  The numerical solution of the generalized eigenvalue problem for a singular
matrix pencil is challenging due to the discontinuity of its eigenvalues.
Classically, such problems are addressed by first extracting the regular part
through the staircase form and then applying a standard solver, such as the QZ
algorithm, to that regular part. Recently, several novel approaches have been
proposed to transform the singular pencil into a regular pencil by relatively
simple randomized modifications. In this work, we analyze three such methods by
Hochstenbach, Mehl, and Plestenjak that modify, project, or augment the pencil
using random matrices. All three methods rely on the normal rank and do not
alter the finite eigenvalues of the original pencil. We show that the
eigenvalue condition numbers of the transformed pencils are unlikely to be much
larger than the $\delta$-weak eigenvalue condition numbers, introduced by Lotz
and Noferini, of the original pencil. This not only indicates favorable
numerical stability but also reconfirms that these condition numbers are a
reliable criterion for detecting simple finite eigenvalues. We also provide
evidence that, from a numerical stability perspective, the use of complex
instead of real random matrices is preferable even for real singular matrix
pencils and real eigenvalues. As a side result, we provide sharp left tail
bounds for a product of two independent random variables distributed with the
generalized beta distribution of the first kind or Kumaraswamy distribution.
