In this note, we consider the problem of determining necessary and sufficient conditions for the existence of a common quadratic Lyapunov function for a pair of stable linear time-invariant systems whose system matrices are of the form A, A - gh(T), and where one of the matrices is singular. A necessary and sufficient condition for the existence of such a function is given in terms of the spectrum of the product A(A-gh(T)). The technical note also contains a spectral characterization of strictly positive real transfer functions which are strictly proper. Examples are presented to illustrate our results.