We derive various properties of the operator matrix A = 0 I −A 0 −D , where A 0 is a uniformly positive operator and A −1/2 0 DA −1/2 0 is a bounded non-negative operator in a Hilbert space H. Such operator matrices are associated with second order problems of the form z(t) + A 0 z(t) + Dż(t) = 0 which are used as models for transverse motions of thin beams in the presence of damping.