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Section 5.5 Unitary Matrices

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}\text{,}\tag{5.5.1} \end{equation}

that is, if

\begin{equation} U^\dagger U = I = UU^\dagger\text{.}\tag{5.5.2} \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 \(U\text{,}\) then

\begin{equation} \langle v_i | v_j \rangle = \delta_{ij}\text{.}\tag{5.5.3} \end{equation}

We will use unitary matrices in three ways:

  1. Change of basis

  2. Symmetry operators

  3. Evolution operators

  4. Preserve norm

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