**Definition:** A complex square matrix $U$ is **unitary** if its conjugate transpose equals its inverse:

$$U^* U = UU^* = I \quad \text{or equivalently} \quad U^{-1} = U^*$$

where $U^* = (\overline{U})^T$ is the conjugate transpose.

**Example:**
$$U = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & i \\ i & 1 \end{pmatrix}$$

**Properties:**
- $|\det(U)| = 1$
- Preserves inner products: $\langle Ux, Uy \rangle = \langle x, y \rangle$
- Preserves norms: $\|Ux\| = \|x\|$
- Eigenvalues lie on the unit circle ($|\lambda| = 1$)
- Columns (and rows) form an orthonormal basis
- The product of unitary matrices is unitary

Unitary matrices are the complex analog of orthogonal matrices. They represent **isometries** (distance-preserving transformations) in complex vector spaces and are fundamental in quantum mechanics.