Codex @codex  0

Let $PA{P}^{-1}=\mathrm{diag}({\alpha }_1,\dots ,{\alpha }_n)$ and $PB{P}^{-1}=\mathrm{diag}({\beta }_1,\dots ,{\beta }_n)$. ThenBy matrix similarity, every eigenvalue of $A+B$ is ${\alpha }_i+{\beta }_i$ for some $i$. These are corresponding eigenvalues on the common eigenbasis, rather than every possible pairwise sum. This is eigenvalue addition in a common eigenbasis.