Eigenvector Phase Retrieval: Recovering eigenvectors from the absolute value of their entries

We consider the eigenvalue problem Ax=λx where A∈R^n×n and the eigenvalue is also real λ∈R. If we are given A, λ and, additionally, the absolute value of the entries of x (the vector (|x_i|)_i=1ⁿ), 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 λ is a simple eigenvalue. The problem should become easier when |λ| is large and we discuss another algorithm for that case as well.