Consider the following real symmetric matrix:

0 1 -1

1 0 1

-1 1 0

it has eigenvalues t1=-2,t2=1,t3=1

with corresponding eigenvectors:

v1=(1,-1,1)^T

v2=(1,0,-1)^T

v3=(0,1,1)^T

However the inner product of v2 and v3, <v2,v3>=-1

Can anyone explain why this isn't 0?