A wide class of matrix pencils connected with dissipative Hamiltonian descriptor systems is investigated. In particular, the following properties are shown: all eigenvalues are in the closed left half plane, the nonzero finite eigenvalues on the imaginary axis are semisimple, the index is at most two, and there are restrictions for the possible left and right minimal indices. For the case that the eigenvalue zero is not semisimple, a structurepreserving method is presented that perturbs the given system into a Lyapunov stable system.

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Let A be a fixed complex matrix and let u, v be two vectors. The eigenvalues of matrices A + τ uv ⊤ (τ ∈ R) form a system of intersecting curves. The dependence of the intersections on the vectors u, v is studied.In other words, for each eigenvalue λ j (j = 1, . . . r) only the largest chain in the Jordan structure is destroyed and there appears a structure of simple eigenvalues instead.