Let $A,B \in \mathcal M_n(\mathbb R^n)$ be two square real matrices of dimension $n$. Suppose that the eigenvalues of $A,B$ are real and strictly positive. Can $A+B$ have a strictly negative eigenvalue?
I know that if $A,B$ commute and are diagonalizable, then they are diagonalizable in a shared basis and therefore the answer to the question is negative.
However, I don’t see how to tackle the general case.
Best Answer
Take $A=\pmatrix{1&0\cr n&1}$ and $B=\pmatrix{1&n\cr 0&1}$, $A+B=\pmatrix{2&n\cr n&2}$, the polynomial characteristic of $A+B$ is $(2-X)^2-n^2=X^2-4X-n^2+4$. If $n$ is enough big, it has a negative root.