  We consider the eigenvalue problem $Ax = \lambda x$ where $A \in
\mathbb{R}^{n \times n}$ and the eigenvalue is also real $\lambda \in
\mathbb{R}$. If we are given $A$, $\lambda$ and, additionally, the absolute
value of the entries of $x$ (the vector $(|x_i|)_{i=1}^n$), is there a fast way
to recover $x$? In particular, can this be done quicker than computing $x$ from
scratch? This may be understood as a special case of the phase retrieval
problem. We present a randomized algorithm which provably converges in
expectation whenever $\lambda$ is a simple eigenvalue. The problem should
become easier when $|\lambda|$ is large and we discuss another algorithm for
that case as well.
