Suppose $B = A+D$, where all the eigenvalues of $A$ are already known and $D$ is a diagonal matrix, how to compute the eigenvalues of B without diagonalizing $B$ directly?
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Modulo a similarity, it is like asking: if I know the eigenvalues of $A'$ and if $D'$ is a diagonalizable matrix, what can I say about the eigenvalues of $A'+D'$. Nothing, I'm afraid. The diagonalizable matrices are dense, so the eigenvalues of your matrix $A'+D'$ could be pretty much anything when $D'$ ranges over diagonalizable matrices.