# Showing unitary equivalence

How can it be shown that $UU^{*} = I$ where $U$ is a square matrix of an operator on a complex vector space implies that $\langle Ux, Uy\rangle = \langle x, y\rangle$?

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Hint: For adjoint operators you have $$(Ax,y)=(x,A^{*}y)$$
You may try to expand the summation as $(UX,UY)=\sum_{i}(\sum_{j}u_{ij}x_{j}\overline{\sum_{j}u_{ij}y_{j}})$, etc. But this is usually quite messy. – Kerry Oct 10 '11 at 17:06