PSD rank 1 matrix decomposition into product of two vectors Assume that I'm given with matrix $\textbf{Q} = \textbf{q}\textbf{q}^H$. Assume I only have $\textbf{Q}$, is there any way for me to find vector $\textbf{q}$?
Thanks!

Here's the quickest approach I can think of: let $\mathbf x$ be any non-zero column of $Q$.  Then, it suffices to take
$$
\mathbf q = \frac{\sqrt{\operatorname{trace}(Q)}}{\|\mathbf x\|} \cdot \mathbf x
$$
Note, however, that if $z$ is a complex number with $|z| = 1$, then $(z\mathbf q)(z\mathbf q)^H = \mathbf q \mathbf q^H$.
