I am working in a problem that involves multivariate normal distributions and, at a given point, I need to solve the following matrix equation:
$x=\sqrt{x^{\prime}\Sigma^{-1}x} \cdot y$
Where $x$ is a $N\times1$ vector, $\Sigma$ is a $N\times N$ variance-covariance matrix (therefore symmetric, positive definite and with each element being positive), and $y$ is a different $N\times1$ vector ($y$ can take positive and negative values).
I am interested in a (hopefully explicit) solution for $x$ as a function of $\Sigma$ and $y$, which are known parameters.
There is a trivial solution $x=0$, and this is the only solution when $N=1$. My question is: Are there any other solutions when $N>1$? How can they be characterized?