Generalized Eigenvalue Problems with Specified Eigenvalues

TL;DR

This paper introduces a singular value optimization approach to measure the distance from a matrix pencil to the nearest pencil with specified eigenvalues, with applications in control theory and eigenvalue problems.

Contribution

It provides a novel singular value formula for eigenvalue-constrained pencils and addresses the problem of finding the nearest pencil with a complete eigenvalue set.

Findings

Derived a singular value optimization characterization.

Enabled computation of the nearest stable descriptor system.

Partially solved the eigenvalue set distance problem.

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] 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 BFGS and Lipschitz-based global optimization algorithms.