in-exercises-9-18-construct-the-general-solution-of-mathbfxprimea-mathbfx-involving-complex-eigenf-5

Question

In Exercises $9-18$ , construct the general solution of $\mathbf{x}^{\prime}=A \mathbf{x}$ involving complex eigenfunctions and then obtain the general real solution. Describe the shapes of typical trajectories. 
$$
A=\left[\begin{array}{ll}{-2} & {1} \ {-8} & {2}\end{array}ight]
$$

Bobby Barnes · Accepted Answer

so for us to find the solution to this differential pressure, remember, the first thing we&#x27;re going to do is we want to write this in the form of our solution being so some constancy one times our first vector or Eigen vector and then e to the Lambda won t where Lambda One is our first, um, Aiken value. And then we just repeat. But just with tubes everywhere. So this is the form gonna end up wanting to get in, and they tell us our solution, they&#x27;re gonna be complex. And then we could talk about how to turn them into the real ones after we get the complex solution. Right. So let&#x27;s go ahead and write that down. Eso we first need for what are Eigen? Values are going to be. So we do a my slammed I, which is going to be negative two minus lambda One negative eight to minus lambda. And remember, if we take the determinant of this, that should be equal to zero. So we get negative two minus lambda tu minus lambda. And then I would be plus eight zero. And if we were to expand this out now, we would get? Um, actually, let&#x27;s pull that negative out. So would be negative here. And then this is just gonna be the difference of squares. So this would be four minus lamb to square plus eight. And that&#x27;s gonna be zero. And now we can just go ahead and distribute the negative. So that would be Lambda Square to B minus eight plus eight to the B plus. Or is he go to zero? And then we could go ahead and move everything over. So Lambda Square, it is going to be made before. And then we take the square root on each side. And so then that gives us Lambda is going to be able to plus or minus two I and actually from that not x lambda. And actually, from this here, we can already go ahead and figure out what the trajectory off these were going to be. Because if we have something in the form of a plus B, I so if a is zero so that if a zero, then we know we will have ellipsis of Lipsky&#x27;s of round the origin. So this is going to be the trajectory of it. Since over here are a is zero. So that&#x27;s the trajectory. But now let&#x27;s go ahead and find one of our Eigen vectors of and go ahead and pick this up. And remember, we only really need to find one of the icon vectors since for the complex ones thes second Eigen vector is just going to be its complex conjugate. So if we were to go ahead and do lambda or no, not Lambda but a minus, uh, to I I this is going to give minus two minus two. I negative 81 to minus to I and we could go ahead and do like the road reduction to make sure that the bottom row is going to be zero. But since these are complex, we know it will be the case. So we could just be a little bit lazy. And I&#x27;m just going to use that last one there. Eso I&#x27;m going to rewrite this as negative eight x one and you can also use the first one. Doesn&#x27;t really matter. But I just think the numbers down there, uh, well, solve for X one a little bit easier, since that&#x27;s normally will be solved for okay, next to me said that he was zero. And now I&#x27;m gonna add X one over. So be eight x one. There&#x27;s a two to minus two I next to and then we divide the eight over. So x one is going to be equal to So the one minus one I over four, uh, to. So when we come over here and create our Eigen vector, this is going to be so one minus I over four, uh, X two and then we have X two here. Then we could pull the excuse out so we&#x27;d be left with that. And remember, now we could multiply this by anything we want. So I&#x27;m just gonna multiply this by four. And so our first Eigen vector is going to be one minus I for all right. And so, actually, let&#x27;s write this down here. So we have when Lambda is equal to two I we get B one to B one, minus I for and that means for our complex conjugate or second hike in vector is just gonna be one plus I or because we just make it the complex conjugation. So those are going to be our sectors now. And what we can do is go ahead and plug it into like, that equation. We had a top there, so are complex. Um, valued one is going to be X is equal to So it was C one and then be one. So one minus side for and then e to the one or not 12 i t. And then plus C two one plus I or and then eat negative to I teeth. And so this is going to be the complex solution. And then we can go ahead and get the rial valued solution. Um, by expanding this and then rewriting everything. But they actually give us a nice way in the book to do this. And they say that we can write this as so the X is equal to C one y one plus C two y two, where why one is equal to e riel of one of the Eigen vectors. So it doesn&#x27;t matter which one we use, plus cosine Bt and then minus the imaginary portion of one of our Eigen vectors and the sign of Bt. And then all of this is multiplied by E to the 80 and then for y two we like, switch the signs and co signs we keep the vector the same And then the subtraction is going to become addition. You could get this just by, like, using the E to the I data is co signed data, plus I sine data. But I think doing it this way is a little bit, uh, cool quicker, because then we don&#x27;t have to, like, think about, like, combinations of things. But you could do it that way as well. You&#x27;ll get a similar answer. It might just be a little different due to, like, combinations and stuff, but it&#x27;ll be the same thing in, like, the grand scheme of things. But let&#x27;s go ahead and plug everything in, although so I&#x27;m just gonna use Lambda One and we want So in this case, slammed of one is going to be. And remember, this is supposed to be like a plus B I. So that&#x27;s where our A&#x27;s and the&#x27;s air coming from. So that was just zero plus route to I, and now we can get our riel portion of the vector and imagining by breaking that up. So the real portion of this is just one or and then we add this to would be negative 10 Then we have either, So we could just come down here. So this is the rule sourcing. This is the real and this is the imaginary. So let&#x27;s go ahead and plug those in. So the reels are going to be 14 and depending on if you started with a different Eigen vector, um, you would get something slightly different here, but again, it should all give you the same thing as long as you&#x27;re consistent with everything else. And then the imaginary should be negative. 10 negative 10 for our B that&#x27;s going to be not route to just to. So let&#x27;s go ahead and replace all these bees here with twos. So to to to to And then a was supposed to be zero. So actually, this is just e 20 which that&#x27;s just one. So we can kind of just ignore that on. Then we just need to multiply the stop one. I see one c one, c two, C two, and we can go ahead and, um, collect all of the collect all the coastlines together and the signs so that gives us X is going to be so let&#x27;s see. Um, so this would be the cosign here along with this and doing that, we will end up with, um, are very I&#x27;ll just add these together. I&#x27;ll keep the coastlines and signs the same. So this is just gonna be C one. And so this is going to be host sign to t and then minus minus, actually, let&#x27;s just go ahead and make that a plus there and there will be plus sign two teeth and then this would be just four co sign to teak and then plus to see, And then it would be so sign of two t and then minus co sign of two teeth and then down below it before sign of two teams. So this would be our solution with the rial Once, um, I guess you could try toe, make this in tow. One bigger one. But I&#x27;m just gonna leave it like this here and so again, you might have slightly different numbers, um, in front of them. But as long as you can, like, pull all those factors out, you should get exactly what we have here.

Ibrahima Barry · Answer

Okay, so we&#x27;re given to me a goes my human. It&#x27;s 111 arrest solve assist in X equals. It&#x27;s probably because first we have to find value in values and collectors of a one of the idea though salted value that we have with you Geico to my team. Plus I already you see that the the real part of the even though he negative it implies that you will have spirals in your feet portrait and it take a note been inward to the second part of the question that we can automatically solve right now Analysis pointing the general solution. So the first point are you vector two bodies keep I is one minus I want have you can construct our submission the one time e V as in Victor in my eye Im ki you. The other solution is just because you get a solution This one just minus que mind If I v one you hear and then no, you know we can write Need the money What I times t using orders formula we get I need the money. Cuchi co sign. He was fine. This is the word for it. And then we can just spoke this into imaginary in real part. First, we&#x27;ll expand this out. We get U minus Q t co sider He I signed he I defector the one defector I infected. You know, when you do that you get Was he minus I Co sci fi. What? I find p and my ice We&#x27;re fine tea. The second rotation A coast I he off site then when you split into the imaginary in real part, you got so sorry people sign t over co sci fi and you have the legend report 70 my co sign T psyche. Really? We use the one that of this port years because you get solution that air con you There&#x27;s, you know, we have a real and imaginary part. I mean, our solution Just the one times Mike cuchi coastline api flying you, he and you, you can like see, I&#x27;m signed he my coast he and you know before, Like I said, the this is the spirals because we have complex like in value and they get a spiral towards or you because the idea like the real part, I think like value value, negative

Ibrahima Barry · Answer

obstructed general solution of this Justin ex prime. Because there so value though one of the ideas that you get should be to close. I and you should get the curse upon the like in vector one plus i to no a construct the general solution, you know, solution gonna be He won. Well, you know, maybe GTO two plus I dissector v people here equal to plus I he time one plus I life too. Then we can use toilers formula to turn this into the beauty. Khost is key, I find here Director one plus I money get me. You know he I co signed t i t. And we have mice on Sunday My son in the second world we have ice to co sign he might have to find Let me just take the real and imaginary part separate them beef, tea and the real Korean My sanity My co sign of tea in the imaginary parties sign he no sign of t what is key side and that we should Jaros motion exit TV Could you he won t Kiki place this factor here first solution you either to be going second solution each to he. And now we know the real part of the again. We two. Plus I sport here really is all that is, Miss Trajectories. We&#x27;re gonna be spiralling of the Bible. Spiral inward, spiral outward.

James Chok · Answer

okay in this question, you want to solve this differential equation Where a is this metrics? So in this question, we can use and make use Capital of the Sun&#x27;s gonna skip a love the cap elections. But I&#x27;ll explain what we do along the way. The first thing that we have to do is off course computer ideal balance and Ivan function. I&#x27;m 1000 island vectors. So these are your values and your ID in our values here. And I&#x27;m in Victor&#x27;s every year. Now get general solution will then look like this up here. So then substituting B one b two B three, as as the&#x27;s eyes were here and land or two. And then the three up in Holland one, then two. Then the three. You get this solution right here? Yes. Get this solution they got Okay, So now this is your complex general solution. But we want it in really values. So to look at it from real values, all you really need to do is to look at just one of these. So this is already really so the question is why do anymore at oneself to Well, if you look at both of these two, you kind of just get in multiple of each other. But these multiples off each other cancels out later as you&#x27;ll see. So just looking at this one, expanding the reach of life I e five plus I t get five tea cozy on T I 70. So then splitting up into its real and imaginary parts. The first real part that you get from this first entry is six times curse NT. Next next play is to talk about I 70 which is minus two scientists. So the next part for this one is just nine coast auntie and three I type of ice. I t So that&#x27;s this entry and this is simply really so It&#x27;s just 10 Christ, aunty. So to get the imaginary part, the first thing that you do is you get this six any. Multiply it by the AI scientist. So got to this one and then you get this to my entire by the co sign. So that&#x27;s why we&#x27;re on the same thing for this night time tete with this one and this one and three course anti this one and this one. So that&#x27;s how you get this. So If you do this for minus, I t. Or you&#x27;ll simply get easier. Simply get a negative here and you probably get a plus plus plus plus year on dhe. Really? DC are going to differ by constant, but that doesn&#x27;t really matter. Because if you put it back into your general solution, which is 68 Martucci, that&#x27;s everything here. When you put it back, you already have Constance here anyways. And so the constants will just really readjust when you put the initial condition. So this is your final solution.

James Chok · Answer

Okay. This question. You want to solve this already with this matrix? So this question they let us use capital is to find a convict is an iron values. So here we get land up. One is equal to two in the corresponding vector, is it too? 21 lambda too easy to three. 32 is equal to three minus one and one on and off. Three is into four with the three using itude seven minus two and three. So, you know, as the general solution then, is this simply x function of tea is just see someone 21 you to the true t So see some, too. Three marks, 11 each of the three T See some three, seven months. 23 You two for two. Now, to describe the trajectories, you just have to recognize that one or three Aryan values are positive. And so the origin is an unstable No. So everything just go away, Move away from the origin

Ibrahima Barry · Answer

you&#x27;re asked if I d generous ex prime clues e X and whereas to draw feats poacher describe the feast of dissolution. So now you know you should get when you take it to really be minus land I is to be I notice there&#x27;s no real part No determine how the feast scorch it looks like And I get vector You should get to be my street mystery I to solution It&#x27;s gonna be even be I may be useful to zero when you use always forego times V which is my screen must be I to So we get So what was Formula toes is that he has e he I be not just equal to it would be times co sign a Finally. So you know the group. The real court. Use your we get each with zero times suicide 15 plus I sine et times a director. Maestri you No, it&#x27;s just a place to get my three co sign Creepy. My I re sign creepy or stream. I co signed Clukey and then point we I squared said it&#x27;s minus any three and second world. We have to co side plus two. I sign spoken to zero and let&#x27;s just split the real imaginary part. First solution to get my speak Ozai taking my 5 30 and to co star with a freaky three pp My plus three co sign +33 and second row We had 22 sides off. Creepy. There are general solutions. One comes this solution What&#x27;s your time each year and are you gonna be? Is he I Euro rial part of you that implied our face portrait It&#x27;s gonna be look sees around and origin

Bobby Barnes · Answer

linear system X Prize was given the Matrix a first week eight determinative a money lender setting zero que two by two matrix. We get my lander to my anger. No, my because the euro we get Get lying to squint here. Okay, Like for here. And And landed square plus four. It was a hero. Your land is our Let&#x27;s remind you I hold your this will make a difference later on in the problem. So look, use Lendl one Q r, We have a my to I identity zero Probably systems that we have. Why do I 1 to 2 i you zero looks reduced matrix first first look, you top row We can make that first century boots one. So it&#x27;s multiply first by my one I before So we should get first one here in the second Entry should get minus one plus before zero to life too I euro And then now we have one for first to the country you can multiply that by eat to cancel out the entry place I love it Times and attitudes Second row one My one eye Before you go I just can&#x27;t becomes zero now we had my eight plus I before plus two minus two I No one is the give you before we have my feet Plus I eat my p I your before your that entry. Here&#x27;s zero. I&#x27;m gonna skip the work, but get your Eigen vector. Well, first obesity, Aiken speed, You go to my one so I over for one. Then you choose the variable executes people for just so you can have a nice looking like inspector, you get, I vector Be equal to mice one i for Oh, sorry. You move this you saw for excellent. You actually get one minus I before that should be one minus I, you know, find the solution. We have eaten the to I team 40 p. I mean vector one I for using for Know what? We get you to zero times hillside pt I side you can Times Inspector one life I were You get Khost. I find you t my co sign my I squared my mind. That&#x27;s fine. We have a second really have so called suicide duty. What&#x27;s for you? You get the general solution, you see one e minus. I won&#x27;t even Txeroki just one times 1 10 times riel, part Mrs Side of Q T que for Petey and vaginal part a few times. This is a sign, Yuki, this foot here minus sign T and four people. And then, you know, I value people too. I hear. So the real part equals zero that applies retreats. It&#x27;s gonna be spirals around origin. She gets something like this.

Problem

In Exercises $9-18$ , construct the general solut…

Problem 14 Hard Difficulty

In Exercises $9-18$ , construct the general solution of $\mathbf{x}^{\prime}=A \mathbf{x}$ involving complex eigenfunctions and then obtain the general real solution. Describe the shapes of typical trajectories.
$$
A=\left[\begin{array}{ll}{-2} & {1} \\ {-8} & {2}\end{array}\right]
$$

Answer

$\mathbf{x}(t)=c_{1}\left[\begin{array}{c}{\cos 2 t+\sin 2 t} \\ {4 \cos 2 t}\end{array}\right]+c_{2}\left[\begin{array}{c}{\sin 2 t-\cos 2 t} \\ {4 \sin 2 t}\end{array}\right]$

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Linear Algebra and Its Applications

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$\mathbf{x}(t)=c_{1}\left[\begin{array}{c}{\cos 2 t+\sin 2 t} \\ {4 \cos 2 t}\end{array}\right]+c_{2}\left[\begin{array}{c}{\sin 2 t-\cos 2 t} \\ {4 \sin 2 t}\end{array}\right]$

Linear Algebra and Its Applications

Lily A.

Anna Marie V.

Kayleah T.

Samuel H.

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.

Samuel H.

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

Linear Algebra and Its Applications