quick factorization of rank-one matrices (generally, and of a particular form)

Question

Let $Q$ be an arbitrary non-zero matrix and let $x$ be a column vector.

It should be true that $xx^TQ$ and $Qxx^T$ are both rank 1 matrices.

It is a fact that all rank-one matrices can be factorized as $uv^T$, where $u$ and $v$ are vectors. This can be accomplished by means of the singular value decomposition rather slowly. Is there a quick way to factorize the above rank-one matrices with small time complexity? I would think there should be a quick way to factorize all one-rank matrices...

Answer 1

But they are already factorized as wished. We have $$ xx^tQ = x(Q^tx)^t, \quad Qxx^t = (Qx)x^t. $$