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Let $A$ be an $m\times k$ matrix with iid $\mathcal{N}(0,1)$ entries and $a_{i}$ and $a_{j}$ be its $i$th and $j$th columns, respectively. I would like to compute the following quantity: \begin{equation} \mathbb{E}[(a_{i}^{\top}AA^{\top}a_{j})^{q}], \end{equation} for $q\in\mathbb{N}$.

One can expand $a_{i}^{\top}AA^{\top}a_{j}$ to get \begin{equation} \|a_{i}\|_{2}^{2}a_{i}^{\top}a_{j}+\|a_{j}\|_{2}^{2}a_{i}^{\top}a_{j}+\sum_{t\in[k]\backslash\{i\}\cup\{j\}}(a_{i}^{\top}a_{t})(a_{j}^{\top}a_{t}) \end{equation} and explicitly compute the expectation for $q=1, 2$ etc. But how can I generalize for larger $q$?

[Posting this here since couldn't get answer on Math Stack Exchange and Cross Validated]

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    $\begingroup$ I don't think it is reasonable to expect an exact answer in closed form; consider the simpler problem where instead of asking for moments of the matrix element $B_{ij}=a_i^T AA^T a_j$ you would ask for moments of ${\rm tr}\,B={\rm tr}\,W^2$ with Wishart matrix $W=AA^T$; for large $m,k$ you could use Marcenko-Pastur and make analytical progress, but exact results for any $m,k$ seem out of reach. $\endgroup$ – Carlo Beenakker Dec 6 '19 at 20:17

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