Hey ianchenmu.
Have you tried proof by contradiction? (Assume it has different properties and find a contradiction).
The question is:
Let be a symmetric matrix of rank one. Prove that must have the form , where is a scalar and is a vector of norm one.
(I think we can easily prove that if has the form , then is symmetric and of rank one. But what about the opposite direction...that is what we need to prove. How to prove this?)