Distance Evaluation to the Set of Defective Matrices

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 $$\mathcal{F}(z)=0,where\mathcal{F}(z):={\mathcal{D}}_{\lambda }(\det [(\lambda I-A)(\lambda I-A^{\mathrm{\top }})-z{I}_n])\mathrm{/}{z}^n$$ 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.