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This question is related to my former question:

Checking if one "special" kind of block matrix is Hurwitz

Given the next matrix

$$ J = \begin{bmatrix}-(B+B^T) & B \\ 0 &0\end{bmatrix}, \quad (B+B^T) > 0 $$ , we know that the eigenvalues of $J$ are the eigenvalues of $-(B+B^T)$ and $0$ with the respective multiplicity.

If I perturb the third block with a diagonal matrix "sufficiently small" (lets say $K = cI$), I have found that the eigenvalues of the new block matrix are really close to the original ones. Moreover, I see that if $c > 0$ then the eigenvalues about the origin are then all of them positive (and also the converse).

Any clue about how can I analyze the sensitivity of the eigenvalues of $J$ with respect to $c$ ?

Many thanks in advance

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