1. ## finding eigenvectors

For the given matrix

2 0 -2
0 -1 0
-2 0 -1

I want to find eigenvectors with a length of 1.

So the eigenvalues are -1,-2,3

solving for each one i got x=0, z=0 for -1
x,y,z = 0 for -2
and x = -2z, y=0 for 3

so i guess the only eigenvector with length 1 would be the when the norm of the last solution equals to 1?

2. ## Re: finding eigenvectors

Originally Posted by Kuma
For the given matrix

2 0 -2
0 -1 0
-2 0 -1

I want to find eigenvectors with a length of 1.

So the eigenvalues are -1,-2,3

solving for each one i got x=0, z=0 for -1
x,y,z = 0 for -2
and x = -2z, y=0 for 3

so i guess the only eigenvector with length 1 would be the when the norm of the last solution equals to 1?
Whatever eigenvector has a norm of 1. There is only 1 the other two of a norm of $\sqrt{5}$

3. ## Re: finding eigenvectors

if v is an eigenvector, so is av, for any non-zero a.

so if a given eigenvector doesn't have norm (length) 1, just consider v/|v|. since v is an eigenvector, v is non-zero, so |v| is non-zero.

4. ## Re: finding eigenvectors

Originally Posted by Kuma
For the given matrix

2 0 -2
0 -1 0
-2 0 -1

I want to find eigenvectors with a length of 1.

So the eigenvalues are -1,-2,3

solving for each one i got x=0, z=0 for -1
x,y,z = 0 for -2
Nonsense! The definition of "eigenvalue" is that there exist a non-zero vector such that Av= lambda v.

and x = -2z, y=0 for 3

so i guess the only eigenvector with length 1 would be the when the norm of the last solution equals to 1?
You can find a non-zero eigenvector for each eigenvalue. Divide each by its norm to get length 1.