Definition of unitary matrix

Definition. A complex square matrix $U$ is unitary if its conjugate transpose equals its inverse:
$$U^* U = U U^* = I$$
where $U^* = \overline{U}^T$ denotes the conjugate transpose.

Equivalently, the columns of $U$ form an orthonormal set in $\mathbb{C}^n$.

Example. $U = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ i & -i \end{pmatrix}$ is unitary, since:
$$U^* U = \frac{1}{2}\begin{pmatrix} 1 & -i \\ 1 & i \end{pmatrix}\begin{pmatrix} 1 & 1 \\ i & -i \end{pmatrix} = I$$

Unitary matrices are the complex analogue of real orthogonal matrices.