Section 5.4 Properties of Unitary Matrices
The eigenvalues and eigenvectors of unitary matrices have some special properties. If \(U\) is unitary, then \(UU^\dagger=I\text{.}\) Thus, if
then also
Combining (5.4.1) and (5.4.2) leads to
Assuming \(\lambda\ne0\text{,}\) we thus have
Thus, the eigenvalues of a unitary matrix are unimodular, that is, they have norm 1, and hence can be written as \(e^{i\alpha}\) for some \(\alpha\text{.}\)
Just as for Hermitian matrices, eigenvectors of unitary matrices corresponding to different eigenvalues must be orthogonal. The argument is essentially the same as for Hermitian matrices. Suppose that
Then
or equivalently
Thus, if \(e^{i\lambda}\ne e^{i\mu}\text{,}\) \(v\) must be orthogonal to \(w\text{.}\)
As with Hermitian matrices, this argument can be extended to the case of repeated eigenvalues; it is always possible to find an orthonormal basis of eigenvectors for any unitary matrix.