In this article, we find necessary and sufficient conditions to identify pairs of matrices X and Y for which there exists Δ ∈ C n,n such that Δ + Δ * is positive semidefinite and ΔX = Y . Such a Δ 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,

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