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In Exercises $37-40$ , use a matrix program to find the eigenvalues of the matrix. Then use the method of Example 4 with a row reduction routine to produce a basis for each eigenspace.$$\left[\begin{array}{rrrr}{9} & {-4} & {-2} & {-4} \\ {-56} & {32} & {-28} & {44} \\ {-14} & {-14} & {6} & {-14} \\ {42} & {-33} & {21} & {-45}\end{array}\right]$$

$\left\{\left[\begin{array}{l}{2} \\ {7} \\ {7} \\ {0}\end{array}\right],\left[\begin{array}{c}{0} \\ {-1} \\ {0} \\ {1}\end{array}\right]\right\}$

Calculus 3

Chapter 5

Eigenvalues and Eigenvectors

Section 1

Eigenvectors and Eigenvalues

Vectors

Harvey Mudd College

University of Nottingham

Idaho State University

Boston College

Lectures

02:56

In mathematics, a vector (…

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21:29

[M] Orthogonally diagonali…

25:16

00:34

[M] Diagonalize the matric…

21:28

23:19

02:05

Let each matrix in Exercis…

24:32

01:45

Diagonalize the matrices i…

06:02

08:50

Okay, so we are given This are four by four matrix, which I will hard light in green. So what, we're gonna do it be going to use a computer package? So I used Mabel. If you want to use Matt, Matt or Mathematica, I feel free to do so or something else off that, like again, feel free to use whatever computer package you desire. So once I have this four by formation, so I want to find the Eiken values and the corresponding Aiken vectors, or sort of a basis for the Eiken space, so to speak. All right, so two. So, once I used the computer package, so let me just sort of figure out what this character is. It polynomial is. So if I obtained the characteristic polynomial, it's going to be our to the four miles to are cute, minus 311 are square plus 312 aw, plus, 24,000 336. And I'm sure that this fact is nicely, um, so factors into, uh, plus 12 times our minds 13. A love that square good. And when we solved the zeros off the characteristic polynomial which is in this case minus 12 and 13 are then what we called the Aiken Valleys. So for this particular matrix, we have to Aiken values minus 12 and 13 case are so that our oppressed it yet. Okay, So for the Eiken battle, it's for the Eiken value. Say Lambda equals two. I'm with these arms. That lab, just to make it easier, are equal to 13. Then if we compute, uh, our time, the identity matrix minus the matrix that we have, let's call it a Okay, so let's call it a Are I minus A. This ends up being for 4 to 4. 56 minus 19 28 minus 44. And then we just raised this just to make make it need us to be minus 19. Um, just dish it black. It's minus 19. Then you got 14. 14 seven on 14. That should be 14 uh, minus 42. 33 Mona's 21 and 58. Okay, now the Eiken space for our the Eigen value, our eagles 13 is pretty much just the null space off our times. I minus a or in other words, the Colonel. Okay, if you want you can. You can interject two things. Although I reserve the word colonel for lenient maps and no spaces for matrices. Okay, Now this matrix reduces too to to want. So I've done it in its simplest form. So it ends up reducing to something like this. Okay, so at this point, you can use back substitution and you'll end up for the basis for the null space ends up being It will be minus 10 to 0 and one minus 40 and three. So obviously, if you did back substitution, you would set these two quantities to be one, and you work backwards, but I've just sort of normalized. It's another normal, but I've made it fraction free just for our simplicity purposes. Okay, so we've got a basis for the eye Gance base for articles 13 and similarly, I will do in the next page, sir. I'll keep it in black financing for the Eigen value. Let our used ob for I mean, it doesn't really matter. I just like using are because we're sticking to English letters rather than moving to Greek letters. But in any case, so for the Eiken value are equal to minus 12. Are I modest, eh? You know, when you write out what its people to sort of figure that out on your own. But it reduces to Mona's 21 for 24 Sara Wan. Modest one of one. And in two bars, off zeros on a basis for the null space. So for the Eigen space for article 2 12 minus 12. Sorry is 2770 and zero. Modest one, sirrah. One. Okay, so I'll leave it there. Uh, please verify older cat places that I've made Adlai Vilo on the computer or by hand, But in any case, thank you very much.

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