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Consider a $2n-$dimensional symmetric random matrix $M$ of form, $M = \begin{bmatrix} aa^T & ab^T \\ ba^T & bb^T \end{bmatrix}$ where $a$ and $b$ are $n$ dimensional random vectors.

  • Are there conditions known on $a$ and $b$ s.t we have the following property : that for any $\hat{x} \in S^{2n-1}$ and any $R \in SO(2n)$, $\Vert M \hat{x} \Vert$ be equidistributed as $\Vert (R M R^T)\hat{x} \Vert$ ?
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  • $\begingroup$ Cross-posted from MSE. $\endgroup$ Commented Jan 9, 2021 at 22:13
  • $\begingroup$ I believe the two questions are different. I expect that if this question has an answer it should be something more general than the other one. If you suggest not then I can as well delete the other one. $\endgroup$ Commented Jan 9, 2021 at 22:15

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