Distance Evaluation to the Set of Defective Matrices

03/13/2023

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by   Alexei Yu. Uteshev, et al.

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We treat the problem of the Frobenius distance evaluation from a given matrix A ∈ℝ^n× 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 √(z_∗) where z_∗ is a positive (generically, the least positive) zero of the algebraic equation ℱ(z) = 0, ℱ(z):= 𝒟_λ( [ (λ I - A)(λ I - A^⊤)-z I_n ] )/z^n and 𝒟_λ stands for the discriminant of the polynomial treated with respect to λ. 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.