Orthogonal transformation on random matrices

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

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Unless I'm completely misreading the question, the answer is obviously no; just take $A$ (and therefore $B$) to be constant. –  Steven Landsburg May 29 '13 at 15:24