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. 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. In this work we analyze these methods and 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 shows that these condition numbers are a reliable criterion for detecting 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.

翻译：奇异矩阵束的广义特征值问题因特征值的不连续性而具有挑战性。经典方法通常先通过阶梯形式提取正则部分，再应用标准求解器（如QZ算法）处理。近年来，多种通过相对简单的随机化修正将奇异矩阵束转化为正则矩阵束的新方法被提出。本文分析了Hochstenbach、Mehl和Plestenjak提出的三种方法，这些方法分别通过随机矩阵对矩阵束进行修正、投影或增广。所有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。我们通过分析表明，变换后矩阵束的特征值条件数不太可能显著大于Lotz与Noferini引入的原矩阵束的$\delta$-弱特征值条件数。这不仅证明了这些方法具有良好的数值稳定性，也表明该条件数是检测有限特征值的可靠判据。此外，我们从数值稳定性角度提供证据：即使对于实奇异矩阵束及实特征值，使用复随机矩阵也优于实随机矩阵。