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

We consider the eigenvalue problem $Ax = λx$ where $A \in \mathbb{R}^{n \times n}$ and the eigenvalue is also real $λ\in \mathbb{R}$. If we are given $A$, $λ$ 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 $λ$ is a simple eigenvalue. The problem should become easier when $|λ|$ is large and we discuss another algorithm for that case as well.