Structured eigenvalue/eigenvector backward errors of matrix pencils arising in optimal control

TL;DR

This paper develops formulas for backward errors of eigenvalues and eigenpairs in matrix pencils from optimal control, emphasizing structure-preserving perturbations and highlighting cases where these errors exceed unstructured ones.

Contribution

It introduces new formulas for structured backward errors of eigenvalues/eigenvectors in matrix pencils relevant to optimal control, considering block and symmetry structures.

Findings

Structured backward errors can be significantly larger than unstructured errors.

Formulas are derived for block-structure-preserving perturbations.

Symmetry-preserving perturbations are also analyzed.

Abstract

Eigenvalue and eigenpair backward errors are computed for matrix pencils arising in optimal control. In particular, formulas for backward errors are developed that are obtained under block-structure-preserving and symmetry-structure-preserving perturbations. It is shown that these eigenvalue and eigenpair backward errors are sometimes significantly larger than the corresponding backward errors that are obtained under perturbations that ignore the special structure of the pencil.