let-a-be-as-in-exercise-9-use-the-inverse-power-method-with-mathbfx_0100-to-estimate-the-eigenvalue-

Name: let a be as in exercise 9 use the inverse power method with mathbfx_0100 to estimate the eigenvalue
Uploaded: 2019-08-15
Duration: 2 min 20 s
Channel: Numerade
Description: Let $A$ be as in Exercise 9 . Use the inverse power method with $\mathbf{x}_{0}=(1,0,0)$ to estimate the eigenvalue of $A$ near $\alpha=-1.4,$ with an accuracy to four decimal places.

Question

Let $A$ be as in Exercise 9 . Use the inverse power method with $\mathbf{x}_{0}=(1,0,0)$ to estimate the eigenvalue of $A$ near $\alpha=-1.4,$ with an accuracy to four decimal places.

James Chok · Accepted 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, 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 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

Nicholas Barvinok · Answer

we want to find an Ivan Vector of the Matrix 9023 with wagon value. Nine. Well, let&#x27;s say our factories X y. On one hand, this equals nine x and two x plus three. Why, on the other hand, because the night in Vector with Ivan value nine. This equals nine x and nine. Why so nine X equals nine. That&#x27;s no matter what X is. And we have two x plus three. Why equals nine. Why? Which we can simplify to two. X equals six. Why and X equals three. Why or better yet, why equals X over three. So when I can vectors anything is any vector which has any value of X and 1/3 that value of Why So, for example, 31 would be such a nag factor. No. For the next Matrix, we do the same thing we have 1257 We wanted Ivan Vector that has an icon value of four plus square root of 19. So we have X. Why on one hand this equals X plus five. Why and two X plus seven. Why? And on the other hand, this equals work whispered of 19 x times for four plus squared of 19. Why so we have two equations. X plus five. Why equals four plus swear it of 19 x and X two x plus seven Why equals four plus squared of 19? Why? So what can we do? What would be the easiest way to solve this? Well, let&#x27;s say Let&#x27;s solve this first equation for Let&#x27;s actually solve the second equation for X. That seems to be the simplest thing. So we have X equals four plus squared of 19 minus seven times. Why divided by two, which simplifies to squared of 19 minus three. Why over two? And plug that back into here. So we have, I swear it of 19 minus three. Why all of that over to Plus five. Why equals four plus squared of 19 and then multiplied by squared of 19 minus three all over, too. Why so no, we can isolate why we can factor out why and actually get why multiplied by squared of 19 minus three over two minus five minus four plus squared of 19 squared of 19 minus three all of that over too. And in fact, this equal zero, which means that Why? Well, let&#x27;s see. Does this equal zero? If this factor is not zero, then why must be zero? So let&#x27;s take a look at this factor. It&#x27;s squared of 19 minus three and I want to write everything over to someone to make the minus five minus 10 and then minus for swear it of 19 plus 19 minus. I&#x27;m sorry. That should be a minus 19 and then plus three squared of 19 and finally plus 12. So all of this over to now this numerator simplifies to wave squared of 19 minus four, squared of 19 plus three squared of 19. So those actually canceled and we have minus three minus 10 is minus 13 plus 12 is minus one minus 19 is minus 20. Divided by two is actually minus 10. So we do. In fact, get that Why equal zero? And if y equals zero, then we have that to X equals zero. And, uh, from this equation, and if two X equals here, then X equals zero, and the vector 00 is not considered an Eigen vector. So we don&#x27;t have an I in vector for this value. Next, we have 103200 0 to 1, with bye in Vector, which has Aiken Value. Three. Let&#x27;s do X y zem. It gets us X plus three z two x and two X plus z should equal three x three Why three z We have X plus three z equals three x two x equals three Why and two x plus z equals three z So from this first equation, we can simplify thio three z equals two x or X equals three halves Z um, from the second equation, we have X equals three halves. Why? So we in fact have three halves? Why equals three have Z which means that why equals e And so now in this 30 equation, we can substitute in X three halves Z for X so I get three z plus Z equals four z So we get four z Abel&#x27;s for zine. So what we get is that Z can be any value. Why has to equal to Z that X has to equal to three have busy and any Eigen vector of this form. We&#x27;ll work the next one 0312 We want Aiken value of minus one. So again. We do the same thing I get. Why three X plus two Why? Which should equal minus X minus y. So you get why equals minus x three X plus two y equals minus y so I can substitute in minus X for y. In the second equation, I get three X minus two X equals X. Yeah, so X, as we see can be This gets X equals X so acts could be any value. And why just has to be the negative Any vector of this form will satisfy will be an I in vector with wagon value minus one. Next we have 0111 times x y with Well, let&#x27;s first write it out as why X plus y. And this should equal one plus square to five over too. X one plus square to five over two. Why? So I get why equals one plus square to five over to X and X plus y equals one plus square to five over too. Why so again I can plug in wantto plus square to five over to X in tow. Why here? And get X plus one plus square to five over too X equals one plus Where 25 over two Squared X And so what do I get? I can factor out the X and get X one minus one plus square to five over too. I&#x27;m sorry. That&#x27;s a plus. Minus one plus square to five. Over too. Squared. Equal zero. So again, we need to check whether this factor is zero. If it&#x27;s not zero than X, must be zero. So let&#x27;s see, we have I&#x27;m gonna write everything. Well, let&#x27;s write everything over four The end. So we have four plus two plus two square to five, minus one minus two square to five plus five. And this simplifies to four till two. Square to five, minus two square to five. Cancels four plus two minus one is five and then I&#x27;m sorry, this should be a minus five. So, in fact, this factor is zero, which means that ex convey anything and then why just has to be one close square to five over too X. And this is our wagon Vector or anything of this form is an I in vector and our last one. 123017 0 to 1 We wanted I can value of one plus square to 14 x y z equals X plus two y plus three z Why plus seven z to why plus z equals And I&#x27;m just gonna keep his land for now. Lambda X Landa y land Izzy, where land is one plus square to 14. So we have X plus to why plus three z equals one plus square to 14 x Why plus seven z equals one plus square to 14. Why? To why? Plus c equals one plus square to 14. Izzy. So what can we do? We can solve for why, in terms of Z in this last equation and we get that, why equals I&#x27;m gonna use Lambda instead? Could shorter land of minus one Z over too now. So we know what wise in terms of Z and so we can now substitute that into the first equation. Get Lambda minus one Z over too. Plus lambda minus one Z plus three Z equals Land X. And if we expand that where if we simplify that factor out the sea, we get Z land of minus one over too, plus lambda minus one plus three and then all of that divided by Lambda equals X. So again we can pick axe to be anything. And then I&#x27;m sorry. The opposite. We can pick we pig zine. Then why is Lambda minus one over two Z and finally X is Z times lambda minus one over two, plus Lambda minus one plus three. She&#x27;s really land. Ah, plus two and all of that divided by Lambda And that&#x27;s our final Agon vector.

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

In Exercises 19 and $20,$ find $(a)$ the largest …

Problem 18 Medium Difficulty

Let $A$ be as in Exercise 9 . Use the inverse power method with $\mathbf{x}_{0}=(1,0,0)$ to estimate the eigenvalue of $A$ near $\alpha=-1.4,$ with an accuracy to four decimal places.

Answer

an estimate for the eigenvalue to four decimal places is $-1.4244$ . The actual eigenvalue is $(7-\sqrt{97}) / 2$ or $-1.424429 .$ to six decimal places.

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an estimate for the eigenvalue to four decimal places is $-1.4244$ . The actual eigenvalue is $(7-\sqrt{97}) / 2$ or $-1.424429 .$ to six decimal places.

Linear Algebra and Its Applications

Anna Marie V.

Samuel H.

Michael J.

Joseph L.

Vectors Intro

Vector Basics - Example 1

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

Anna Marie V.

Samuel H.

Michael J.

Joseph L.

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

Linear Algebra and Its Applications