Say I have two matrices $A$ and $B$. I know that they each have real eigenvalues. Clearly if they're Hermitian, then their sum would also have real eigenvalues, since then $A+B$ is Hermitian. However, I'm wondering what happens if they are not necessarily Hermitian. Can we say anything about $A+B$? Are the eigenvalues of $A+B$ real? I have tried to come up with a counterexample but didn't manage and didn't get very far manipulating the characteristic equation. What if just one of them is Hermitian?
Are the eigenvalues of the sum of matrices with real eigenvalues still real
eigenvalues-eigenvectorslinear algebramatrices
Best Answer
As an example, look at $A=\begin{bmatrix} 1 & 0 \\ 1 & 0 \end{bmatrix}$ and $B=\begin{bmatrix} 0 & -1 \\ 0 & 0 \end{bmatrix}$.