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