Streamlines for the Transition Matrix

  One can use matrix methods to simplify the expression P^k so that one does not have to do a large number of matrix multiplications. The result which is a consequence of factoring P into a product of a rotation and a contraction is,

 P^k = Κ + ρ^k [Η cos(k·theta) + Γ sin(k·theta)]
where Κ, Η and Γ are a set of matrices defined below. The expression can also be used to compute "streamlines" for the transition matrix using fractional values for k.

Starting at some initial point p^<0> on the edge of the probability state space or anywhere else one can compute the steps p^<k> for the integers k and then compute the streamlines as shown above. To get the coordinates of a point, (x,y), in the plane one needs to subtract the origin e^<0>=(1,0,0)^T which is the vertex at the lower left from p^<k> before calculating the projections onto the x and y axes of the plane.

One can also replace the matrices in φ by their product with p^<0> to get a vector equation for an individual streamline.