  Given an appropriate class of structured matrices S; we characterize matrices
X and B for which there exists a matrix A \in S such that AX = B and determine
all matrices in S mapping X to B. We also determine all matrices in S mapping X
to B and having the smallest norm. We use these results to investigate
structured backward errors of approximate eigenpairs and approximate invariant
subspaces, and structured pseudospectra of structured matrices.
