# Perturbing/sensitivity of the eigenvalues of a block tridiagonal matrix

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 $c$ ?

Of course, if $c$ is a small real number, then the eigenvalues of $J$ are close to the original ones (because the eigenvalues are continuous functions of the entries). Of course, the eigenvalues about the origin are not necessarily real because $0$ was a multiple eigenvalue. The case $n=2$ is easy: indeed, with a change of orthonormal basis, we may assume that $B+B^T=diag(2b_1,2b_2)$ where $b_1,b_2>0$, that implies $B=\begin{pmatrix}b_1&u\\-u&b_2\end{pmatrix}$. Easily, we find that the eigenvalues of $J$ are $\approx -2b_1,-2b_2,\gamma_1,\gamma_2$ where $Re(\gamma_i)=c/2$. But the following is interesting: numeric computations seem to "show" that (for any $n$) all eigenvalues of $J$, that are in a neighborhood of $0$, have a real part that is approximately $c/2$.