Normal Modes: Finding Eigenfrequencies

[Sum Guy]

If I have a system where the following is found to describe the motion of three particles:

The normal modes of the system are given by the following eigenvectors: $$(1,0,-1), (1,1,1), (1,-2,1)$$
How can I find the corresponding eigenfrequencies? It should be simple... What am I missing?

 

[berkeman]

Is this for schoolwork?

 

[Sum Guy]

Is this for schoolwork?

This is something I found in a pdf online where it simply asserted what the eigenfrequencies were... and not how to find them.