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When $x$ is an '$n$' dimensional standard Gaussian, we have $x'x \sim \chi^2$ with $n$ degrees of freedom.

Now if I have a symmetric matrix $C$, what will be the distribution of $x'Cx$ ?

$C$ is the inverse of a positive definite matrix, like an inverse covariance matrix for example.

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    Are your "*" meant to denote matrix multiplication? That is usually written without any sign; using a star as you do in this context will make many readers think of adjoint matrices instead. (Also, does the apostrophe denote transposition?)2012-01-30
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    yea sorry ... '*' is matrix multiplication and apostrophe is transpose. So x'Cx is what I'm asking about.2012-01-31
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    A real symmetric matrix can be diagonalized by an orthogonal matrix. I'm assuming by "standard Gaussian" you mean in particular that the variance is the identity matrix (and the expected value is the zero vector). If you multiply an $n$-dimensional standard Gaussian vector by an orthogonal matrix, you don't change its probability distribution. So the problem reduces to the case where $C$ is a diagonal matrix.2012-01-31

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