An example of unitary matrix which is $3\times 3$ and complex

Question

Please give me an example of unitary matrix which is $3\times 3$ and complex. If I get this example, i will finish my thesis.

Answer 1

Is this sufficient?

$$ A = \begin{bmatrix} i & 0 & 0\\ 0 & i & 0\\ 0 & 0 & 1 \end{bmatrix} $$

or the matrix $$\begin{bmatrix} \sqrt{1/3}+i\sqrt{1/3} & \sqrt{1/6} - i\sqrt{1/6} & 0\\ -i\sqrt{1/6} & \sqrt{1/3} & \sqrt{1/6} +i\sqrt{1/3}\\ \sqrt{1/6} & i\sqrt{1/3} & \sqrt{1/3}-i\sqrt{1/6} \end{bmatrix} $$ which can be found here.

Answer 2

A matrix with $\imath$ on the diagonal, since $\imath \overline{\imath} = 1$. Or a Discrete Fourier transform matrix: $$ D = \frac{1}{\sqrt{3}}\begin{bmatrix} 1 & 1 & 1\\ 1 & \jmath & \overline{\jmath}\\ 1 & \overline{\jmath} & \jmath \end{bmatrix} $$ with $\jmath$ the standard complex root of unity: $\jmath = e^{2\pi \imath/3}$