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Diagonalize the matrices in Exercises $7-20,$ if possible. The eigenvalues for Exercises $11-16$ are as follows: $(11) \lambda=1,2,3$ (12) $\lambda=2,8 ;(13) \lambda=5,1 ;(14) \lambda=5,4 ;(15) \lambda=3,1 ;(16)$ $\lambda=2,1 .$ For Exercise $18,$ one eigenvalue is $\lambda=5$ and one eigenvector is $(-2,1,2) .$$\left[\begin{array}{rrr}{0} & {-4} & {-6} \\ {-1} & {0} & {-3} \\ {1} & {2} & {5}\end{array}\right]$

$ \begin{pmatrix}-2&-2&-3\\ \:\:-1&1&0\\ \:\:1&0&1\end{pmatrix} \ \begin{pmatrix}1&0&0\\ \:\:0&2&0\\ \:\:0&0&2\end{pmatrix}\begin{pmatrix}-1&-2&-3\\ \:\:-1&-1&-3\\ \:\:1&2&4\end{pmatrix} $

01:25

Amy J.

Calculus 3

Chapter 5

Eigenvalues and Eigenvectors

Section 3

Diagonalization

Vectors

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so they want us to go ahead and diagonal eyes this matrix and they already give us our Eigen values. So let's first go ahead and check out what we get when we do. Um, a minus. So it's supposed to be Lambda I and so I'll do first where Land A is equal to two. So in this case, we get negative to negative two, three, and then everything else stays the same for this would be negative. Four negative six negative one negative three 12 And if you were to reduce this, um, this should give us a 123 And then the rest of these were just zeros, because, I mean, this is actually not too hard to do by inspection. You could see here 123 that cancels out with second row and then multiply that by negative, or just do two times rotary onto that. That also council's up. So even if you don't have a calculator, this one's not too bad to see. So let's go ahead and get our Eigen vectors out of this. Remember, this should technically be equal. Uh, 20 matrix. Um, So what? This is going to give us now is the expression x one plus two x two plus the three x three is a go to zero and then that means X one is gonna be negative two x two minus three x three And so if we were to take some random variable Yeah, so x And then we had x one x two x three Well, we come over here and plug that in for X ones that be negative two x two minus x three and the just x two x three And then remember, we can break this up into just the excuse in the X threes, which is going to be so the X twos. So we have one in column are in rogue one. So that being negative two, we have one in Rome to which is just one and then not in row three. So zero and then plus X three times eso We have negative three x three in that first room Second nothing. And the first roar Third very. We have one. So so far we have these two hi conductors on those who look pretty independent. So no problems there. Then we're going to repeat this, but with lamb day equal to what? So, actually, let me move this down. So over here. Yeah, we did. A minus slammed I with Lambda equals one that is going to give a negative one. Negative four negative six Negative. One zero Negative. Three, 1 to 5. And if you were to go ahead and reduce this one, uh, this should give us 102 011000 And so from this, we end up with two different one. So the first one being, um it's all right to sell it here, x one plus two, x three physical to zero, and then we have x two plus X three is equal to zero. And so if we were to solve for X one and x two, that tells us X one is negative. Two x three x two is equal to negative x three And then if we were to do that same thing, set up that little vector. So, um, X is equal to ex wine up next to next one next two and x three So x one is supposed to be negative two x three x two is supposed to be negative X three and then x three is just x three. So we can pull that X three out when we get negative to negative 11 And so this is going to be our third Eigen Vector eso Let's go ahead on pull both of these down here. I was just actually picked this up. Scoot it down. Okay, so our Eigen vectors with So this is Lambda is equal toe one and then these are Lambda is equal to two. So when we go about constructing P now, so we have p is going to be actually let's say what D is first before we do that. So d I'll say is one, 22 Because remember, it will just be our Eigen values along the diagonal and now key. We just need the what is called the Eigen value to match up with the Eigen factor. So for one, we're going tohave negative to negative 11 and then for two we can use either of these Doesn't matter, so I'll do negative to one zero and then negative 301 So this is P and now you go about finding p inverse. However, you want. Um, the way I normally do it, though, is I set up a augmented matrix equal to the identity. And then I just wrote, reduce it. So I went ahead and actually already did that. So let's go ahead and set that up really quickly. So we do. Negative two. Negative to negative three. Negative. 110101 And then I want this to be 10 not 10010001 And now we're going to reduce this, and, uh, let me see where I wrote this down. Eso This should give so 111 So we have the identity on the left now and then. Over here, we get negative one negative to negative. Three negative one negative. One negative. Three. 124 like that. And we can go ahead and use this over here as p in verse. And now to get our actual diagonal ization. Remember, we're saying it's supposed to be a is equal to P. D. P. In verse. We just go ahead and write this out. So P was negative. Two negative to negative. Three Negative. 110101 de was 1 to 2 along the diagonals everywhere else. Zero and then p inverse is negative. One negative to negative. Three negative one negative. One negative. Three, 12 Right, That but better. 124 And so this is going to be our diagonal ization of that matrix that we started with. And again, This is not a unique diagonal ization. You could get something different. Uh, if you put your Eigen values in a different order. Um, since we have one Eigen vector that goes with or to begin vectors that go with one Eigen value, you could switch those two, and you should still get an equivalent one. Um, you just need to make sure that whatever makes whatever way you set it up, you're just being pretty exact. But again, it doesn't really matter how you have it set up, because again, there is multiple ways to write this Diagon localization. And as long as you plug this into your calculator and it spits out the same answer, that's all that really matters.

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