In this note, 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 A, and A:, are in companion form is considered. It is shown that a necessary and sufficient condition for the existence of such a function is that the matrix product A(1) A(2) does not have an eigenvalue that is real and negative. Examples are presented to illustrate the result.