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:
$$= \lambda^3+\lambda^2i+i+\lambda=0$$