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[M] In Exercises 27 and $28,$ find a factorization of the given matrix $A$ in the form $A=P C P^{-1},$ where $C$ is a block-diagonal matrix with $2 \times 2$ blocks of the form shown in Example $6 .$ (For each conjugate pair of eigenvalues, use the real and imaginary parts of one eigenvector in $\mathbb{C}^{4}$ to create two columns of $P$.)$$\left[\begin{array}{rrrr}{-1.4} & {-2.0} & {-2.0} & {-2.0} \\ {-1.3} & {-.8} & {-.1} & {-.6} \\ {.3} & {-1.9} & {-1.6} & {-1.4} \\ {2.0} & {3.3} & {2.3} & {2.6}\end{array}\right]$$

$C = \left[ \begin{array} { c c c c } { 4 } & { - 1 } & { 0 } & { 0 } \\ { 1 } & { - .4 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { - .2 } & { - .5 } \\ { 0 } & { 0 } & { .5 } & { - .2 } \end{array} \right]$

03:13

Pagadala K.

Calculus 3

Chapter 5

Eigenvalues and Eigenvectors

Section 5

Complex Eigenvalues

Vectors

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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.

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