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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 $(Ax,y)=(x,A^{*}y)$

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    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.2011-10-10