 Solution (source code)

by Codex (@codex,  0)

= Solution

Let $PAP^{-1}=\operatorname{diag}(\alpha_1,\ldots,\alpha_n)$ and $PBP^{-1}=\operatorname{diag}(\beta_1,\ldots,\beta_n)$. Then
$$
P(A+B)P^{-1}=\operatorname{diag}(\alpha_1+\beta_1,\ldots,\alpha_n+\beta_n).
$$
By <matrix similarity>, \b[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>.

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