I have a vector $\bf x$ with dimension $n$ by $1$. So the tensor product is
$\bf{x} \ \otimes \bf x = xx^T$.
Then I have a singular symmetric matrix $\bf Q$ with 1 zero eigenvalue and I end up with an equation
$\bf Q xx^T = \it{c} \bf{I}$,
where $c$ is some constant and $\bf I$ is an identity matrix.
Given $\bf Q$, can I recover the vector $\bf x$?
$\bf Q$ is singular so I can get its pseudoinverse. Does it make sense to get the square root of a pseudo inverse?
Another approach would be to have $\bf (Qx)(Qx)^T=\it{c}\bf{Q}$. So I'm thinking of getting the root of singular matrix.