Doubly structured mapping problems of the form $\hat{I}”x=y$ and $\hat{I}{”}^{\ast }z=w$

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 $\hat{I}”=[\hat{I}{”}_1\hat{I}{”}_2]$ with $\hat{I}{”}_1\in \mathbb{S}$ and $Δ_2 \in \C^{n,m}$ such that $\hat{I}”x=y$ and $\hat{I}{”}^{\ast }z=w$. We also characterize the set of all such mappings $\hat{I}”$ and provide sufficient conditions on vectors $x,y,z$, and $w$ to investigate a $\hat{I}”$ 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.