I have got a problem with matrices like, for example: $\left( \begin{array}{cccccc} 1 & 1 & 2 & 1 & 1 & 2 \\ 1 & 1 & 1 & 1 & 3 & 1 \\ 2 & 1 & 1 & 2 & 1 & 2 \\ 1 & 1 & 2 & 1000 & 1 & 2 \\ 1 & 3 & 1 & 1 & 1000 & 1 \\ 2 & 1 & 2 & 2 & 1 & 1000 \\ \end{array} \right)$
All elements are rather small compared to the diagonal entries in one of the 3x3 sub matrices.
If the matrix was 3x3 block diagonal (upper right, and lower left 3x3 subs set to zero) one would get two independent eigenproblems and eigenvectors of the form $(a,b,c,0,0,0)$ and $(0,0,0,d,e,f)$.
For my example matrix this doesn't work (obviously) but one gets eigenvectors that look very similar, e.g. $(a,b,c,d,e,f)$ with $a,b,c << d,e,f$ or the other way around.
Is there any way to quantify the error if one would solve the block-diagonal case instead of the full problem?
PS: I don't know if it helps with the problem but I'm predominantly interested in the case of hermitian matrices.