# Definition of unitary matrix

Put content here**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.
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Equivalently, the columns of $U$ form an orthonormal set in $\mathbb{C}^n$.
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**Example.** $U = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ i & -i \end{pmatrix}$ is unitary, since:
$$U^* U = \frac{1}{2}\begin{pmatrix} 1 & -i \\ 1 & i \end{pmatrix}\begin{pmatrix} 1 & 1 \\ i & -i \end{pmatrix} = I$$
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Unitary matrices are the complex analogue of real orthogonal matrices.

# Parents

* Unitary matrices