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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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If $D$ is a multiple of the identity matrix, easy. Otherwise, no idea. – Will Jagy Mar 9 '13 at 3:09
Yeah, but I'd like to know the case when $D$ is only a general diagonal matrix. – Hugo Mar 9 '13 at 3:12
we can look the problem in another way, we know the eigen values of $A$ and since $D$ is diagonal matrix, we also know the eigenvalues of $D$. Knowing eigen values of two matrix, how can we deduce the eigen value of $A+D$. [There is an ambiguity moving from eigen value to matrix] – Learner Mar 9 '13 at 3:20
@Learner In that case, only when $A$ and $D$ can be diagonalized simultaneously can we add their eigenvalues together to compute that of $B$. – Hugo Mar 9 '13 at 3:34
@Hugo Agreed. Im just repharsing the question to find an answer. – Learner Mar 9 '13 at 3:37
up vote 5 down vote accepted

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.

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