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的著名论调表明如何用一个一级更新来修改一个矩阵的单一电子值,而不改变其余的电子值。该理论的概括化(由于R. Rado)被用来用结构保护的方式,将一个共振矩阵S的一对电子值改变为理想的目标值。给出了S与新建立的带有修改频谱的静脉矩阵S之间的相对距离的通用界限。S的S电子电子值特性与S有关,并得出了S的静脉值条件号的说明。主要结果扩展至矩阵铅笔。