We treat the problem of the Frobenius distance evaluation from a given matrix $ A \in \mathbb R^{n\times n} $ with distinct eigenvalues to the manifold of matrices with multiple eigenvalues. On restricting considerations to the rank $ 1 $ real perturbation matrices, we prove that the distance in question equals $ \sqrt{z_{\ast}} $ where $ z_{\ast} $ is a positive (generically, the least positive) zero of the algebraic equation $$ \mathcal F(z) = 0, \ \mbox{where} \ \mathcal F(z):= \mathcal D_{\lambda} \left( \det \left[ (\lambda I - A)(\lambda I - A^{\top})-z I_n \right] \right)/z^n $$ and $ \mathcal D_{\lambda} $ stands for the discriminant of the polynomial treated with respect to $\lambda $. In the framework of this approach we also provide the procedure for finding the nearest to $ A $ matrix with multiple eigenvalue. Generalization of the problem to the case of complex perturbations is also discussed. Several examples are presented clarifying the computational aspects of the approach.

翻译：本文研究了给定具有不同特征值的矩阵 $A \in \mathbb R^{n\times n}$ 到多重特征值矩阵流形的Frobenius距离评估问题。通过将考量限制在秩为1的实扰动矩阵上，我们证明了该距离等于 $\sqrt{z_{\ast}}$，其中 $z_{\ast}$ 是代数方程 $$ \mathcal F(z) = 0, \ \text{其中} \ \mathcal F(z):= \mathcal D_{\lambda} \left( \det \left[ (\lambda I - A)(\lambda I - A^{\top})-z I_n \right] \right)/z^n $$ 的一个正零点（通常为正的最小零点），而 $\mathcal D_{\lambda}$ 表示关于 $\lambda$ 的多项式的判别式。在此方法框架下，我们还给出了寻找与 $A$ 最近且具有多重特征值矩阵的步骤。同时讨论了该问题向复数扰动情形的推广。通过若干算例阐明了该方法的计算特性。