How Do You Normalize Eigenvectors for an Observable Matrix?

[KeyToMyFire]

Homework Statement

An observable is represented by the matrix

0 $\frac{1}{\sqrt{2}}$ 0
$\frac{1}{\sqrt{2}}$ 0 $\frac{1}{\sqrt{2}}$
0 $\frac{1}{\sqrt{2}}$ 0

Find the normalized eigenvectors and corresponding eigenvalues.

The Attempt at a Solution

I found the eigenvalues to be 0, -1, and 1

and the eigenvectors to be (1,0,-1), (1,-$\sqrt{2}$,1), and ((1,$\sqrt{2}$,1) (respectively)

I'm pretty sure these are right.


My problem comes with the word "normalized". The only place my lecture notes and book mention normalized eigenvectors is after the matrix is diagonalized, which seems unnecessary.

 

[micromass]

You want to find eigenvectors (x,y,z) with norm one. So you want eigenvectors (x,y,z) such that $\sqrt{x^2+y^2+z^2}=1$. That is what normalized means.

The eigenvectors you list are correct, but they are not yet normalized.