Convergence of a sequence of powers
Ralph E. DeMarr · Proceedings of the American Mathematical Society · 1969
A well-known theorem states that if a stochastic matrix (definition below) of finite order has all positive entries in it, then the sequence of its powers (or iterates) converges to a limit; see [3, p. 173]. In this paper we will give a new proof of this result using elementary ideas from the theory of partially ordered linear algebras. Our proof does not use the internal structure of the given matrix; therefore, it can be applied to nonnegative operators. The basic definition of a partially ordered linear algebra (pola) is as follows. A pola A is first of all a linear algebra with real numbers as scalars. Real numbers will usually be denoted by small Greek letters. Multiplication of elements of A is assumed to be associative, but not necessarily commutative. Next, the linear algebra A is a partially ordered set subject to the following conditions (x, y, z denote arbitrary elements of A and a denotes an arbitrary real number under the specified restrictions in each condition): (a) if x_y, then x+z 0 such that x =y-z. We may also introduce a form of order completeness described as follows. The pola A is said to be Dedekind a-complete if it satisfies the following condition: if {x Xn} is a sequence of elements from A such that xI > X2 >* * > 0, then inf { Xn } exists. See [4, pp. 9-11 ]. Of course, inf{ x} denotes the infinum (greatest lower bound) of the sequence {xn}. It is defined as follows: inf { xn} =x means that (1) x z2 > . . . >0, inf {Zn} =0, and -Zn?yY-y 0