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.

翻译：奇异矩阵束广义特征值问题的数值求解因其特征值的不连续性而具有挑战性。经典方法通过阶梯形式提取正则部分，再对正则部分应用标准求解器（如QZ算法）来处理此类问题。近期，有学者提出通过相对简单的随机修正将奇异矩阵束转化为正则矩阵束的新颖方法。本文分析了Hochstenbach、Mehl和Plestenjak提出的三种此类方法，这些方法通过修改、投影或扩增矩阵束并借助随机矩阵实现。三种方法均依赖于正常秩，且不改变原矩阵束的有限特征值。我们证明，转换后矩阵束的特征值条件数不太可能远大于Lotz和Noferini提出的原矩阵束的$\delta$-弱特征值条件数。这既表明算法具有良好的数值稳定性，也再次确认该条件数是检测简单有限特征值的可靠准则。我们还提供了数值稳定性角度的证据：即使对于实奇异矩阵束和实特征值，使用复随机矩阵也优于实随机矩阵。作为衍生结果，我们给出了服从第一类广义贝塔分布或Kumaraswamy分布的两个独立随机变量乘积的精确左尾界。