Assume $A, B$ are real matrices. Weyl's inequalities provide bounds on the eigenvalues of $A + B$ if both are symmetric. Is there any bound if neither are symmetric?
I am particularly interested about the case where $A$ and $B$ are positive stable, that is, have eigenvalues with positive real part. For instance, can one always produce $A, B$ positive stable such that $A+B$ has eigenvalues with arbitrarily negative real part?
Take -for example- $A=\begin{pmatrix}125&-61\\20&-9\end{pmatrix},B=\begin{pmatrix}6&24\\-44&-5\end{pmatrix}$.
$A,B$ are $>0$ stable. Note that $Re(tr(A+B))>0$; then $A+B$ has at least one eigenvalue with $>0$ real part.
Here $spectrum(A+B)\approx \{137,-20\}$. By multiplying $A,B$ by a real $>0$ number, we can obtain a second eigenvalue which can be as small as we want.