Eigenvalues of rank one perturbations of unstructured matrices

Abstract

Let A be a fixed complex matrix and let u, v be two vectors. The eigenvalues of matrices A + τ uv ⊤ (τ ∈ R) form a system of intersecting curves. The dependence of the intersections on the vectors u, v is studied.In other words, for each eigenvalue λ j (j = 1, . . . r) only the largest chain in the Jordan structure is destroyed and there appears a structure of simple eigenvalues instead.

Eigenvalue Placement for Regular Matrix Pencils with Rank One Perturbations

“…The spectral theory of matrix pencils is a generalization of the eigenvalue problem for matrices [17,26,31,35]. Recently, there is a growing interest in the spectral behavior under low rank perturbations of matrices [2,12,27,30,33,34] and of matrix pencils [1,13,14,15,29]. For matrices it was shown in [12] that under generic rank one perturbations only the largest Jordan chain at each eigenvalue might be destroyed.…”

Matrix methods for Padé approximation: Numerical calculation of poles, zeros and residues

“…Remark 17. Although it is not stated above, one may easily show using the results of [26] that the curves λ j (τ ) are analytic and do not intersect each other. Therefore, Theorem 16 provides a method of joining the zeros z n j of Q n (w) with the zeros λ n j of P n (w) (an one with infinity) with analytic curves.…”

“…The proof of statement (i) is a standard reasoning based on the Rouche theorem, see e.g. [26,Theorem 4.1]. Statement (ii) follows from the fact that λ n j (τ ) (τ ≥ 0) satisfies the equation…”

Rank two perturbations of matrices and operators and operator model for t-transformation of probability measures

“…In particular, we will be interested in large parameter limits of the spectrum. We extend here some ideas from [28] for finding the limits of rank one perturbations to the rank two case. However, we will refrain from investigating the generic Jordan structure of rank two perturbations, as this problem was addressed in the paper [3].…”

“…We recall that several papers have studied the topic of rank one perturbations of matrices (e.g. [26,21,22,28,31,30,32]) and operators (e.g. [10,11,33,34,39]), also rank k perturbations of matrices were recently considered in [3,38].…”