The left characteristic matrix equation (lcme), introduced in earlier work, has proved to be a useful tool in the development of a feedback stabilization theory for delay systems. In this work, the dual notion of the right characteristic matrix equation (rcme) of a delay system is introduced via the state feedback stabilization of a dual system. As an application of the rcme, an observer theory is developed for spectrally detectable delay systems using well-established finite dimensional system tools. Employing the resulting observer to generate the state feedback controls of the above-mentioned stabilization theory, it is shown that the separation property holds. This work thus renders the state feedback stabilization theory implementable by input-output data.