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?