An extension of the method used to find transition matrices was used to find one for a 3 state system. Since the sum of a probability vector is 1 all possible probability vectors lie on a plane going through the unit position of the axes that represent the states of the system. Values were selected for the reduced matrix so that the initial changes were along the lines connecting the unit vectors of the axes of the coordinate system. The transition matrix found was:
When the unit vectors for the axes are taken as the initial states of the system the action of the transition matrix produces a set of points that spiral inwards to the stationary point (1/3,1/3,1/3).
To get a better picture of what the transition matrix does one can select a set of points along the border of the plane for the probability vectors and apply the transition matrix one step at a time. In this case the set of points spirals in towards the center maintaining their relative positions along the way. It might help to repeatedly click on the beginning of the progress bar while playing to restart the video.