  We discuss different cases of dissipative Hamiltonian differential-algebraic
equations and the linear algebraic systems that arise in their linearization or
discretization. For each case we give examples from practical applications. An
important feature of the linear algebraic systems is that the (non-Hermitian)
system matrix has a positive definite or semidefinite Hermitian part. In the
positive definite case we can solve the linear algebraic systems iteratively by
Krylov subspace methods based on efficient three-term recurrences. We
illustrate the performance of these iterative methods on several examples. The
semidefinite case can be challenging and requires additional techniques to deal
with "singular part", while the "positive definite part" can still be treated
with the three-term recurrence methods.
