I start with background and then ask my question, background is a brief description of wishart distribution.

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Background
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The Wishart distribution with $\nu$ degrees of freedom and positive definite $p \times p$ scale matrix $V$, $\mathcal{W}_p(\nu,V)$, has the pdf

$$p(S\mid V,\nu) = \frac{|S|^{(\nu - p - 1)/2}}{2^{\nu p/2}|V|^{\nu/2}\Gamma_p(\nu/2)} \exp(-\frac{1}{2}\text{tr}(V^{-1}S))$$

Draws from this distribution will be $p \times p$ positive semidefinite matrices so long as $\nu > p$, with expectation $\mathbb{E}[S]= \nu V$ and variance $\operatorname{Var}[S_{ij}] = \nu(V_{ij}^2 + V_{ii}V_{jj})$.

If $\nu$ is integer valued, we can write a Wishart random variable as a sum of outer products of $\nu$ i.i.d multivariate Gaussian random variables:

$$S = \sum_{i=1}^{\nu} \mathbb{u}_i \mathbb{u}_i^\top \sim \mathcal{W}_p(\nu,V), \text{ where } \mathbb{u}_i \sim \mathcal{N}(0,V).$$

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Question
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Suppose $A$ is a **symmetric** $n\times n$ constant matrix and $S$ is a $n\times n$ matrix distributed from $W(v,V)$, I want to compute:

$$ \operatorname E[S^TAS] $$