Let $A$ and $B$ be two arbitrary real matrices of the same dimension. If the eigenvalues of $A$ and $B$ are all in the left half of the complex plane, can we estimate the the location of the eigenvalues of matrix $A+B$?


Assume that there is no common $P$, such that $PA+A'P \prec 0$ and $PB+B'P \prec 0$.


From some simple experiments, I found that the eigenvalues of $A+B$ are still on the left half of the complex plane. But I can not prove it. Is there any proved result I can relate to?



Is the sum of two stable matrices also stable?
Consider the example
$$ A = \pmatrix{0 & 0\cr 1 & 0\cr},\ B = \pmatrix{0 & 1\cr 0 & 0\cr} $$
$A$ and $B$ have eigenvalues $0$, but $1$ is an eigenvalue of $A+B$. 


I'm not sure in what sense you mean $< 0$ (element-wise? negative definite?). But for any $P$, $(P A + A' P)\_{1,1} = 0$ and $(P B + B' P)\_{2,2} = 0$, so there is no such $P$ in either sense.


EDIT: To make $A$ and $B$ have **strictly** negative eigenvalues, subtract some small $\epsilon$ from their diagonal elements. $A+B$ will still have an eigenvalue $1-2\epsilon > 0$. 


