Characteristic polynomial above the complex numbers I need to find the Characteristic polynomial of   $\begin{pmatrix}
    i+1 & 0 & 0 \\
    0 & 3i-1 & 2-2i\\
    0 & 2-2i & 3i-1
\end{pmatrix}$
I know that there is not different in $det(xI-A)$ or $det(A-xI)$ is it different over the complex numbers?
$det(xI-A)=(x-i-1)*((x-3i+1)^2-(2-2i)^2)$
$det(A-xI)=(1+i-x)*((3i-1-x)^2-(2-2i)^2)$

In general it holds $det(xI-A) = det(-(A-xI)) = (-1)^{dimV} det(A-xI)$, no matter whether the entries of $A$ are real or complex numbers. $(-1)^{dimV}$ can be omitted for determining the characteristic polynomial.
Therefore there is no difference over the complex numbers.
