Unitary Matrices

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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:

Unitary matrices preserve the norm of vectors, Section 5.5
Unitary operators can be used to change the basis in a vector space, Section 5.10
Evolution operators are unitary, Section 5.12
Some types of symmetry operators are unitary, Section 5.13