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I managed to prove that if $A$ is unitary than for any base orthonormal base $\{u_i\}$: $\{A^*u_i\}$ is also orthonormal base.

I need some help proving that it's an if and only if, and not only if. I only managed to say that $A$ is invertible since $\{A^*u_i\}$ is also a base.

Edit: I'm using the definition "$A$ is unitary if $AA^*=A^*A=I$.

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There are many equivalent properties to define unitary. What is the definition of "unitary" that you are using? – Arturo Magidin Mar 23 '12 at 16:08
@ArturoMagidin I edited the question to add this – Belgi Mar 23 '12 at 16:08
... and obliterated the other fixes... re-added now. – Arturo Magidin Mar 23 '12 at 16:09
thanks for the edit, I really need to learn latex... – Belgi Mar 23 '12 at 16:11
up vote 3 down vote accepted

HINT. Note that $$ AA^*u_i = \sum_{j=1}^n \langle AA^*u_i,u_j\rangle u_j = \sum_{j=1}^n \langle A^*u_i,A^*u_j\rangle u_j. $$

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