Matrices such as
$$
\begin{bmatrix}
\cos\theta & \sin\theta \\
\sin\theta & -\cos\theta
\end{bmatrix}
\text{ or }
\begin{bmatrix}
\cos\theta & i\sin\theta \\
-i\sin\theta & -\cos\theta
\end{bmatrix}
\text{ or }
\begin{bmatrix}
\pm 1 & 0 \\
0 & \pm 1
\end{bmatrix}
$$
are both unitary and Hermitian (for $0 \le \theta \le 2\pi$). I call the latter type trivial, since its columns equal to plus/minus columns of the identity matrix.
Do such matrices have any significance (in theory or practice)?
In the answer to this question, it is said that "for every Hilbert space except $\mathbb{C}^2$, a unitary matrix cannot be Hermitian and vice versa." It was commented that identity matrices are always both unitary and Hermitian, and so this rule is not true. In fact, all trivial matrices (as defined above) have this property. Moreover, matrices such as
$$
\begin{bmatrix}
\sqrt {0.5} & 0 & \sqrt {0.5} \\
0 & 1 & 0 \\
\sqrt {0.5} & 0 & -\sqrt {0.5}
\end{bmatrix}
$$
are both unitary and Hermitian.
So, the general rule in the aforementioned question seems to be pointless.
It seems that, for any $n > 1$, infinitely many matrices over the Hilbert space $\mathbb{C}^n$ are simultaneously unitary and Hermitian, right?
Best Answer
Unitary matrices are precisely the matrices admitting a complete set of orthonormal eigenvectors such that the corresponding eigenvalues are on the unit circle. Hermitian matrices are precisely the matrices admitting a complete set of orthonormal eigenvectors such that the corresponding eigenvalues are real. So unitary Hermitian matrices are precisely the matrices admitting a complete set of orthonormal eigenvectors such that the corresponding eigenvalues are $\pm 1$.
This is a very strong condition. As George Lowther says, any such matrix $M$ has the property that $P = \frac{M+1}{2}$ admits a complete set of orthonormal eigenvectors such that the corresponding eigenvalues are $0, 1$; thus $P$ is a Hermitian idempotent, or as George Lowther says an orthogonal projection. Of course such matrices are interesting and appear naturally in mathematics, but it seems to me that in general it's more natural to start from the idempotence condition.
I suppose one could say that Hermitian unitary matrices precisely describe unitary representations of the cyclic group $C_2$, but from this perspective the fact that such matrices happen to be Hermitian is an accident coming from the fact that $2$ is too small.