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Let $A$ and $B$ be two random matrices such that $A=U^T B V^T$. Here $U$ and $V$ are deterministic and orthogonal matrices. The random matrices $A$ and $B$ have the same distribution?

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This depends on the distribution of the entries of $B$. When entries of $B$ are i.i.d. standard normal, entries of $A$ are also i.i.d. standard normal. However, if entries of $B$ are i.i.d. $U(0,1)$, then in general, entries of $A$ and $B$ have different distributions. For instance, suppose $V^T=I$ and $U^T$ is a $2\times2$ rotation matrix for the angle $\frac{\pi}{4}$. Then the entries of $A$ can be as large as $\sqrt{2}$ (because $U^T\pmatrix{1\\ 1}=\pmatrix{\sqrt{2}\\ 0}$), but entries of $B$ are bounded by $1$.

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