Deriving matrix definition of unitary matrices from a definition based on norms

Question

Let a matrix $A$ be unitary if $$||Ax||_2=||x||_2$$ for any $x\in\mathbb{C}^n$. We see that $$||Ax||_2=x^*A^*Ax=x^*x=||x||_2$$ which implies $A^*Ax=x\longrightarrow A^*A=I$.

How do you derive the fact that $AA^*=I$?

Answer 1

If $A^\ast A = I$ then multiplying $A$ to the left on both sides yields $A A^\ast A = A$ and multiplying $A^{-1}$ to the right gives $A A^\ast = I.$