How do you find the eigenvalues (hence the eigenvectors too) of a matrix with complex bits like this:
$\hat{H}=\epsilon \begin{vmatrix} 0&i&0 \\\\ -i&0&0 \\\\ 0&0&-i \end{vmatrix}$
With $\epsilon$ real. How do you find the eigenvalues (hence the eigenvectors too) of a matrix with complex bits like this:
$\hat{H}=\epsilon \begin{vmatrix} 0&i&0 \\\\ -i&0&0 \\\\ 0&0&-i \end{vmatrix}$
With $\epsilon$ real.
I get so far as this:
$|\hat{H}-\lambda{}I | = \epsilon \begin{vmatrix} -\lambda&i&0 \\\\ -i&0-\lambda&0 \\\\ 0&0&-i-\lambda \end{vmatrix} =0$
$|\hat{H}-\lambda{}I | =\epsilon(-\lambda(-\lambda(i+\lambda))+i(0-i(i+\lambda))$
$=\epsilon(\lambda^2 (i+\lambda)+(i+\lambda))=0$
Then I am not sure what to do next. I think the $\epsilon$ cancels and I can do this:
$= \lambda^3+\lambda^2i+i+\lambda=0$