So I basically tried to calculate the eigenvectors of the given matrice just like I did it with the first matrice with real numbers. But now with complex numbers my calculation is kind of going nowwhere and my answers don't match with those from WolframAlpha. Is my method the right way or am I doing it completly wrong?
Few things which may help you to ease out your calculations:
-Characteristic equation for a square matrix of order $3$ is $t^3-tr(A)t^2+(\sum_1^3 A_{ii})t-det(A)=0$ where $tr(A)=$sum of diagonal entries of $A$ and $A_{ii}=$ cofactor of element $a_{ii}$.
-To find roots of a cubic equation $f(x)=x^3+ax^2+bx+c=0$, we seek an integer which divide $c$ and is a zero of $f(x)$. If we are lucky enough (if integer roots occur) and $p$ be such number then $x=p$ is first root. Then if complex roots occur they must occur in pairs. Let $r\pm i s$ be the complex conjugates which are roots of equation.
Use $p+r\pm is= -a$ to get $r$ and then $p.(r\pm is)=-c$ to get $s$.