A unified approach to the transition matrices of DNA substitution models

For a reversible finite-state continuous-time Markov chain containing similar states, the computation of the transition matrix can be expressed quite elegantly in terms of the transition matrix of an associated lumped Markov chain. This result is immensely useful for obtaining explicit transition matrices for many DNA substitution models, without diagonalizing a matrix or solving a differential equation. Furthermore, the technique works for the analogous problem in the discrete-time DNA substitution models.

► Algebraic formula of transition matrices is presented for reversible Markov chain models of DNA substitution.
► The proof works through matrix algebra, not matrix diagonalisation.
► The same method applies to discrete-time Markov chains.