Szalai, Stépán, and Hogan [SIAM J. Sci. Comput., 28 (2006), pp. 1301–1317] gave a general construction for characteristic matrices for systems of linear delay differential equations with periodic coefficients. First, we show that matrices constructed in this way can have a discrete set of poles in the complex plane, which may possibly obstruct their use when determining the stability of the linear system. Then we modify and generalize the original construction such that the poles get pushed into a small neighborhood of the origin of the complex plane.