Diagonalize unitary matrix

__Problem statement:__

Diagonalize this unitary matrix $\displaystyle \textbf{V}$ to reach $\displaystyle \textbf{V}=\textbf{U}\Lambda\textbf{U}^*$.

$\displaystyle

V=\frac{1}{\sqrt{3}}\left[\begin{array}[pos]{cc}

1 & 1-i \\

1+i & -1 \\

\end{array}\right]

$

__Pathetic attempt:__

The trace of $\displaystyle \textbf{V}$ is zero and since it is unitary it has eigenvalues with absolute value one. This mean that $\displaystyle \lambda = 1,-1$

What is confusing me is the fact that I am supposed to make a matrix with orthonormal column vectors out of a unitary matrix (which has orthonormal column vectors)...

Any hints are greatly appreciated, thanks!