  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.
