Distance Evaluation to the Set of Defective Matrices

Abstract

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 $$ \mathcal F(z) = 0, \ \mbox{where} \ \mathcal F(z):= \mathcal D_λ \left( \det \left[ (λI - A)(λI - A^{\top})-z I_n \right] \right)/z^n $$ 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.

arXiv comment28 pages, 1 figure