The paper deals with linear systems of differential equationswith symmetric system matrices M,D, and K.The mass matrix M and the stiffness matrix K are both assumed to bepositive definite. The damping matrix D is indefinite. Three questionsare of interest: 1) When is the system unstable? Apparently not always,if the matrix D is indefinite. 2) What can we say about conditions whichensure that an unstable system can be stabilized by adding a gyroscopicterm Gdx/dt? 3) What is, in this case, a suitable or optimal matrixG? The questions are answered in the frame of a first order perturbationapproach.