So they want us to or thought analyzes. And since they already give us the Eigen vectors arming the Eigen values, we could just go ahead and find the Eigen vectors and then normalize them and then plug everything in from there. So let's first do for when Lambda is equal to negative three. So we do a minus minus three. I. So that's really just like we're gonna add three along the diagonal. So this is going to be for negative 64 negative 65 negative to four negative 20 And if we go ahead and roll reduces, this will end up giving us one zero negative one half 01 negative one and then all zeros down here. And so what this implies is that x one minus one half x three is equal to zero. So that's just x one is even told one half x three And then over here X two minus x 30 or X two is a good X three so we could go ahead and write. So we have x one next two x three here, so x one we said Waas one half x three x two is just x three and the next three is X three. So we go ahead and pull that out. So it be x 31 half 11 Now, before we actually normalize this, I don't have toe square fraction. So I'm just gonna multiply this by two. So it won't really change anything since when we normalize it. That'll get accounted for. So just be 1 to 2. So now let's normalize this one. So to do that, it would be one squared plus two squared plus two squared spur voted. So that would be, um Well, what I do to here, Right that to square. So that would be four plus 48 plus 19 So that's actually just gonna be three. So we divide all of this by three. So that actually gives us when Lambda One is even to negative three. Our first Eigen vector is going to be one third, two thirds and two thirds. All right, now we'll do the same thing. But with Lambda equal to negative six. Um, let me pick this up and scoop this down. Just we kind of have it with us. Eso would be a minus minus six eyes. So again, it's gonna be like we just add six to all of this. So this is going to be equal to s 0785 and everything else stays the same. And now, if we were to go ahead and roll reduce this, we should end up with. Actually, this is in a five. Here. Um, there's maybe three plus three are negative. Three plus six is three, right, but only very reduced. Now, this one that should give us 101 are 011 half and then 000 So then this implies that x one plus x three is equal to zero. Um, which is gonna be next one is equal to negative x three and then for the next one. This is saying x two plus one half x three is going to be zero or X to negative one half x three so that we can use that to build our Eigen vector. So we have x one x two x three So x one was negative x three x two is negative one half x three and x three is just x three So this is x three negative one negative one, House one. And just like before, I'm gonna multiply all this by two. Uh, just cause I don't wanna have toe square a fraction. So this is going to be negative, too. Negative one to. So now we can normalize this one on doing that would get negative two squared plus negative one squared plus two squared, all square rooted. So that would be four plus 48 plus one. So, again, three once described it. So then that tells us when lambda, uh, two is equal to negative six. We get our second director being so we just divide all that by three. So negative two thirds negative. One third and to third. And then our last one. We're going to do this with nine. Let's go ahead and scoot this town and going to now do a minus nine I. So we just tracked nine along the diagnosis. That be negative. Eight negative seven and negative 12. And then everything else stays the same. And then when we rode use this, we should end up getting eso. It's 10 negative too. 012 and then 000 So then this implies that x one minus two x three is equal to zero or x one is equal to x three. And in that second equation is saying. X two is x two plus two x three is equal to zero or x two is negative two x three and then we go ahead and plug that into x one x two x three. So then again, X one is, uh, it's just a equal sign, uh, two x three x two is negative two x three and x three is just x three. Pull that out. So we have to negative 21 and then we'll go ahead and normalize this one. And doing that, we should get it's gonna be two squared plus negative two squared plus one square rooted. So again, that's just going to be, uh, Route nine or three. So then that would give us when Lambda is equal to nine. Uh, our third Eigen vector is going to be too negative to one, or we need to divide all these by three first, so it would be two thirds negative two thirds and one third. So let's go ahead and write out with our diagonal ization matrix system. This is gonna be D is equal to, um So we had negative six negative. Three and nine. Remember? Doesn't remember the order you write these a song is when you write your or thought little matrix. You kind of line everything up. But I'm just used to draw writing it from smallest to largest like this. All right. And so now let's write what are vectors are actually the one associate with nine here, So if we write, p is the one we just found. So this is that two thirds negative. Two thirds. One third now for negative three. Uh, this should be actually, let me just pick these up and scoop them all down. Then there will be easier for me to just plug it. So he's down. Okay, so now for negative three, we have of one third, two thirds to third and then or negative six, we have negative two thirds negative. One third to third. And if you want, you could go ahead and pull out that one third just to make it look a little bit prettier. So this isn't really needed, but it will make it look better s that we have negative too negative. 12 1 to 2. And then to negative. To what? So now these are going to be our diagonal ization, along with far orthogonal matrix P.