Friday, February 7, 2014

Streamlines for the Transition Matrix

  One can use matrix methods to simplify the expression Pk 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,

 Pk = Κ + ρ[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.

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