<s>[INST] Eigenvectors of a complex matrix Given the following matrix 
$\begin{pmatrix}
0 & 1-i & 0\\
1+i & 0 &1-i\\
0& 1+i &0\\
\end{pmatrix}$
I have found the Eigenvalues $0, 2,-2$. But I have no idea how to calculate the corresponding Eigenvectors and I failed with Gaussian method. What could you recommend?
Thanks in advance!
 [/INST] For the eigenvalue $-2$:
\begin{align}
\begin{bmatrix}
2 & 1-i & 0\\
1+i & 2 &1-i\\
0& 1+i &2\\
\end{bmatrix}
&\to
\begin{bmatrix}
1 & (1-i)/2 & 0\\
1+i & 2 &1-i\\
0& 1+i &2\\
\end{bmatrix} && R_1\gets \tfrac{1}{2}R_1
\\[6px]&\to
\begin{bmatrix}
1 & (1-i)/2 & 0\\
0 & 1 &1-i\\
0& 1+i &2\\
\end{bmatrix} && R_2\gets R_2-(1+i)R_1
\\[6px]&\to
\begin{bmatrix}
1 & (1-i)/2 & 0\\
0 & 1 &1-i\\
0& 0 &0\\
\end{bmatrix} && R_3\gets R_3-(1+i)R_2
\\[6px]&\to
\begin{bmatrix}
1 & 0 & i\\
0 & 1 &1-i\\
0& 0 &0\\
\end{bmatrix} && R_1\gets R_1-\tfrac{1-i}{2}R_2
\end{align}
Thus the equations can be written
$$
\begin{cases}
x_1=-ix_3\\
x_2=(i-1)x_3
\end{cases}
$$
so an eigenvector is
$$
\begin{bmatrix}
-i\\
i-1\\
1
\end{bmatrix}
$$
</s>