## Section6.5Unitary Matrices

A complex $$n\times n$$ matrix $$U$$ is unitary if its conjugate transpose is equal to its inverse, that is, if

$$U^\dagger = U^{-1}\text{,}\tag{6.5.1}$$

that is, if

$$U^\dagger U = I = UU^\dagger\text{.}\tag{6.5.2}$$

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 $$U\text{,}$$ then

$$\langle v_i | v_j \rangle = \delta_{ij}\text{.}\tag{6.5.3}$$

We will use unitary matrices in three ways:

1. Change of basis

2. Symmetry operators

3. Evolution operators

4. Preserve norm