Abstract
General properties of eigenvalues of A+tau uv^* as functions of tau in {mathbb {C} } or tau in {mathbb {R} } or tau ={{,mathrm{{e}},}}^{{{,mathrm{{i}},}}theta } on the unit circle are considered. In particular, the problem of existence of global analytic formulas for eigenvalues is addressed. Furthermore, the limits of eigenvalues with tau rightarrow infty are discussed in detail. The following classes of matrices are considered: complex (without additional structure), real (without additional structure), complex H-selfadjoint and real J-Hamiltonian.
Highlights
The eigenvalues of matrices of the form A + τ uv∗, viewed as a rank one parametric perturbation of the matrix A, have been discussed in a vast literature
Moro, Burke and Overton returned to the results of Lidskii in a more detailed analysis [32], while Karow obtained a detailed analysis of the situation for small values of the parameter [17] in terms of structured pseudospectra
As is well-known the problem does not occur in the case of Hermitian matrices where an analytic function of τ with Hermitian values has eigenvalues and eigenvectors which can be arranged such that they are analytic as functions of τ (Rellich’s theorem) [35]
Summary
The eigenvalues of matrices of the form A + τ uv∗, viewed as a rank one parametric perturbation of the matrix A, have been discussed in a vast literature. To understand the global properties with respect to the complex parameter τ we will consider parametric perturbations of two kinds: A + tuv∗, where t ∈ R, or A + ei θ uv∗, where θ ∈ [0, 2π ) The former case was investigated already in our earlier paper [34], we review the basic notions in Sect. Similar results can be found in the literature we have decided to provide a full description, for all possible ( generic) vectors u, v This is motivated by our research, where we apply these results to various classes of structured matrices. This idea came to us through multiple contacts and collaborations with Henk de Snoo (cf. in particular the line of papers on rank one perturbations [1,13,14,15,38,39]), for which we express our gratitude here
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