The complex Schur decomposition goes as follows:
For all $A \in \mathbb{C}^{n \times n}$ there exists a unitary matrix $U$ such that $U^{*}AU$ is triangular say, $U^{*}AU=T$. I have seen it being written as $UTU^{*}=A$ as well. Are these two statements always equivalent? It seems to me the answer should be YES, but I am a bit unsure to why this is the case.
A matrix $A$ is said unitary if $AA^* = A^*A= Id$ ie $A^* = A^{-1}$. Now $U^*AU = T \iff UU^*AU = UT \iff AUU^* = UTU^* = A $
So yes the two statements are equivalent.