in-exercises-1-4-the-matrix-a-is-followed-by-a-sequence-leftmathbfx_kright-produced-by-the-power-m-2

Question

In Exercises $1-4,$ the matrix $A$ is followed by a sequence $\left\{\mathbf{x}_{k}ight\}$ produced by the power method. Use these data to estimate the largest eigenvalue of $A,$ and give a corresponding eigenvector.
$$
\begin{array}{l}{A=\left[\begin{array}{rr}{1.8} & {-.8} \ {-3.2} & {4.2}\end{array}ight]} \ {\left[\begin{array}{l}{1} \ {0}\end{array}ight],\left[\begin{array}{c}{-.5625} \ {1}\end{array}ight],\left[\begin{array}{c}{-.3021} \ {1}\end{array}ight],\left[\begin{array}{c}{-.2601} \ {1}\end{array}ight],\left[\begin{array}{c}{-.2520} \ {1}\end{array}ight]}\end{array}
$$

James Chok · Accepted 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

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

Gennady Notowidigdo · Answer

Okay, so we are given This are four by four matrix, which I will hard light in green. So what, we&#x27;re gonna do it be going to use a computer package? So I used Mabel. If you want to use Matt, Matt or Mathematica, I feel free to do so or something else off that, like again, feel free to use whatever computer package you desire. So once I have this four by formation, so I want to find the Eiken values and the corresponding Aiken vectors, or sort of a basis for the Eiken space, so to speak. All right, so two. So, once I used the computer package, so let me just sort of figure out what this character is. It polynomial is. So if I obtained the characteristic polynomial, it&#x27;s going to be our to the four miles to are cute, minus 311 are square plus 312 aw, plus, 24,000 336. And I&#x27;m sure that this fact is nicely, um, so factors into, uh, plus 12 times our minds 13. A love that square good. And when we solved the zeros off the characteristic polynomial which is in this case minus 12 and 13 are then what we called the Aiken Valleys. So for this particular matrix, we have to Aiken values minus 12 and 13 case are so that our oppressed it yet. Okay, So for the Eiken battle, it&#x27;s for the Eiken value. Say Lambda equals two. I&#x27;m with these arms. That lab, just to make it easier, are equal to 13. Then if we compute, uh, our time, the identity matrix minus the matrix that we have, let&#x27;s call it a Okay, so let&#x27;s call it a Are I minus A. This ends up being for 4 to 4. 56 minus 19 28 minus 44. And then we just raised this just to make make it need us to be minus 19. Um, just dish it black. It&#x27;s minus 19. Then you got 14. 14 seven on 14. That should be 14 uh, minus 42. 33 Mona&#x27;s 21 and 58. Okay, now the Eiken space for our the Eigen value, our eagles 13 is pretty much just the null space off our times. I minus a or in other words, the Colonel. Okay, if you want you can. You can interject two things. Although I reserve the word colonel for lenient maps and no spaces for matrices. Okay, Now this matrix reduces too to to want. So I&#x27;ve done it in its simplest form. So it ends up reducing to something like this. Okay, so at this point, you can use back substitution and you&#x27;ll end up for the basis for the null space ends up being It will be minus 10 to 0 and one minus 40 and three. So obviously, if you did back substitution, you would set these two quantities to be one, and you work backwards, but I&#x27;ve just sort of normalized. It&#x27;s another normal, but I&#x27;ve made it fraction free just for our simplicity purposes. Okay, so we&#x27;ve got a basis for the eye Gance base for articles 13 and similarly, I will do in the next page, sir. I&#x27;ll keep it in black financing for the Eigen value. Let our used ob for I mean, it doesn&#x27;t really matter. I just like using are because we&#x27;re sticking to English letters rather than moving to Greek letters. But in any case, so for the Eiken value are equal to minus 12. Are I modest, eh? You know, when you write out what its people to sort of figure that out on your own. But it reduces to Mona&#x27;s 21 for 24 Sara Wan. Modest one of one. And in two bars, off zeros on a basis for the null space. So for the Eigen space for article 2 12 minus 12. Sorry is 2770 and zero. Modest one, sirrah. One. Okay, so I&#x27;ll leave it there. Uh, please verify older cat places that I&#x27;ve made Adlai Vilo on the computer or by hand, But in any case, thank you very much.

Samuel Goyette · Answer

Okay, so in this problem, we want to diet on ice. This matrix a three by three matrix. We know that five is one of its hag in values and that the vector minus 212 is an egg in vector off. So the first thing we&#x27;re gonna do to die agonizing Matrix is to find the Agon vectors for blander equals five. In order to do that, we&#x27;re going to solve the following linear system. So a minus five times identity matrix time vector X equals zero. So we solve this equation for X, we&#x27;re gonna get the following system. So minus 12 minus 16 and four. Ah, six eight minus two. 12. I raised a second line and we right there, just a bit more nicely. Eight. Okay, so they&#x27;re very light is 12. 16 and minus four times x one x two and x three, which is 00 Um, so all these lines are literally dependent. So we have We can see that line one is minus two times 92 and that 93 is two times 93 So the solution of the system will basically just look like mine, too. Which would which we will divide by 24 Simplicity shake and so ocean off The system will have to look something like this so three x one plus four x two minus X three equals hero. Therefore, x three is equal to three x one plus four x two So that means that are against pay the basis of our econ space will look something like this so x one x to three x one plus four x two as calor x one times 103 plus s collar x two times 01 That&#x27;s not 60 014 So these two vectors are the again the basis for the Agon space for Lambda equals five. So now we have three Eigen vectors since one was already getting in tow us now, as we must find his, we must find the egg and values associate ID with the Eggen Victor that was giving tow us so fine. The egg and value for the egg and vector Ah, minus tool one to eso No, we solve a new system. Except this time we saw for Lambda. So we have a minus land. Uh, times Identity times minus 212 equals zero zero&#x27;s and we need to solve four limbed that will give us minus seven minus and a minus 16. My enough. Minus four Just for, uh six 13 minus the Lambda and minus two, 12 16 and one Minus Glenda Times X one x two x three No, this is not x one x two x three We know these values already minus 212 equals toe a vector full of Xerox. I&#x27;m just going to write zero because I&#x27;m starting to elect space. Oh, we can rewrite our system. Ah, slightly differently. By multiplying I&#x27;m the 2nd and 3rd line. Uh, no. By multiplying by simply they computing this matrix vector products are. So now we just multiplied his vector by this matrix, which will give us this. So 14 plus two. Ah, then, uh, minus 16 plus a equals zero minus 12 to us on this. This is not in your system. This is a was the same thing. But this is a vector. You Well, much better. Minus 12 plus 13 minus and minus for and minus 24 plus 16 first to minus tool and equal. So that&#x27;s equal to a vector for the euro. So now we have a vector equal to another victor. And we get something like this when we simplify everything. So tool them. The plus six, minus lambda minus three and minus six minus tool and US equals 000 And all of these have one solution, which is Lambda equals minus three. So that&#x27;s our the remaining Hagen value. And therefore, we can write a like this. So a is equal to 10301 four. So those are the two alien values for them equals five plus the again Victor that was given. So minus +212 That&#x27;s our matrix. Pee wee writer died on matrix with the Hagan values. So 55 and not six, but minus three. So 55 minus three and a bunch of zero outside of diagonal and re simply inverse. Okay, zero for minus 212 and my sworn We know this matrix is convertible. Since all those pictures are literally independent

Samuel Goyette · Answer

Okay, so we want to die. I generalize this matrix a ah, and to give it the pagan values are given. So we have Randall want is one number two is to and then the three is through. So let&#x27;s start Dagon izing. So what are we going to do? We&#x27;re gonna find De wagon Victors, because since we have the egg and values when we want a equals p d p minus one, we already know the values off de since we have de again values. So let&#x27;s find the egg and vector so we can fill p how to find the AIG in Victor&#x27;s will for the first Hagen value of them, they equals one. We we try to solve this linear system so a minus identity times a vector X equals zero. So zero here&#x27;s a vector for zero and X will be the a convict Age and victor were looking for one of the again vectors were looking for. This will give us assistant within our system that looks like this minus 24 to minus two minus 330 minus 312 times a vector x one x two x three from the components X and which is equal to a vector full of zeros. Well, that will give us three Viniar equation. So if you want to solve using matrix or in your equations, it&#x27;s up to you as long as you solve it properly and that will give us the following system. So minus two x one plus four x two minus two x three equals zero minus three x Want Last three x two equals zero and minus three x one plus x two first two x three equals zero. If we look at the second equation here, that means that X one and X two are equal, so excellent he caused x two. If we inject that into the 1st and 2nd equation, we will get minus two x two plus four x two minus two x Treece of two x two minus two x three equals zero, and we&#x27;ll get the same thing. Something very similar here except here. The sun will be opposed to minus three x two plus X two equals minus two x two. Thus, a two x three equals zero and, well, that basically means that there&#x27;s no need for debt. You hear that just means, uh, let&#x27;s start on in your name that x one and x two and x three are all equal. So x will look something like x one x equals x one time 111 So every component is equal and we&#x27;re gonna use 111 as the basis forties dragon space. So I&#x27;m going to circle it and Green X one will be the first taken vector off. Well, they the first basis for Reagan space of a and it will be the first column of the major Expedia Natural. Now we continue for Lambda equals two. What party? Eggen vectors. So we solve it in the near system a minus two times and into tee times extra. We self that linear system. This will give us, uh, this new system minus 34 minus two minus 3 to 0 minus three one juan times x one x two x three For every component of Erector X, she is equal to a vector full of zero. It can be written as a linear system personally, a prefer to solve in your system for smaller dimensions. So for dimension two or three, I think it&#x27;s always morning easy to visualize system that maitresse ease Trust foreks to minus two x three equals hero minus three X one plus two ext. Who cause zero and minus three x one person X two last X three equals zero. Well, and what are we gonna do here? Ah, we&#x27;re gonna get sub strapped some light. So first of all, we that we have this second line here&#x27;s a minus three x one plus towards two equals zero that give us that two X two equals three x want. And here we have four x two So four x two will be six x one So we will have two x two x two minus two x three equals zero and ah, here we have minus three x one. So two X two Shall we have, uh, minus X two plus X three equals zero. So here we just replaced minus three. Excellent. By minus two up to. And here you have it. So last linear system looks like this. So x two equals x three two X two equals three x one and again the last line will give us the same thing. So x two equals x three Ah, which is the same as saying that two x two recourse to x three two x two equals three x one and two x two equals two x three So the 1st and 3rd line are the same. Which means that the basis for his against space will look something like this so x two times, uh, to three through. So So this is the basis for his second against pay, which would Which is gonna be the second column of Matrix P And we can just senator process for a Lambda equals three. So we have another in their system a minus three times identity times X equals zero. Now you should be a bit familiar, so I&#x27;m going to skip right away today. The near system minus four x one my plus four x two minus two x three call zero minus three x one plus X two equals zero and minus three x one plus X two equals zero. So the 2nd and 3rd line are the same, so we don&#x27;t need to rewrite them, which is great. There&#x27;s already a lot of writing here, So we have minus four x one crush and here we&#x27;re gonna use this Second entered lying to write X two equals my equals three x one So the first time will be minus four x one plus four times three x one uh minus two x three equals zero. Uh, we still gonna have it two equals three x one and up here we&#x27;re gonna have eight x one minus two x three equals hero and it would put everything in the most simple for possible means. X three equals four x Ron and X to course three x one Their floor x is well, look something like this 134 So this is the third basis for the third hike in space and the third call off p and just give us the following diagonal ization. So a cool The 1st 1st Eggen Victor, who found was 111 The second basis was 233 and then 13 four for the last one. Find it 38 and values were 12 threes. We put them on the diagonal matric, Steve. And here we I won&#x27;t computed and verse if you want. That&#x27;s fine, but just writing it like this is a the most important part So here you have it. A in the form of P times diagonal matrix d, times P minus one. Pretty simple. We found the values Diegan Victor&#x27;s based on the Hagen used implicating in the question.

Problem

In Exercises $1-4,$ the matrix $A$ is followed by…

Problem 2 Easy Difficulty

Answer

Eigenvector $x_{4}=\left[\begin{array}{c}{-2520} \\ {1}\end{array}\right]$ or $A x_{4}\left[\begin{array}{c}{-1.2536} \\ {5.0064}\end{array}\right]$
Eigenvalue $\lambda \approx 5.0064$

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

Linear Algebra and Its Applications

Lily A.

Anna Marie V.

Kayleah T.

Caleb E.

Vectors Intro

Vector Basics - Example 1

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Eigenvector $x_{4}=\left[\begin{array}{c}{-2520} \\ {1}\end{array}\right]$ or $A x_{4}\left[\begin{array}{c}{-1.2536} \\ {5.0064}\end{array}\right]$Eigenvalue $\lambda \approx 5.0064$

Linear Algebra and Its Applications

Lily A.

Anna Marie V.

Kayleah T.

Caleb E.

Vectors Intro

Vector Basics - Example 1

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

Lily A.

Anna Marie V.

Kayleah T.

Caleb E.

Eigenvector $x_{4}=\left[\begin{array}{c}{-2520} \\ {1}\end{array}\right]$ or $A x_{4}\left[\begin{array}{c}{-1.2536} \\ {5.0064}\end{array}\right]$
Eigenvalue $\lambda \approx 5.0064$

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

Linear Algebra and Its Applications