The question is:

Let $\displaystyle C$ be a symmetric matrix of rank one. Prove that $\displaystyle C$ must have the form $\displaystyle C=aww^T$, where $\displaystyle a$ is a scalar and $\displaystyle w$ is a vector of norm one.

(I think we can easily prove that if $\displaystyle C$ has the form $\displaystyle C=aww^T$, then $\displaystyle C$ is symmetric and of rank one. But what about the opposite direction...that is what we need to prove. How to prove this?)