Matrix Equation with Quadratic form 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?

You must have $x \in \text{sp}\{y\}$. Setting $x=\lambda y$ gives the equation $\lambda y =|\lambda | \left(\sqrt{y' \Sigma^{-1} y} \right) y$ which has a solution iff $\lambda = 0$ or if $ y' \Sigma^{-1} y =1$.
