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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}{7} & {4} & {16} \\ {2} & {5} & {8} \\ {-2} & {-2} & {-5}\end{array}\right]$

$A=\left(\begin{array}{rrr}-1 & -4 & -2 \\ 1 & 0 & -1 \\ 0 & 1 & 1\end{array}\right)\left(\begin{array}{lll}3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 1\end{array}\right)\left(\begin{array}{rrr}-1 & -4 & -2 \\ 1 & 0 & -1 \\ 0 & 1 & 1\end{array}\right)^{-1}$

Calculus 3

Chapter 5

Eigenvalues and Eigenvectors

Section 3

Diagonalization

Vectors

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Yeah. Okay, so we have a three baby by three matrix eight. As we want to die, guys, this matrix has to bag and values so three and one for a to be diagonal sizable one of thes again values much must lead to an against place of dimension to And so we're gonna figure out right now if that's the case, by finding the egg and Victor's off A on the basis of the egg in space, which is basically the same thing. So for lender One equals three. We saw this than your system of a minus three times identity times a vector X times of it with that times a minus threatening to tee times a vector X, which is equal to a vector full off zeros. That gives us a system that looks like this. So four for 16 and to to eight minus two minus two minus a times X one x two x three So the components of vector X and every team equals two 000 and basically all their line of this other lines of this matrix are the same. Uh, is there just multiple of each other So that first night is the second line, multiplied by two, and the third line is the second line multiplied by minus one. So we can only consider the second line if we won't want to and divided by two cents everything in the Chinese. A multiple of two. And get this equations or the solution of this system were must satisfied his equation. So x one plus x two plus four x three equals zero. So the deck means that the basis of the against pace will look. Something's this. So what? The X one of the bases will look like this minus x two, minus four x three, which means that X is equal to minus X two minus for expiry as the first component and x two ended x three as the 2nd and 3rd components. Therefore, X is equal to x two times, minus 110 at plus x three times minus +401 So those two factors here, minus 110 and minus force you want, are the basis for the egg again space. Associating with Brenda equals three. Therefore, it's the Huygens space of dimension to so a is daikon advisable to complete and find two diagonal eyes form of A. We can compute another Hagen victor associated with Lambda one by solving just in their system. So a minus identity Times X equals zero. We can skip directly and write it as a linear system so six x a one plus four x two plus 16 x three equals zero two x one plus for X two, class eight x three equals zero and minus two x one plus not plus minus minus two x two minus six x three equals zero. So since we have four x two and 44 x two in the 2nd and 3rd line, I'm going to try have minus four x two in the third line. So if you find another way with more a quicker way to solve this system, it's perfectly fine as long as you get to the right answer. So for me, I'm gonna multiply the third line by tools I'm gonna have minus four x one minus for extra minus 12 x three And which is of course, of course, to zero. And the second and third line will remain the same at the 2nd and 1st time were made the same. So two examples for ex troops plus eight x three equals zero and a six x one plus four x two plus 16 x three equals zero. Ah, Now where we doing? Um, I'm gonna have a line three. He called to 93 uh, plus line to and gonna have lying. One equals 91 minus nine to and that will give us the following equation. So four x one plus, uh, eight x three equals zero for, like, the first line that they're lying We minus two x one minus four x three equals zero second and my sister I'm so two x one plus four x two plus eight x three equals zero. The 1st and 3rd line of this system are the same. So if we multiply their third line by minus two, we get the first line, so we don't need to rewrite everything we were, and that will lead us to this equation. So X, uh, one equals minus two x three. So, basically, we just to the ext. Three to the right hand side of the equation and divided by four. And, um, the second nine will be will therefore become minus four x three plus four x two plus eight x three equals zero which becomes this so x two schools minus x three Therefore or effect er are Vector X will look something like this so x three times my next you'll minus want one, and that's the basis for the egg hiding space. Associate it with it. Lender equals one weekend. Therefore, write a equals so minus one once hero, first basis for first against face and minus four one. So did second places for the first time space and minus two minus 11 We write our agent values in so through pet 33 cents to egg and factors are so TT three and one with a bunch of zero outside of the diagonal and minus 110 minus four. CEO one minus two minus one poll minus one and a reality. We have to agonized A with P Times D, Times P minus one

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