Reconstruct a vector from outer product with itself

Question

Suppose I am given the matrix $A = xx^\top$ where $x \in \mathbb{R}^n$ is some vector that is unknown. How can I reconstruct the vector $x$ from $A$?

Answer 1

With physicists' notation (and assuming that there is a scalar product, which helps identity the vector space with its dual), one has $$ A = | \mathbf{x} \rangle \langle \mathbf{x} |$$

Up to the factor $\lVert \mathbf{x} \rVert^2$, it's the projection on $\mathbb{R}\, \mathbf{x}$.

The factor $\lVert \mathbf{x} \rVert^2$ coincides with the norm of the operator $A$, so one can recover $\pm \mathbf{x}$ by just picking a unit vector in the image $\mathrm{Im}(A)$, and multiply it by $\sqrt{\lVert A\rVert}$.

Remark: I would rather call the "product" the tensor product of a vector with a linear form