Doubly structured mapping problems of the form $\Delta x=y$ and $\Delta^*z=w$

TL;DR

This paper characterizes the existence and structure of certain structured matrix mappings that satisfy specific linear equations, with applications in computing backward errors of matrix pencils in control theory.

Contribution

It provides necessary and sufficient conditions for structured matrices to solve specific linear equations and characterizes all such solutions, including minimal norm solutions.

Findings

Conditions for existence of structured solutions to $ ext{Delta} x=y$ and $ ext{Delta}^* z=w$

Characterization of all solutions and minimal Frobenius norm solutions

Application to backward error analysis in control-related matrix pencils

Abstract

For a given class of structured matrices $\mathbb S$, we find necessary and sufficient conditions on vectors $x,w\in \C^{n+m}$ and $y,z \in \C^{n}$ for which there exists $\Delta=[\Delta_1~\Delta_2]$ with $\Delta_1 \in \mathbb S$ and $\Delta_2 \in \C^{n,m}$ such that $\Delta x=y$ and $\Delta^*z=w$. We also characterize the set of all such mappings $\Delta$ and provide sufficient conditions on vectors $x,y,z$, and $w$ to investigate a $\Delta$ with minimal Frobenius norm. The structured classes $\mathbb S$ we consider include (skew)-Hermitian, (skew)-symmetric, pseudo(skew)-symmetric, $J$-(skew)-symmetric, pseudo(skew)-Hermitian, positive (semi)definite, and dissipative matrices. These mappings are then used in computing the structured eigenvalue/eigenpair backward errors of matrix pencils arising in optimal control.