In one of my classes, we showed that, given a 2×2 matrix $\mathbf{A}$ with eigenvalues $\lambda_\pm$,
$\Re[\lambda\pm]<0 \iff \text{Tr}[\mathbf{A}]<0$ and $\det[\mathbf{A}]>0$
by computing the eigenvalues as
$\lambda_\pm = \text{Tr}[\mathbf{A}]\pm\sqrt{(\text{Tr}[\mathbf{A}])^2-4\det[\mathbf{A}]}$
It seemed to me that from the derivation, we were assuming the trace and determinant were both Real numbers; however, this isn't the general case.
Does this statement hold for $\text{Tr}[\mathbf{A}],\det[\mathbf{A}]\in\mathbb{C}$?
Best Answer
You can show the first result by noting that trace is real in the case of a $2\times 2$ matrix since the characteristic polynomial has either real roots or 2 complex conjugate roots. The determinant is the product of the eigenvalues so again will be positive in any case (you can check) if real parts of the eigenvalues are positive. Your question then maybe should be if the statement holds for a matrix of size $n\times n $, or for which $n $?