Structured Pseudospectra for Small Perturbations

Abstract

Abstract. In this paper we study the shape and growth of structured pseudospectra for small matrix perturbations of the formIt is shown that the properly scaled pseudospectra components converge to nontrivial limit sets as δ tends to 0. We discuss the relationship of these limit sets with μ-values and structured eigenvalue condition numbers for multiple eigenvalues.

On the computation of structured singular values and pseudospectra

“…Indeed, a closer inspection of the real pseudospectrum reveals that for smaller ǫ the eigenvalue −1 is still moved to almost the same extent but its movement is mainly confined to the six spikes at −1. This observation agrees well with the fact that the shape of real pseudospectra for Jordan blocks converges to spikes as ε → 0 (Karow 2008;Chaitin-Chatelin and Frayssé 1996).…”

Eigenvalues of rank one perturbations of unstructured matrices

“…with the parameter t > 0. It should be noted that the sets σ(A, u, v; t) are strongly related to the pseudospectral sets as introduced in e.g., [16], Definition 2.1. In fact they can be viewed as the boundaries of pseudospectral sets for the special case of rank one perturbations.…”

“…We mention the classical works of Lidskii [22], Vishik and Lyusternik [39], as well as the more general treatment of eigenvalues of perturbations of the matrix in the books by Kato [17] and Baumgärtel [3]. Recently, Moro, Burke and Overton returned to the results of Lidskii in a more detailed analysis [31], while Karow obtained a detailed analysis of the situation for small values of the parameter [16] in terms of structured pseudospectra. Obviously, parametric perturbations appear in many different contexts.…”

Computing Extremal Points of Symplectic Pseudospectra and Solving Symplectic Matrix Nearness Problems

“…where M = C n×n . In recent years there has been substantial interest in structured pseudospectra [KKK10, Kar11,Ru06], where A is in some linear subspace M ⊂ C n×n and the perturbed matrices B in (1.1) are restricted to lie in the same subspace M. Cases of particular interest include M = R n×n (real pseudospectra [BRQ98, GL13,Ru06]) and M = Ham(n), the space of complex or real Hamiltonian matrices (Hamiltonian pseudospectra [ABKMM11,GKL13]). There are, however, also structures of interest where M ⊂ C n×n is not a linear space, but a matrix Lie group or, more generally, a matrix manifold.…”