describe-how-you-might-try-to-build-a-solution-of-a-difference-equation-mathbfx_k1a-mathbfx_kk012-ld

Question

Describe how you might try to build a solution of a difference equation $\mathbf{x}_{k+1}=A \mathbf{x}_{k}(k=0,1,2, \ldots)$ if you were given the initial $\mathbf{x}_{0}$ and this vector did not happen to be an eigenvector of $A \cdot\left[	ext { Hint: How might you relate } \mathbf{x}_{0} 	ext { to eigenvectors of } A ?ight]$

Dharmendra Jain · Accepted Answer

Hello. I want to see in question. We have toe. We have to find how to try to write the equation. Difference equation. Right. So we could try to write X not as linear combination off Eigen vectors. We want we toe upto VP. If Lambda one if lamb down upto lambda pr the corresponding Aiken various. And if x not is a questo Si won, we won plus c two e two plus upto cp weepy right. Then we can define XK is a questo. C one lambda, one k, we went plus Saito Lambda. Okay, we to let&#x27;s have toe cp Lambda Big, weepy like this case for judo 12 Okay, right. So we can multiply. He both sides. You can write it as see even Lambda One. Okay, everyone, let&#x27;s see. Two lambda. Okay. Every two plus up to say p lend a big A V p right. We can like this. See, one lambda gave from the definition we can like toe. From the definition we can write with us on Stephen Lambda K plus one. We even let&#x27;s say, do Lambda Tau K plus one way to similarly this&#x27;d. If we see, we can write it. Us XK plus one. Okay, So, like like this, we can try to write the difference equation. Okay. Thank you.

Jason H · Answer

idea for this question were given Matrix eight on the edge of values and workers off it, although do not know the Matrix itself on then we know the area solution off a difference equation on Dhere For the first question, we&#x27;re going to compute acts one because eight times at zero, I wouldn&#x27;t know. Oh, here, let&#x27;s just set a to be a B, c and D since it is two times two matrix Astro s So we have this Diaco&#x27;s too well, no, I a plus B and I see hoops fast be mocha. This is a So this got here is our X one and we&#x27;re going to compute it. So that&#x27;s what we want to the information. We already have things. We know that for metrics a It so satisfied this equation it shouldn&#x27;t be. It should be. I&#x27;ll be here for our angel lepers eight. House, we because Lambda comes be So this is our that initial for it on values and entering batters on Let&#x27;s just right up. So here it should be a B C and the palms V for first be it is 11 and it equals two Lambda Tom&#x27;s V O, which is three and three. It is very obvious, Onda, for our next equation, we will have A, B, c and D. And here we&#x27;re gonna use the second Adam Factors, which I think should be minus 11 here on the equals to one of those three Toms O V two, which is Nega to 1/3 and 31. Overstreet. Okay, so what do we get for this? Let&#x27;s just kind of rewrite it. We know that here for this part, because to a plus B, uh, then C plus D. Okay, uh, you Khowst 33 And here, similarly for this part is not It&#x27;s in a B for this part is now that you&#x27;ve seen a plus d equals negative one or three and then one or worse street. OK, so actually we can compute a, B, c and D according to these four equipments, because because here we have foran noise and Warrick Regis wolf. But actually we do not really need to compute a, B, c and D. And here that&#x27;s because we only need to know night a plus B at Nice a plus B a trustee in older the computer that&#x27;s one. So that&#x27;s just, you know, this could interview one than this to be true. And let&#x27;s see how, for example, I want to use a plus B minus A plus B and combined is two to get not eight months. So how can we do that? It&#x27;s actually quite easy. Let&#x27;s just use five times one, uh, minus war comes to let&#x27;s see what we got. Uh, nine times a minus four grams minus times minus a gives us night. And here for B, we have five B minus four b ecause speed. So we get exactly not a plus B. So we have, uh, nine a plus b because, uh, street cams file, which is 15. And here I run it as 45/3 them minus four times 91/3, which is plus four, or street. So it gives us, uh, 49 or street, okay. And for our second 2nd 1 element off these weapons, we have nicely plus city on five times one minus four times the second ingredient. Also give us a nice C plus d equals to file come street, which is 15 which are right here as what if I over three the minus four times one was three. So it here I write here as four. Illustrate minus roast. So it gives us street. Uh, okay. 41 Overstreet 41. The worst. That here is 14 hours. So way have solved the first problem, which is to get x one. So acts one Here is 40 night or three and 41 or three. Okay, so that&#x27;s We&#x27;ll launch our second problem, which is here. We want to compute axe. Okay, let&#x27;s see how to do that. Well, we already know XK because to eight humps x k minus one. But it&#x27;s not enough here. We only know x zero. Right? So we need to write it or at or it&#x27;s like a recurring process. Soul. Let&#x27;s do it. Well, right. XK minus one as eight times a x Q minus two. Or, uh, or that&#x27;s just write a to the power off to hear. All right. Okay, so we have a to the power of three hubs, x k ministry and so on and so on and so on. Do it. Finally gets a to the power off K ax there. Since at zero, we already know, right? At zero. It is ny one we already know. Accidental. So here we only need to compute eight to the power of K in order to get access came. Okay, let&#x27;s see how to do that. As we know, they has to a distinct and rail vaginal values three on one or three. Right? So we know that a is die ago, Nall Izabal. So in other words, we can rewrite ax to the pharaoh minus one times a to the Times X because the and here acts is, uh well, that right here is This is the first Adam Wachter is the second panel doctor. And so off a on for D, it is ah, Matrix comprised off all aging values away on it is the diagonal matrix. So here it is. Lump one long up to and so on. A racing else zero. Okay, so here for this we can also rewrite it as a equals two AKs, the AKs, English. There we get a to the power of K because to, uh, axe Tom&#x27;s d to the power of Kate Times x inverse. It is Because, for example, if we want to compute a to the power of to then we simply have axe English times acts. Then these two term cancels out and we have a the tooth east combined together. So we have X times d to the power of two times actually were. So it is the same for all of the other order. No meals off a. So we have eight of our Okay, it goes to this quantity. And that&#x27;s now compute ate the problem, k. So it is Attari equal to ax here. We already know it is 11 not the one on one times d are okay, Well, it is every, uh quite either. Here it is ST to the power of K one or street for Okay. Since it is a diagonal matrix, we have everything else that goes to zero. So here it is. Two terms of mixtures. In this case, we have another one 111 the worse. Okay, so the only thing remains to be done is to compute this quality, which is the anglers off this matrix? Well, since it is a two times two matrons, it is not that complicated to compute the inverse off. Mr. Well, let&#x27;s just use. Uh, this way. The compute, for example. That set this quantity to be, for example, a one C one. Do you want the war? And And when it comes 11 Now, if you want one, it must decode to identity matrix, which is 1100 So here is this quantity is exactly the thing worse off. Uh, this matrix here, Right? And after, uh, computing the corresponding terms for anyone. Dont want anyone. Let me just give you the results off these quantity or this matrix equals to 1/2, uh, minus one. No. 1 to 1 or two and one or two here. So we have, uh, XK Okay. Yuko&#x27;s, uh, age the part age, the ROK comes back zero chase. Let me just roger Stone 1111 three to the power of K one or three to the four K than their own hero pumps. Uh, one or two narrative one or two lower to one or two. Then Tomczak 00 is 91 Okay, so here a racing is known, we can use this one here to compute a couple. The results we need and it turns out, uh, that&#x27;s XK equals to fly. Homs three you could part of K. Then comes 11 minus four palms one or three to the power of K. In times negative 11 Here we know that 11 is our We want no 11 is over. Be too. So we have expressed at the part of K with okay, We want on the way to here. And it can be obtained from by computing this chronic because we have a racing we need for for this column Here it is. We want Just call him here is if we two we have k here and everything else is no already. So we can&#x27;t get this quantity. And that&#x27;s all for computing at the power. Okay. Thank you.

Runpeng Li · Answer

Yeah. Okay, so four problem, too. Work even three I can values with the Cory&#x27;s money, I get vectors and we need to find a solution off the of the equation. Escape plus one. He was a Xscape. So the first thing is to find accidental and by definition, in our textbook, we have X zero, these people to see one. She&#x27;s a scaler times. Uh, V one C, two times you too. See? Three times feet. Okay, so four problem too work even because we only have three inspectors in this problem way only write down three terms. Okay, so it turns out to be x not X, not a Steven, which is connected to negative five and three. And we have First I get vector which it iss 10 negative. Three and three. I can values with the Cory&#x27;s money, I get vectors and we need to find a solution off the of the equation. Escape plus one. He was a Xscape. So the first thing is to find accidental and by definition, in our textbook, we have x zero, these people to see one. She&#x27;s a scaler times uh, V one C two times. You too. buzz See? Three times feet because I can. Again Victor, which is 21 negative five. Excuse me. Now that last again, Victor, she&#x27;s an active 33 negative. Three of seven. So that is equivalent to find out a solution off this, then your assistance. We on Lee have three of inspectors in this Problems way only write down three terms. Okay, so it turns out to be x not X, not a Steven, which is connected to negative five and three. And we have first I get vector which it iss 10 connective three and second 12 collective three 01 Collective three Elective 3 to 5, seven times the column vector with C one C two C three, which is equal to yeah, effective too. Elected five. I mean it back and again. Victor, which is 21 negative five. Excuse me. Now that last are you Victor? She&#x27;s an active 33 negative. Three of seven. So that is equivalent to find out a solution off this, then your sister is solving this system. I was keeping the process. So the solution is see Juan si tu c three iss to want to so that we fund. We have courage on our wishes. Either of her ridge. Excuse me for you. Check on Victor. It turns out to be why so? One, 12 collective three 01 collective three elective 3 to 5, seven times the column vector with C one C to C three, which is equal to yeah, effective too. Next, it&#x27;s five. If either or negative three US. One times 21 Act if I and ah, see three. See the reason too. Times did you again, Victor Nectar three Negative. Three at seven. Me solving this system, I was keep the process. So the solution is see Juan Si tu c three iss 212 So that we fund we have heard on our agenda wishes either off Ridge. Excuse me for each item, Victor, it turns out to be. Why? Said one so that our x k we just need to add our uh huh. Give value to dead. So we have two times. First I can value wishes. Three. So I have a three to the power of K times again. Victor 10 Negative. B plus 23 us one times to one negative five and ah, see three. See the reason too. Times to you again, Victor. Nectar three Negative three at seven. Uh, the second I get back that way, it is four. Fifth to the power of K to one active fight. And it&#x27;s me or so I have two times 3 50 to the power. Okay, times again, Victor. So that our x k we just need to add our, uh, give value to dead. So we have two times. First I can value wishes. Three. So I have a three to the power of K times. Yeah. Give Victor 10 negative. B plus negative. Three nectar three and seven. Okay, so here&#x27;s our That&#x27;s the solution off of the equation. X capers won was because I have a ex king. And so when we take K to be infinity, then the first term will be infinity. But the last two turns to terms will be zero. So our when the second I get back that way is four. Fifth to the power of K to one negative five and it&#x27;s me or so have two times 3 50 to the power. Okay, times again. Victor, cake tends to be infinity. Are x k Will just be two times 32 power Okay. Times 10 Negative three

Nicholas Barvinok · Answer

we want to solve the Rikers in and T plus one equals a uh huh Times and sip tea where the end some teas are to buy one column vectors and a is the matrix 0110 Now, if we can Diagonal eyes a meaning a equals p D. P. Inverse, then our solution is going to look like right where D is. A diagnose matrix is going to look like Auntie equals P D to the t p inverse and subzero for some initial column vector and zero. So all we have to do in order to get the solution is diagonal ized, a meaning find D, after which one will be easy to find e to the T and then find P and thus P inverse So D is going to have the wagon values of a on its diagonal. So let&#x27;s find the Agon values of A by taking the determine a minus landed times the identity and then finding the roots of that determine it. Since the turn is lame, two squared minus one and the roots of that are lambda equals plus remind this one So d we have found the because one minus one on the diagonal, which means that de to the t is just 100 minus one to the team. Because exponentially ating dago matrix just involves exponential hating each element on the diagonal. So now all we have to do is find p p is going to be made up off The columns of P are going to be the Eigen vectors of a So let&#x27;s find the Eiken vectors of aid. Let&#x27;s start by finding the Eiken vector corresponding to the Eiken value of one. So if we let that I can vector b x y, then on one hand, we get why X by matrix multiplication. And on the other hand, because an eye in vector with Aiken value one, we get X y so y equals x X equals Y, which means that I want this Aydin vector the ease the simplest version the simplest Aiken vector, where X equals y y equals X would be 11 not 00 because 00 is not a night in vector. So I&#x27;m gonna put that is the first column of P. So now, in order to find the Agon value corresponding to negative one we just put a negative there and we see that why? It was negative. X X equals negative. Why again, the simplest Aiken vector satisfying. That would be one minus one. That isn&#x27;t the zero in Vector. So now we have P and now all we have to do is find P inverse. So how do we find P inverse? Well, let&#x27;s take P. Multiply it by the unknown p inverse A B C d. And what does this get us? This gets us a plus C B plus de a minus C, the minus d. And then, on the other hand, because it&#x27;s the in versus gets us the identity matrix. So we have a plus. C equals one B plus d equals zero a minus. C zero b minus d equals one. So we have a equals C and a plus c equals one. So a equals C equals 1/2 then B plus d a zero so b equals minus d dreams that b plus B equals one. So be was 1/2 d equals minus 1/2. So finally, our last piece of information P inverse equals a is 1/2 be is 1/2 si is 1/2 dia&#x27;s minus 1/2. And so we now have salt. We&#x27;ve solved this recurrence recursive relation because we now know what P is. We know what p in verses, and we know what d to the tea is.

Matthew Allcock · Answer

in this video will go through the answer to question number 17 from chapter 9.5. So we&#x27;re given a matrix. A Onda were told that, uh, told consider the system Ex prime is equal to a thanks. So it&#x27;s two by two matrix on We asked Thio, do a series of, uh, turns a series of questions about this system. Simply a rusted verify the Eiken values. And I can vectors so we don&#x27;t actually to drive them. We just need to verify that they are indeed at the Eiken vectors and I can blow you off this matrix. So all we need to do is show that a minus one times I That&#x27;s that times the identity matrix Times by the fist. I&#x27;m vector zero. Thanks. Let&#x27;s do that. So a minus R one, which is to have that and the matrix is minus one, 33 33 minus three times by you want, Which is the column factor, Ruth 31 This is gonna be the talk. It&#x27;s gonna be minus three aside. Minus week three. Close Route three, which is zero on DDE 33 times 33 which is three minus three, which is ever so At least I wanted you. One other are and I can vector. And I could buy you night and I connected. Let&#x27;s try to show the same for are too on you too. Artoo is minus two. So one minus minus two is three. They&#x27;re got ring three 33 minus one plus two. This one on DDE. We&#x27;re told that the Eiken vector is one minus 33 Okay, so three minus 33 sides withdrew. Three, which is 33 months to me is ever on every three months. 33 zero. So yeah, two on dhe you to also, and I value and I respect it. It&#x27;s too muddy matrix. So these could be the only I&#x27;d buy youse, like, practice off the matrix. OK, Poppy, this is gonna be helpful. Now, before we&#x27;re get ahead with the sketching off the trajectories, it could be helpful to write down the general solution off this equation, which we do because we have, uh, a linear system linearly in many way independent system of high Quebec list and the corresponding back in values. So it&#x27;s gonna be a constant C one times by E to the first I am value, which is to times to you times by jag Vector, which is big three one who was a second constant times e to the second I compare value Remind us too. T times a second, Aiken Vector one minus for you. See? Okay. So I have a mind to sketch the trajectory solution with initially a vector reminds you want. Okay, this grid. So what is minus? You won&#x27;t look like so you want. Is this guy so minus that is gonna be minus root three rue three&#x27;s between one and two. What would it be? About 1.7 ish. Um, so minus 33 minus one is gonna be somewhere like here. That&#x27;s gonna be where, where? Initially starting. And since we&#x27;re starting with a particular item factor far from multiple events just to scare the multiple of annoying factor, that means that weaken only move along as time goes on, we can only move along that I connect it. So for the minute we can forget about this guy. And as time goes on, this is just gonna get bigger and bigger and bigger. So as time goes on, we&#x27;re going to go more and more in the direction off. AA minus. You won&#x27;t. And the doctor minus the one is this way. So we&#x27;re just gonna keep going. Keep going. Keep good. Bond is gonna keep going in that direction. Faster and faster rate as time goes on. No hotsy. What about when we stopped with initial value of you too? So you two is the vector one minus route three. So one minus 33 is here somewhere again, looking at this general solution. As time goes on, we can only stay in the direction off. You, too. On. As time goes on, we&#x27;re gonna slowly get closer and closer to the origin because e to the minus two tost gets bigger, Bigger, bigger, closer, closer to zero. So that means we&#x27;re gonna start here and we&#x27;re gonna get closer and closer to the origin. As time goes on, we&#x27;ll never reach it because he&#x27;s the power of Manitou. T will never actually hit zero. We&#x27;ll just get closer. Closer, Closer. Okay. What about pot? De So says we start, Huh? You two minus you want So you two remember Woz here on Dhe minus. You won&#x27;t is over here somewhere. So you two minus you won&#x27;t. It&#x27;s gonna take us to Well, if this is you too, and this is a minus. You want then up those two together and you gotta get over here somewhere. So that&#x27;s our That&#x27;s why you two minus you want. Okay, so it&#x27;s time for his own. We&#x27;re gonna have more and more of this guy that&#x27;s gonna get bigger, bigger, bigger. And we&#x27;re gonna have less unless of this guy. So we&#x27;re gonna start off with a bit of both of them, so we&#x27;re gonna start off going kind of more in the direction of both of them. It&#x27;s kind of this way away from dark day away, regularly away from the origin. But then the component off you, too. It&#x27;s gonna be much, much, much less than you want. So it&#x27;s going to start curving into the direction off. You want something like that direction

Nicholas Barvinok · Answer

we want to solve the Riker jh in and T plus one equals A and T where a is the matrix 0110 and we have an initial condition of zero cools 11 So if we can right A as a diagnosed matrix as as ah, if a similar to a diagonal matrix so we can write it as P D. P inverse where d is a diagonal matrix, then our solution becomes and t equals instead of a to the n a to the tee times and zero as p. D to the t p inverse times and zero. And so and then that this will be our solution. If we can find D and we confined Pete, So D is a diagonal matrix is going to have wagon values. It&#x27;s going to have its dago matrix b the Eigen values of a So let&#x27;s find the Eiken values bay. Yeah, good values of a will be the roots of the determinant of a minus landed times the identity This determinant is Lambda squared minus one. So the roots are lambda equals plus or minus one. So our dago matrix D is going to look like one minus one. Now let&#x27;s find the so to find P. The columns of P are going to be the Eigen vectors of a. So if we lost, find the eye in vectors of eight. So let&#x27;s start with the Eiken Vector correspondents the Agon value of one. So if I have 0110 and I apply it to the Eiken factor x y by matrix multiplication I get why ex? But because it&#x27;s an icon value and I even Victor, if I can tell you one, we get X y as well. So what? You have this? Why was X and X equals? Why? So the simplest Aiken vector that I can think of would be 11 because 00 is not an icon vector. So the first column of P is going to be 11 So what&#x27;s the second column to find the second column? We do the same thing, but for the even value negative one, which means that we get why equals negative X and X equals negative wide, which is the same equation stated twice, and the simplest Eigen vector. For that. That&#x27;s not 00 because that&#x27;s not nice to make your is minus one one. So now we found P. All we need to do is find p inverse. So how do we find that I have pee here One minus 111 multiplied by the inverse a B C D. This equals a minus. C b plus D A plus C and B plus D or wait and I made a steak. No, no, I did not write a plus C and B plus D. Right, So this should be being minus d right. And on the other hand, because this is the inverse, this equals the identity matrix. So what does this mean? This means that a minus C equals one. B minus d equals zero a plus c equals zero b plus d equals one so B minus d equals zero, so B equals D and B plus d equals one to both. BND are 1/2 soapy in verse equals eso B is 1/2 D is 1/2. So a plus c is zero So a equal c and a minus. I&#x27;m sorry. A equals negative C and a minus C equals one. So if I plug in negative, see here or if I just plug in a here I get a plus A equals one, which means that a equals 1/2 is two equals one and the sea peoples minus 1/2. So our final solution is that and some tea equals P one minus 111 D to the tea. Now D is 100 minus one. And if you exponent, if you raise if you exponentially eight diagnose matrix, you just do one to the U. It&#x27;s just the diagonal entries to that power. So one to the T is one than minus one to the team and finally multiplied by p inverse and multiplied by n zero, which is 11 and we

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

Linear Algebra and Its Applications

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Vectors Intro

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

Catherine R.

Kristen K.

Samuel H.

Michael J.

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

Linear Algebra and Its Applications