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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?

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You must have $x \in \text{sp}\{y\}$. Setting $x=\lambda y$ gives the equation $\lambda y = |\lambda| (\sqrt{y'\Sigma^{-1} y} ) y$ which has a solution iff $\lambda = 0$ or if $y'\Sigma^{-1} y = 1$ –  copper.hat Nov 17 '12 at 1:21
    
Thanks copper.hat. You comment is very helpful; I hadn't thought about doing that. If you want to write it as an answer I'll give it as correct. –  hulp10 Nov 17 '12 at 1:33
    
Good suggestion! –  copper.hat Nov 17 '12 at 1:36
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up vote 4 down vote accepted

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$.

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