Eigenvectors of 3*3 matrix

[ricreads]

Hi guys,

 

I am struggling with finding eigenvectors of a 3x3 matrix in a change of coordinates problem. Basically,
I have taken the equation x^2 - z^2 - 4xy +4yz, converted it into matrix [1 -2 0; -2 0 2; 0 2 1] and found the eigenvalues

 

 

1

1/2 + sqrt(33)/2
1/2 - sqrt(33)/2

But I am truly struggling with the eigenvectors here. The first one (l=1) is easy, but the other ones not. I plug the eigenvalues in the matrix and using Gauss-Jordan, I only get 1's on the diagonal. But I cannot determine such a vector, as I would think that it is [x1 x2 x3] = [0 0 0].

 

Help?

[timo]

The zero vector 0 is -in some sense- an Eigenvector to every matrix A and every eigenvalue n, since A0 = 0 = n0. From the little you said (in detail) about how you try to determine the eigenvector, I have the gut feeling you don't really know what you are doing. Take the eigenvalue 1: When looking to an eigenvector to this eigenvalue, what you are looking for are all possible solutions [math]\vec x[/math] to the system of equations [math]A\vec x = 1 \vec x[/math].