Understanding Eigen Vectors and Diagonalization for a 2x2 Matrix

[koolrizi]

I am trying to find eigen values and eigen vectors for A
Its 2X2 matrix. A first row (16 -10) second row (-10 24)
I got Eigen values as 30.77 and 9.22 but when i try to find eigen vectors here are the equations I end up with
-14.77v1 - 10v2= 0
-10v1 - 6.77v2 = 0

Kinda confused how to proceed with this.

Thanks

 

[ZioX]

Use 9.23, you rounded wrong.

The procedure for finding associated eigenvectors is to find the nullspace of A-λI. So you have to solve the nullspace of that matrix you wrote up to find the 30.77-eigenspace. Then you'll need to do the process again for 9.23.

Does this clear things up for you? Or do you need help with solving the nullspace? Because that should be easy.

 

[miunhauzen]

these two equations are not compatible, because you have rounded your eigenvalues, but if you would have used the ratios as an eigenvalued, you'd see that these thwo equations are exactly the same.

When you are solveing for eigenvectors you have to use either of these equation, because they are same, and then if your states are normalizabe, you have to normalize it.

v1^2+v2^2=1

that's your second equation in system!

 

[Hurkyl]

(You should use fractions instead of decimals)

You've spent the whole semester solving problems like that, haven't you? So what's the trouble?

 

[koolrizi]

Got it

Thanks I got that part. I didnt realize that v1=-0.677v2 for both equations. I am actually taking a different course which uses linear algebra but its been a while since i studied it. Now I have to diagonalize the matrix and also sketch the unit standard deviation contour. If you know any good sites for that do let me know.

Thanks everyone