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I am struggling with an integral of form

$$ \int_{\mathbb R^n} y\otimes y~ \langle Ay,y\rangle \, \mathrm d N(0,\Sigma)(y). $$

I assume that it will involve the trace of some product of $R$ and $\Sigma$ to some power etc., but I cannot find a reference for it and I am unable to solve it myself. I tried applying a linear map to both sides, e.g. for $f,g\in \mathbb R^n$,

$$ \int_{\mathbb R^n} f^\top y~g^\top y~ \langle Ay,y\rangle \, \mathrm d N(0,\Sigma)(y) = \sum_{i,j,k,l}\int f_i \, g_j\, A_{kl}\, y_i\ y_j\ y_k\ y_l \, \mathrm d N(0,\Sigma)(y),$$

so in principle this should be reducible to all up-to-fourth-order moments of univariate Gaussian distributions, but this is leading me nowhere useful since I cannot really enumerate all combinations leading to odd and even symmetries.

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    $\begingroup$ Have a look at Sec. 1.4 of Janson's book Gaussian Hilbert Spaces There he explains how to compute integrals of the kind your interested using so called Feynman diagrams. $\endgroup$
    – Liviu Nicolaescu
    Commented Jan 17 at 15:00

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$\newcommand\s\Sigma$The $(i,j)$-entry of your matrix integral is $$\sum_{k,l}A_{kl}\,EY_iY_jY_kY_l,$$ where $(Y_1,\dots,Y_n)$ is a Gaussian zero-mean random vector with covariance matrix $\Sigma$. In turn, the expectations $EY_iY_jY_kY_l$ can be computed using the Isserlis theorem: $$EY_iY_jY_kY_l=\s_{ij}\s_{kl}+\s_{ik}\s_{jl}+\s_{il}\s_{jk},$$ where $\s_{ij}$ is the $(i,j)$-entry of the matrix $\s$.

So, the $(i,j)$-entry of your matrix integral is $$\sum_{k,l}A_{kl}(\s_{ij}\s_{kl}+\s_{ik}\s_{jl}+\s_{il}\s_{jk}).$$

So, your matrix integral is $$(\text{tr}(A\s))\s+\s A\s+\s A^\top\s.$$

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