Quaternions and biquaternions for symmetric Markov-chain system analysis

In this conference, we use quaternions, biquaternions, and their related algebras similar to them to model symmetric Markov systems. With these algebras, the original N×N Markov system can be reduced into an (N/2)×(N/2) or (N/4)×(N/4) system. It makes the system easier for implementing and analysis. In addition to Markov chains, the proposed idea is also helpful for simplifying the complexities of other symmetric systems whose interactions between two objects are determined by their distance.