# Definition A unitary operator is one characterized by $U^\dagger U = U U^\dagger= I,$ Where $\dagger$ denotes the complex transpose or Hermitian adjoint, and $I$ is the identity operator. Equivalently, $ U^\dagger = U^{-1} $ # Properties Unitary operators preserve the [[inner product]], i.e. $\braket{U \phi | U \psi} = \braket{\phi|\psi}$ # Unitary Matrices [[Matrix|Matrices]] representing unitary operators are unitary matrices (in an [[Orthogonality|orthonormal]] basis), i.e., they obey $U^\dagger U = U U^\dagger I$ and thus $U^\dagger = U^{-1}$. Note that this means that unitary matrices are [[Non-singular Matrix|non-singular]]. In terms of components: $\sum_{\beta} u_{\alpha \gamma} u_{\beta \gamma}^* = \delta_{\alpha \beta},$ where $\delta_{\alpha \gamma}$ is the [[Kronecker Delta]]. ## Properties Note that $U$ is [[Diagonalizable Matrix|diagonalizable]], i.e., $ U = V D V^\dagger $ where $V$ is unitary and $D$ is unitary and diagonal. We also have $(\det U)^* (\det U) = 1$ where $\det$ is the [[determinant]]. This is straightforward to show. Moreover, The [[eigenspace|eigenspaces]] of unitary matrices are orthogonal. Finally, one can show that $U$ can be written as $ U = e^{i H} $ where $H$ is a [[Hermitian Operator|Hermitian matrix]]. Unitary matrices of dimension $n \times n$ form a group called $U(n)$, the [[Unitary Group|unitary group]] of dimension $n$.