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?