How to prove unitary matrices require orthonormal basis

Question

How to prove unitary matrices require orthonormal basis

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Answer 1

The columns of an $n\times n$ unitary matrix form an orthonormal basis for $\Bbb C^n$ with the usual complex inner product $\langle\vec x,\vec y\rangle:=(\vec x^*)\vec y,$ where $\vec x^*$ is the conjugate transpose of $\vec x$. Conversely, if you have such an orthonormal basis, then a matrix having those basis vectors as columns will be unitary. The key, here, is if we write $$U=(\vec u_1 \:\cdots\: \vec u_n),$$ then the $ij$th entry of $U^*U$ is $\langle \vec u_i,\vec u_j\rangle,$ and for square matrices $A,B$ of the same size, $AB=I$ if and only if $BA=I$.