A formula for the 2-norm distance from a matrix to the set of matrices with multiple eigenvalues

Abstract

We prove that the 2-norm distance from an n × n matrix A to the matrices that have a multiple eigenvalue λ is equal towhere the singular values σ k are ordered nonincreasingly. Therefore, the 2-norm distance from A to the set of matrices with multiple eigenvalues isMathematics Subject Classification (1991): 65F15, 65F35

Nonlinear Eigenvalue Problems with Specified Eigenvalues

“…In this section, we show Δ * 2 = κ r (μ) = σ −r (K(μ, Γ * , T )), which amounts to verifying UV + 2 = 1. As noted in [24,22,19], the latter property follows from the relation U * U = V * V, which we will establish under the multiplicity assumption.…”

“…In view of the result by Malyshev [24] for the linear case, we conjecture that (6.1) holds without requiring the multiplicity and linear independence assumptions. It is well known for the standard eigenvalue problem T (λ) = A − λI that the distance to a nearest matrix with a multiple eigenvalue corresponds to the smallest such that two components of the -pseudospectrum coalesce [3].…”

On pseudospectra of matrix polynomials and their boundaries

“…Then Theorems 18 and 19 imply the following result (see [2,13] for the standard eigenvalue problem). , which corresponds to a damped vibrating system.…”

“…(i) Suppose µ = 0, and recall (13). By virtue of Lemma 4, F ε (x, y) is real analytic in a neighbourhood of µ.…”

μ-Values and Spectral Value Sets for Linear Perturbation Classes Defined by a Scalar Product

“…Sinceμ for all H ; G ∈ HermðnÞ, k ¼ 1; : : : ; n. The proofs of Theorem 3.1 and 3.3 use the same technique as in [9], [17], [19], [22]. We need the following preliminary result on the eigenvalues and eigenvectors of a Hermitian pencil.…”