A complex $n\times n$ matrix $U$ is unitary if its conjugate transpose is equal to its inverse, that is, if \begin{equation} U^\dagger = U^{-1} , \end{equation} that is, if \begin{equation} U^\dagger U = I = UU^\dagger . \end{equation}

If $U$ is both unitary and real, then $U$ is an orthogonal matrix. The analogy goes even further: Working out the condition for unitarity, it is easy to see that the rows (and similarly the columns) of a unitary matrix $U$ form a complex orthonormal basis. Using bra/ket notation, and writing $|v_i\rangle$ for the columns of $U$, then \begin{equation} \langle v_i | v_j \rangle = \delta_{ij} . \end{equation}

We will use unitary matrices in three ways:

1. Change of basis
2. Symmetry operators
3. Evolution operators
4. Preserve norm