Exact expression of variance of gaussian quadratic form

Question

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?

Answer 1

Since $\vec{x}$ is a standard gaussian random vector, and if $\mathbf{A}$ is symmetric, then from 1 the first moment of $\vec{x}^\prime\mathbf{A}\vec{x}$ is $$\mathbb{E}[\vec{x}^\prime\mathbf{A}\vec{x}]=trace(\mathbf{A})$$

The second moment is

$$\mathbb{E}[\vec{x}^\prime\mathbf{A}\vec{x}]^2=(trace(\mathbf{A}))^2+2trace(\mathbf{A}^2)$$

Thus we can obtain $var(\vec{x}^\prime\mathbf{A}\vec{x})=2trace(\mathbf{A}^2)$

1 - The moments of products of quadratic forms in normal variable