Skip to main content

Section 5.4 Unitary Matrices

An \(n\times n\) matrix \(U\) is unitary if its Hermitian adjoint (i.e. conjugate transpose) is equal to its inverse, that is, if

\begin{equation} U^\dagger = U^{-1}\text{,}\tag{5.4.1} \end{equation}

tin which case,

\begin{equation} U^\dagger U = I = UU^\dagger\text{.}\tag{5.4.2} \end{equation}

If \(U\) is both unitary and real, then \(U\) is called an orthogonal matrix.

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

\begin{equation} \langle v_i | v_j \rangle = \delta_{ij}\text{.}\tag{5.4.3} \end{equation}

We will use unitary matrices in four ways:

  1. Unitary matrices preserve the norm of vectors, Section 5.5

  2. Unitary operators can be used to change the basis in a vector space, Section 5.10

  3. Evolution operators are unitary, Section 5.12

  4. Some types of symmetry operators are unitary, Section 5.13