Suppose that $A \in \Re^{d \times d}$, $x \in \Re^d$ and each component of $x$ is independently sampled from $N(0,1)$. I wonder what is the distribution of $x^TAx$. To be more concrete, how will the resulting distribution relate to $A$. Note that $A$ may be not symmetric, thus this may be a different problem with previous ones. (e.g. this)
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We may replace $A$ by the symmetric $(A + A^T)/2$ and then diagonalize, to see that $x^T A x$ can be written as $\sum_{i=1}^d c_j Z_i^2 $ where $Z_i$ are independent standard normal random variables. $Z_i^2$ has a Chi-squared distribution with one degree of freedom (or a Gamma distribution with shape parameter $1/2$).
Robert Israel
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this is a great trick. Can you please explain how can we infer on $x^\top A x$ from the symmetric $(A+A^\top)/2$ ? – sefi Jun 22 '23 at 18:37
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found the explanation in https://math.stackexchange.com/questions/307381/why-do-we-assume-that-a-matrix-in-quadratic-form-is-symmetric – sefi Jun 22 '23 at 19:05