On the Fixed Point Probability Vector of Regular or Ergodic Transition Matrices

A simple form for the fixed point probability row vector of regular or ergodic transition matrices is developed using some of the theory of generalized matrix inversion, together with the well known existence and uniqueness of a fixed point probability row vector. The forms for both the regular and ergodic case are identical and afford the easy calculation of the fixed point without calculating powers of the transition matrix or considering the usual convergence methods.