TITLE: Matrices which are both unitary and Hermitian
QUESTION [39 upvotes]: 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?

REPLY [3 votes]: Such matrices are used in the field of Quantum Computing. States of a Quantum system exist in a Hilbert space of dimension n. The space of operators or "gates" having affect on this Hilbert state space is the space of all n*n dimension Unitary matrices. 
Vectors in this operator space that are Hermitian are extremely useful because they denote gates that are reversible (note how if A is Hermitian A=A^-1, so not only is the Quantum operation A invertible, it is in fact it's own inverse.)