Using eigenvectors to predict state frequencies over time

This applet illustrates the application of a transition probability matrix to starting state frequency vectors in a 2-state, equal-rate continuous-time Markov model. The three 2-by-2 matrices result from diagonalization of the instantaneous rate matrix followed by exponentiation of the diagonal eigenvalue matrix. As the starting state frequency vector is multiplied by each matrix, the effect is shown graphically. Note that the effect of multiplying by eigenvector matrices is always to rotate either to or from the eigenspace, while the effect of multiplying by the (diagonal) eigenvalue matrix is to scale separately along each axis represented by an eigenvector. The end result is that the starting point moves closer to the equilibrium state frequencies (0.5, 0.5), indicated by the red dot.