Given a square complex matrix $A$, we tackle the problem of finding the nearest matrix with multiple eigenvalues or, equivalently when $A$ had distinct eigenvalues, the nearest defective matrix. To this goal, we extend the general framework described in [M. Gnazzo, V. Noferini, L. Nyman, F. Poloni, \emph{Riemann-Oracle: A general-purpose Riemannian optimizer to solve nearness problems in matrix theory}, Found. Comput. Math., To appear] and based on variable projection and Riemannian optimization, allowing the ambient manifold to simultaneously track left and right eigenvectors. Our method also allows us to impose arbitrary complex-linear constraints on either the perturbation or the perturbed matrix; this can be useful to study structured eigenvalue condition numbers. We present numerical experiments, comparing with preexisting algorithms.

翻译：给定一个复方阵$A$，我们研究如何寻找具有多重特征值的最近邻矩阵，或者等价地，当$A$具有互异特征值时，寻找最近邻的亏损矩阵。为实现这一目标，我们扩展了[M. Gnazzo, V. Noferini, L. Nyman, F. Poloni, \emph{Riemann-Oracle: A general-purpose Riemannian optimizer to solve nearness problems in matrix theory}, Found. Comput. Math., 已录用]中描述的基于变量投影与黎曼优化的通用框架，使环境流形能够同时追踪左、右特征向量。我们的方法还允许对摄动矩阵或摄动后的矩阵施加任意复线性约束，这对于研究结构化特征值条件数具有重要意义。我们通过数值实验与现有算法进行了比较。