n} \mathbf{e}$ for any starting vector $\mathbf{x}_{0} .$ Thus, for a doubly stochastic transition matrix, the steady-state vector will assign equal probabilities to all possible outcomes.

Okay, so UNICEF program We've already known that the metrics eight has the following decomposition You Sigma Wien transports. So, um, to show that, um to show that to the columns off the argument actors off agents post Wednesday, we can compute h is post A So it just because they you caused to you Sigma the transpose Onda We put analogy Suppose outside times us Sigmund the transports Because to the sick much is suppose you transpose you seem RV transports and that we know that's both you and, uh me. Uh, I thought, you know. So the definition off talking a matrix will be you just oppose equals to you, you numerous on the re transpose he was to the interest eso by this condition h is posed a equals two v tons. Six matches Post sigma because you transpose times U equals two And I didn't Leah magics. So we can you know this part of the universe Now we move this term the universe to the left that gives us agents pose a V equals toe three times seeking matches pose sigma and we already know that Ah sigma are diagonal matrix So sigma because to something like, Let's check in my one have to see my, um And there are some zero terms here. So sick a ma just post time sick. My close to, um something like sick of my wife. Square Sigma to a square. And the sigma is square. Okay, So with this result, we know that if we write out the close to a lot of column vectors, the one I'll be too the three up to the end. Then we can decompose this equation as ancients post eight times the I equals two sigma High Square times v i. And this is exactly the ag of the definition of the Eigen vector and again values. So Sigma Chi Square, our, um, again values off a chance post 10 say, and the re eyes. Ah, corresponding Eigen vectors. So this proof on the so if Sigma Square Aiken values off, patron supposed time, say, um, there's sticking. My eyes are, ah, singular values off a This proof, the first end of the last statement and full a second statement. We just do it exactly the same thing for a times a transpose. But in this case eight times, they transpose equals Teoh, you sigma The transpose times, the times, stigma chest post temps you And again we used effect that you transport of the transpose equals to the verse. So the protect off this two metrics will be an identity magics. So we can you know this then we have you Sigma Sigma matches post types. You Sorry Decisions. Suppose here. But we can I'll rewrite it as your humorous. Then we have a HS post tempts you. You close to a your time signal my time sticking my transposed. So the column vectors off you have you eyes? Are I get better again? Victor's off a agent. Suppose so. This proof, the second statement and then we finish on proof here.

Cakes for prom 20. We have Hey, she said. Or 12 in all. Or telling a matrix. Now we want to shoot that P Times say, has the same singular values as a now, by our assumption can be decomposed as you times seek my times. U transpose so pee off, eh can be reading There's he times you time Sigma times transpose Now Excuse me Now I know what he said since we know peace and on 1000 matrix and you is an orthogonal matrix. So we combine the product off p and you. Then we get a singular value decomposition of P and a p times safe. But at the same time between we observed these singular values that matrix Seema Sigma doesn't change. So that means our singular values will be the same. So we're done.

Okay, so this properly conceived there the men tricks a transpose times a, uh and they're supposed that a transposed time. Say, has, um I can barrios Lunda one greet the number two and the wouldn't recover by the rest. We only are discussed on the 1st 2 again values. And so Lambda Wise, The greatest again value and the love industrious. A second greatest rag ag value. Are we thanking vectors of you Wanted to be chew. Okay, so bye bye purpose he earned. If you're in section um his section 7.3, we know that if we can see that the quadratic form Q X equals toe extras post times a transpose a time sex If we can see that this quadratic form q x the truth, its maximum value if and only if x equals to be one. And we also know que eggs triffids maximum value, um, subject to eggs perpendicular to V one. He finally if x equals Toby too, which is the second Eigen vector corresponding to the second largest tagging value. Lambda too. So this proof, the statements in this case is cryogenic form queues off V two. He closed to me to transpose times, age, chance, post times a times V two equals to the norm off eight tons V two, which is Lambda two times of the two, The Simpsons. Well, seem that both B one B two are normalized victors. Um, so then use their norms are one. So this equals to dance out, too, which is the second largest I can value.

Okay. In this question, we want to say how scaring off mattresses affects their determinants and condition. So we perform this test on two different Majesty's first will perform on this matrix, and then we'll perform on the identity matrix. So here I've created a python script where the first array, the first Matrix that test is this measure is right here. Then the next one little test is the identity matrix, which is this one, my dear. So what this fourth does is it basically performs the steps first taken, determinant off a have been taken condition number, then take the determinant off 10 times a day and the condition of about 10 times a day. And the same thing for 0.1 time. Okay. And then you perform the same process for the identity matrix. So in this case, we can see right here that are determined scales by 10 to 4 sent to the full in this case, in this case, the scouts to 10 to the minus four. So determinants Well, 10 a scales by a factor off tend to the full, while determinants off 011 Why is that? A scales by 10 to the minus fool. Oh, more 0.1 time before And this is also truthful that Ivan intimations Well, we also see, however, is that condition number is the same for post for both major cities.


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