AbstractIn this article, we find necessary and sufficient conditions to identify pairs of matrices X and Y for which there exists such that is positive semidefinite and . Such a 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 Matrix Anal Appl, 37(4):1625–54, 2016).