Matrix classification for impact problems in layered elastic media

In this work, we develop and categorize the class of matrices representing a discrete dynamical system, previously studied in an impact problem in Goupillaud-type layered elastic media. As the stress wave reflects and transmits through the layers, the composition of the layered medium effects the system matrix entries, while preserving the property of each row summing to one. We identify in the physical context of Goupillaud-type media, necessary and sufficient conditions that lead to Markov, regular Markov and signed Markov system matrices, allowing negative entries in the latter. Our analytical results are supported by numerical examples. This work further contributes to the applications and construction of Markov matrices of any size in wave propagation in layered media.