The transitions are his jumps. We also define the transition matrix P ij . An example of a state-transition diagram. The diagram corresponds to the probability matrix Since at each time a transition must occur, we have that the sum of the elements in a column must be unity. In other words, the row vector is an eigenvector of P with unit eigenvalue.
P times an initial-state vector.
There will be values of Z for which the inverse matrix to does not exist. Z 1 is always unity. In our particular example, one can see by inspection that the steady-state probability vector is so the general solution is of the form Finally, a word of caution must be added. It is the same situation as that applying to ordinary differential equations with constant coefficients.