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Orthogonally diagonalize the matrix , giving an orthogcnal matrix and diagonal matrix D To save time , the eigenvalues are 11 andEnter the matrices and below;...

Question

Orthogonally diagonalize the matrix , giving an orthogcnal matrix and diagonal matrix D To save time , the eigenvalues are 11 andEnter the matrices and below;

Orthogonally diagonalize the matrix , giving an orthogcnal matrix and diagonal matrix D To save time , the eigenvalues are 11 and Enter the matrices and below;



Answers

Orthogonally diagonalize the matrices in Exercises $13-22,$ giving an orthogonal matrix $P$ and a diagonal matrix $D .$ To save you time, the eigenvalues in Exercises $17-22$ are: $(17)-4,4,7 ;(18)$ $-3,-6,9 ;(19)-2,7 ;(20)-3,15 ;(21) 1,5,9 ;(22) 3,5$
$$
\left[\begin{array}{cccc}{4} & {0} & {1} & {0} \\ {0} & {4} & {0} & {1} \\ {1} & {0} & {4} & {0} \\ {0} & {1} & {0} & {4}\end{array}\right]
$$

So they want us toe orthogonal e diagonal eyes. These, uh, matrix here. So since they don't give us the icon values and I get pictures for this, we have to go out and find those. So the first s. So I'm just gonna call this a So to a my Islam die, which is going to give six minus lambda minus two minus 29 minus lambda. And we're going to take the determinant of this now. And remember, this should be equal to zero. Um, so we'll end up with six finest lambda times, nine minus lambda and then negative two times. Second to its four. Uh, so that would still be just minus four. I was. Go ahead and expand this over here of that's 54 and then negative 15 lambda and then plus Lambda squared minus +460 Let's go ahead and combine those like terms. So we have. Why are not why Lambda Squared minus 15 lambda and then plus 50 is equal to zero. And is this factory herbal? Yeah, Could be goes We could dio Lambda minus five. Lambda minus 10 is equal to zero. So that tells us Lambda 05 Lambda is equal to 10. All right, so now we can go ahead and find what are Eigen vectors are. So let's take this Miss scoop this down. So looks Thio five first. So we have a minus five I So that's going to give It would be one negative too. Negative, Thio four. So you can see how this would get reduced down to just one negative 200 And so then this is going to imply that X one plus or plus minus two x two. Is he with zero? So x one is just too two x two. So when we look at our actual picture, actually, we expend screen will, um of x one, x two. This should be two x two x two or ex to let me oh clean out of So it would be X to To what Now we need to normalize this so we can go ahead and do that by finding the length of it. So it be two squared, plus one squared square rooted, so they'll be Route five. So we just need to multiply this by Route five, and then that's going to be our orthogonal I get better. That will be using for this. So, Pete to over Route five and then one over route five. So this will be the one on that goes with Lambda being equal to five. Now, we're going to repeat the same process, but with tin us. Let me Scoop was down. Yeah. Actually, it was 10 for the other one, right? Yeah. So now when Lambda is 10 so we have a minus 10 eyes, so that's gonna be negative. Four minus two and then negative to negative one. And so you can see how this is going toe reduce down to it. Looks like 2100 eso this is going to give x one is our x 12 x one plus X two is equal to zero. And so if we're going to go ahead and solve for X one So the X one is equal to so the negative one half x two So we can go ahead and love this end. So we have x one x two and then this is going to be, um, negative one half x two next to which is going to just be negatives. One half one times X two and we could go ahead and normalize this and doing that, we would get so negative one half squared plus one squared square rooted. So that is going to be 1/4 plus one, which is gonna be five. Force takes squared of that. So that's going to be, uh, Route 5/2. So we need to divide this by Route five over to and in doing that we will get negative 1/5. And then if we divide that by one, that would be too over Route five. As I think this is going to be be, too, when Lambda Two is 10. And now let's go ahead and step are diagonal eyes matrix. So D is going to be so 5. 10 along the diagonals and then our orthogonal matrix P is supposed to be so with five. Let me see what this was again. This was supposed to be to root file for one route five. So to route 5/1 5 and then or 10, this was negative. 15 over to Route five. The mentality that negative one, Route five over to route five. And then this isn't really necessarily. But I'm gonna pull out that one over Route five, since it's in everything just to make it look a little bit prettier. So 1/5. Um, then it's too negative. 112 So this is going to be our diagonal ization de along with that orthogonal matrix P.

Bye. Right. Minus minus My no land one my neck than that you might be then that he was then, Lambert. One that again. My one. And when Lambert truly minus then Wait. We cool One body. You When again That responding even. Really? My, my What? No, not planes again. Victor! Oh, you won my name. One. You all right? You too. One day, Jewel. And you be quiet, My baby! No! Minus one. Jeeva One label well, and one brake and know that I wanna be mine. Three You don't. People, people, people jiggle, jiggle. Thank you for watching.

Three miners well minus two for two. We know that. So they exist. Three alien values. Oh, God. Lambert One equals two minus land to it too. And lamb that he also seven? No. When Lambda One equal to minus. Then I convicted. We want equality minus one one by two. Like what? And when the again Really the seven. Then I can Victor minus one by two. One judo. And when I die again, really, seven. When I get there is also minus one way to one. No done. Normally taken Victor's, uh, you won equals minus two by three minus one by three. And who by and taken? Not much. I can make that use you. Minus one day you fight, Who made a fake? You go and you three equals It is also minus one way to fight the baby's fine deal and let be equals minus two by three one lately go minus on Little fight. We'll wait. Right deal and remains to fight you and they're viable. Metric dee Minus Who? Jiggle Jiggle deedle seven judo deedle deedle Thank you for watching


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