Generalized eigenvalue problems with specified eigenvalues

Abstract

We consider the distance from a (square or rectangular) matrix pencil to the nearest matrix pencil in 2-norm that has a set of specified eigenvalues. We derive a singular value optimization characterization for this problem and illustrate its usefulness for two applications. First, the characterization yields a singular value formula for determining the nearest pencil whose eigenvalues lie in a specified region in the complex plane. For instance, this enables the numerical computation of the nearest stable descriptor system in control theory. Second, the characterization partially solves the problem posed in Boutry et al. (2005, SIAM J. Matrix Anal. Appl., 27, 582-601) regarding the distance from a general rectangular pencil to the nearest pencil with a complete set of eigenvalues. The involved singular value optimization problems are solved by means of Broyden-Fletcher-Goldfarb-Shanno and Lipschitz-based global optimization algorithms.

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.…”

“…Specifically, when S = {μ} and r = 1, this distance has the singular value characterization (1.3). The distance (1.4) extends our previous definitions for (generalized) linear and polynomial eigenvalue problems [22,19] to the nonlinear case. In these earlier works, we derived a singular value optimization characterization for τ r (S), which can be numerically solved using global optimization techniques, at least when r is small [26].…”

“…We have derived a further generalization of earlier works [22,19] for locating nearest analytic matrix-valued functions with specified eigenvalues. For this purpose, we have studied the structure of the nullspace for Sylvester-like operators associated with nonlinear eigenvalue problems.…”

“…As in the derivations for the specialized cases [22,19], which led us to consider this more general setting, we prove P r (μ) = κ r (μ), under mild multiplicity and linear independence assumptions, by constructing a perturbation Δ * such that (i) Δ * 2 = κ r (μ) and (ii) rank (K(μ, Γ, T + Δ * )) ≤ n · r − r for some Γ ∈ G(μ). As discussed in [19], the supremum in (3.2) is always attained at some Γ * , as long as r ≤ n and for generic values of μ.…”

“…For the rest of this paper, we assume that T : Ω → C n×n is regular; that is, its determinant z → det(T (z)) does not vanish identically on Ω. Following the developments in [22,19], we will approach the distance (1.4) by first characterizing the inequality…”