another-estimate-can-be-made-for-an-eigenvalue-when-an-approximate-eigenvector-is-available-observe-

Question

Another estimate can be made for an eigenvalue when an approximate eigenvector is available. Observe that if $A \mathbf{x}=\lambda \mathbf{x}$ , then $\mathbf{x}^{T} A \mathbf{x}=\mathbf{x}^{T}(\lambda \mathbf{x})=\lambda\left(\mathbf{x}^{T} \mathbf{x}ight),$ and the Rayleigh quotient
$$R(\mathbf{x})=\frac{\mathbf{X}^{T} A \mathbf{x}}{\mathbf{x}^{T} \mathbf{x}}$$
equals $\lambda$ . If $\mathbf{x}$ is close to an eigenvector for $\lambda,$ then this quotient is close to $\lambda .$ When $A$ is a symmetric matrix $\left(A^{T}=Aight)$ , the Rayleigh quotient $R\left(\mathbf{x}_{k}ight)=\left(\mathbf{x}_{k}^{T} A \mathbf{x}_{k}ight) /\left(\mathbf{x}_{k}^{T} \mathbf{x}_{k}ight)$ will have roughly twice as many digits of accuracy as the scaling factor $\mu_{k}$ in the power method. Verify this increased accuracy in Exercises 11 and 12 by computing $\mu_{k}$ and $R\left(\mathbf{x}_{k}ight)$ for $k=1, \ldots, 4$
$$
A=\left[\begin{array}{ll}{5} & {2} \ {2} & {2}\end{array}ight], \mathbf{x}_{0}=\left[\begin{array}{l}{1} \ {0}\end{array}ight]
$$

James Chok · Accepted Answer

okay, in this question, we basically want to compare two different ways off calculating Eigen balance. So the traditional way in the power method is that we set you okay? Should be that the entry. You okay with the biggest A k a r a X k. Yeah. I x k was biggest the guest. You guess. Absolutely. Valley, In this case, though, we want to compare this between using this and realize questions which is defined as X transpose a X over x transpose X. So this new case is calculated over here, and this realized politician, which I called here is basically this thing right here. So ex transfers well supplied by a X and X transpose X right there. So the important thing to recognize is that in both cases, we are using the same estimated by Becca. It&#x27;s just we&#x27;re simply comparing the idea of values, so yeah, so I paid a plant that&#x27;s good right here, basically comparing the u. K and R K. And this is the results. And you can see that the realize proficient he has high person

James Chok · Answer

Okay, So in this question, we want to estimate the idea value off this mattress right here using that invest help, Mrs. So once again, I&#x27;ve created a program in quietly and can run this yourself. So just go through This appears This is the A matrix. This is your Alfa that you said to me Mars 1.4 and we want to try this 10 times. Well, are trying sometimes. So this is what it does. So what district does when we performed the inverse power step? We want to solve a minus. Phi out for identity. Why, Kay? Is it ex k? So we want to solve. For why So instead, let&#x27;s move a minus. I the inverse of the over to decide. So then we have Why, Kay, deep into a Mars, a upper agency in verse. Thanks. Okay, So this is what this that does a minus. Alpha times my identity Take the inverse of that and we multiply it by X. So then we&#x27;ll we will get new, Which is the entry in Why the entry in why? Which which has the highest absolute value possible So that Mr Does and then new is it to have a plus one of a mule. Then X K is equal to that X is equal to why I ever knew So then here. So here I print out the I didn&#x27;t baby value. Sorry, as in this movie the item vector. So this is for zero. The news here isn&#x27;t used 1.375 then x once after the first iteration This is your again. This is your approximated. I bet that approximate it approximated I didn&#x27;t value. And so it&#x27;s all this house on this hole. So finally approximated Ivan. Bet that is 10.37 and negative 0.78 How approximated Aiken value is negative. 1.4244

James Chok · Answer

Okay, So in this question, we want to basically approximate that again vectors. And I&#x27;m also Ivan values questions era and life spied in about. So in this example right here, I&#x27;ve only given the I&#x27;ve been very close to zero closest to zero. If you want the Ivan value life value, you can just simply said athletic 20. So just change this line of code right here, and you will gets the lives talking about, which should be 19.18 So what I&#x27;ve done here is basically I&#x27;ve cut it up the invest power method to solve this to to find the ivy value. So in this case, just to go through it in case you don&#x27;t understand what I did or how it works. So we start off without initial X and we we set an Alfa and perform this step. So first of that we do is we have to solve a minus Alfa identity. Why is equal to X? So we want to sell for Why so moving a minus? I after invest, I got to decide you have Why Easy to a minus off a after I Inglis X. So what? The sepia is simply this stuff. You&#x27;re eight minus alpha times. Why? Taken in vessel exit? Inglis and we have multiplied by X. Then we want to find about the entreating. Why? Which absolute value. The absolute value is as light as possible. So then we compute new alcohol, plus one of them you&#x27;ll and then we compute x, which is why divided five new. So that&#x27;s what this step, that&#x27;s what this whole thing does here. So the printed out results is I&#x27;m here as a concerned then the wagon value close to zero is 0.122

James Chok · Answer

Okay, So in this question, we want to basically estimate the lives item value and Becca off this aim Agents. Right here. So what I have right here is a heightened Smoot where we basically do the calculations. So a is 1.818 minus 0.8 minus trip 12 and 412 and then these out out X dollars. Right here. So what we do in this step is we perform a and multiply it by X Y. So for x zero, we have 01 and x one. We have minus 0.5 and one l can t to do that. And then for the same, I will also print out the maximum value in the entry so the results can be seen here, where a eight times zero is split up. This vector and the large sentry being forward to sending for this as any for base. Yes, as a night. You had to be careful when interpreting this because I took the transfers here. So this is a one robot to calm. So one by two matrix. But you should actually be a two by one. But I just put the transfer there because it&#x27;s easy to be so. What you see is that mule zero is for one to new 16 Well, two is 5.1 and so on. So on, so on. And so I. Invicta is Expo, which is negative. 0.2. Factor zero and one, or a X four is equal to minus 1.2536 and 50064 with Reagan Value 5.64

James Chok · Answer

Okay, So in this question, you want to approximate our Ivan values and Ivan vectors off this a metrix right here. And by the way, So this is a This is really important. So if you want to use it, just use plastic. So this is a matrix right here on the day of that is given to us is zero is equal to 11 x zero x one is 10.736 and so on. So on, so on. And so what this step does here is we multiply a and zero x want eggs, Irving this, then you do X one x to x ray and sometimes awe. Then what this stuff does it is that it essentially returns the largest valley in here in absolute terms. Yeah, and so would print out the results. So first thing that would print out is the multiplication together, and we&#x27;ll print out in your second. So just be careful when I wanna print out that I printed out the transpose not be better. So you&#x27;ll get a one by two vector instead of two by two. Victor. The reason why because just by the sanitation is a lot easier to read, so what we&#x27;ll see is x zero is negative. 1.9 and a times by x zero is negative 1.9 and negative 1.4 large entreating next, 1.9 and so on. So on, so on and finally x for a times x four is negative. Zero point negative 0.0.4012 on negative 0.3008 with life&#x27;s entry being negative. 04012

Problem

Another estimate can be made for an eigenvalue wh…

Problem 11 Easy Difficulty

Answer

The actual eigenvalue is 6. The bottom two columns of the table show that $R\left(\mathbf{x}_{\mathbf{k}}\right)$ estimates the eigenualue more accurately than $\mu_{k}$ .

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David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications

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Heather Z.

Kristen K.

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The actual eigenvalue is 6. The bottom two columns of the table show that $R\left(\mathbf{x}_{\mathbf{k}}\right)$ estimates the eigenualue more accurately than $\mu_{k}$ .

Linear Algebra and Its Applications

Lily A.

Catherine R.

Heather Z.

Kristen K.

Vectors Intro

Vector Basics - Example 1

David C. Lay, Steven R. Lay, Judi J. McDonaldLinear Algebra and Its Applications

Lily A.

Catherine R.

Heather Z.

Kristen K.

David C. Lay, Steven R. Lay, Judi J. McDonald

Linear Algebra and Its Applications