Conditioning.

As stated before, eigenvalues of singular pencils are discontinuous functions of the matrix entries. For example, consider the singular pencil in (2.5). Changing $A_{22}$ to $\epsilon_1 \ll 1$ and $B_{22}$ to $\epsilon_2 \ll 1$ makes this pencil regular, with an eigenvalue at 1 and one at $\epsilon_1/\epsilon_2$, which can be arbitrary, no matter how small the $\epsilon$ values are. Furthermore, changing $A_{12}$ to $\epsilon_{A12} \ll 1$, $A_{21}$ to $\epsilon_{A21} \ll 1$, $B_{12}$ to $\epsilon_{B12} \ll 1$, and $B_{21}$ to $\epsilon_{B21} \ll 1$, leads to eigenvalues $\epsilon_{A12}/\epsilon_{B12}$ and $\epsilon_{A21}/\epsilon_{B21}$, both of which can be arbitrary, no matter how small the $\epsilon$ values are. Reducing subspaces can also change discontinuously.

Nonetheless, there are important situations where eigenvalues and reducing subspaces change continuously. This happens because the perturbations may be constrained to keep the dimension of a selected reducing subspace constant. In fact, the algorithms can be told to enforce such constraints when computing eigenvalues and reducing subspaces. This leads to useful bounds presented in detail in §8.7 and [119].