The semigroup of real stochastic matrices and generalized inversion

The theory of generalized inversion plays an important role in numerical analysis, least-squares theory, statistical estimation, network analysis, and many other areas of pure and applied mathematics. In this dissertation properties of the generalized inverse of a real stochastic matrix (nonnegativity removed) are developed, and applications are made to the locally compact affine semigroup of the real stochastic matrices. Necessary and sufficient conditions under which the generalized inverse of a real stochastic matrix is real stochastic are established. Additional results concerning the Brazen inverse, Brazen ordering and group inverse of the real stochastic matrices are included. The set Tn consisting of In (the nxn identity) and all elements of the real stochastic semigroup which do not belong to the core of any idempotent is characterized as those matrices of the form B [equals] In 4 [plus] A, where Ae [equals] 6 and AA[plus]e [equals] e. Other related results and unsolved problems are stated.