  In this paper, we find necessary and sufficient conditions to identify pairs
of matrices $X$ and $Y$ for which there exists $\Delta \in \mathbb C^{n,n}$
such that $\Delta+\Delta^*$ is positive semidefinite and $\Delta X=Y$. Such a
$\Delta$ is called a dissipative mapping taking $X$ to $Y$. We also provide two
different characterizations for the set of all dissipative mappings, and use
them to characterize the unique dissipative mapping with minimal Frobenius
norm. The minimal-norm dissipative mapping is then used to determine the
distance to asymptotic instability for dissipative-Hamiltonian systems under
general structure-preserving perturbations. We illustrate our results over some
numerical examples and compare them with those of Mehl, Mehrmann and Sharma
(Stability Radii for Linear Hamiltonian Systems with Dissipation Under
Structure-Preserving Perturbations, SIAM J. Mat. Anal. Appl.\ 37 (4):
1625-1654, 2016).
