  Recent studies based on the notion of black hole pseudospectrum indicated
substantial instability of the fundamental and high-overtone quasinormal modes.
Besides its theoretical novelty, the details about the migration of the
quasinormal mode spectrum due to specific perturbations may furnish valuable
information on the properties of associated gravitational waves in a more
realistic context. This work generalizes the matrix method for black hole
quasinormal modes to cope with a specific class of perturbations to the metric
featured by discontinuity, which is known to be intimately connected with the
quasinormal mode structural instability. In practice, the presence of
discontinuity poses a difficulty so that many well-known approaches for
quasinormal modes cannot be straightforwardly applied. By comparing with other
methods, we show that the modified matrix method is efficient, which can be
used to solve for the low-lying modes with reasonable precision. Therefore, it
might serve as an alternative gadget for relevant studies.
