Doubly structured mapping problems of the form $Δx=y$ and $Δ^*z=w$

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 $Δ=[Δ_1~Δ_2]$ with $Δ_1 \in \mathbb S$ and $Δ_2 \in \C^{n,m}$ such that $Δx=y$ and $Δ^*z=w$. We also characterize the set of all such mappings $Δ$ and provide sufficient conditions on vectors $x,y,z$, and $w$ to investigate a $Δ$ 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.

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