Let $\textbf{x}\in\mathbb{R}^p$ be a standard Gaussian vector and $\textbf{A}$ be a matrix independent of $\textbf{x}$. Is there an exact expression for the variance of the quadratic form $\textbf{x}^T\textbf{A}\textbf{x}$, i.e., $$\mathbb{E}[(\textbf{x}^T\textbf{A}\textbf{x}\text{tr}{\textbf{A}})^2]$$ I know there are bounds available on the centered moments of this quadratic form (trace lemma in random matrix theory), but I was wondering if there is an exact expression in this case?
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1The variance is $2\operatorname{tr}(A^2)$ provided $A$ is symmetric: https://stats.stackexchange.com/q/427332/119261. Also see https://math.stackexchange.com/q/442472/321264. – StubbornAtom Jan 19 '22 at 14:30