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For a given class of structured matrices $\mathbb S$, we find necessary andsufficient conditions on vectors $x,w\in \C^{n+m}$ and $y,z \in \C^{n}$ forwhich 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 alsocharacterize the set of all such mappings $\Delta$ and provide sufficientconditions on vectors $x,y,z$, and $w$ to investigate a $\Delta$ with minimalFrobenius 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, anddissipative matrices. These mappings are then used in computing the structuredeigenvalue/eigenpair backward errors of matrix pencils arising in optimalcontrol.

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