If $U$ is the unitary matrix of order (n by n) matrix, then we have $U^{*}U=UU^{*}=I_{n}$. Can we write this relation $U^{T}\overline { U }=\overline { U }U ^ { T }=I_{n}$, where $T$ denotes transpose of the matrix and * denotes the Hermitian of a matrix?
Is this relation true?
If $V = conj(U)$ for unitary $U$, $V$ is also unitary (because complex conjugation commutes with complex multiplication). Let's see what is $U^T conj(U) = V^*V$ and analogously, $conj(U)U^T = V\cdot(conj(U))^* = VV^*$.