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If I have a unitary square matrix $U$ ie. $U^{\dagger}U=I$ ( $^\dagger$ stands for complex conjugate and transpose ), then for what cases is $U^{T}$ also unitary. One simple case I can think of is $U=U^{T}$ ( all entries of $U$ are real, where $^T$ stands for transpose ). Are there any other cases ?

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It's going to be true in all cases.

In particular, if $U$ is unitary, then $$ (U^T)^\dagger U^T = [UU^\dagger]^T = I^T = I $$

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$$(U^T)^\dagger = \bar U = (U^\dagger)^T, $$ where $\bar U$ is the complex conjugate of $U$.

Moroever $$(U^T)^\dagger U^T = (U^\dagger)^T U^T = UU^\dagger = I.$$

Therefore, your proposition is always true.

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