In Parts I and II of this series of papers, three new methods for the computation of eigenvalues of singular pencils were developed: rank-completing perturbations, rank-projections, and augmentation. It was observed that a straightforward structure-preserving adaption for symmetric pencils was not possible and it was left as an open question how to address this challenge. In this Part III, it is shown how the observed issue can be circumvented by using Hermitian perturbations. This leads to structure-preserving analogues of the three techniques from Parts I and II for Hermitian pencils (including real symmetric pencils) as well as for related structures. It is an important feature of these methods that the sign characteristic of the given pencil is preserved. As an application, it is shown that the resulting methods can be used to solve systems of bivariate polynomials.

翻译：在本系列论文的第一和第二部分中，针对奇异矩阵束的特征值计算，我们提出了三种新方法：秩完备摄动、秩投影和增广。我们注意到，对于对称矩阵束，无法直接进行结构保持的适配，如何应对这一挑战被留作一个开放性问题。在这第三部分中，我们证明了如何通过使用埃尔米特摄动来规避所观察到的问题。这为埃尔米特矩阵束（包括实对称矩阵束）以及相关结构，导出了第一和第二部分中三种技术的结构保持类比。这些方法的一个重要特性是能够保持给定矩阵束的符号特征。作为一个应用，我们证明了所得方法可用于求解二元多项式方程组。