A famous theorem by R. Brauer shows how to modify a single eigenvalue of a matrix by a rank-one update without changing the remaining eigenvalues. A generalization of this theorem (due to R. Rado) is used to change a pair of eigenvalues of a symplectic matrix S in a structure-preserving way to desired target values. Universal bounds on the relative distance between S and the newly constructed symplectic matrix S' with modified spectrum are given. The eigenvalues Segre characteristics of S' are related to those of S and a statement on the eigenvalue condition numbers of S' is derived. The main results are extended to matrix pencils.

翻译：R. Brauer的一个著名定理展示了如何通过秩一更新来修改矩阵的单个特征值，同时保持其余特征值不变。该定理的推广（归功于R. Rado）被用于以保结构的方式将辛矩阵S的一对特征值修改为期望的目标值。本文给出了S与修正谱后新构造的辛矩阵S'之间相对距离的通用界限。将S'的Segre特征多项式与S的相应特征多项式相关联，并推导了关于S'特征值条件数的结论。主要结果被推广至矩阵束。