algorithms, Vm,n e S':P(m I n)> 0. Thus, by Definition Bl, p is primitive for any integer
k 2 1. Hence, from Section B.3, the stationary distribution of the two and three-operator
simple genetic algorithm exists, is unique and is a left eigenvector of the state transition
matrix corresponding to eigenvalue 1, i.e.
-.P -q
or equivalently
q~j(P- I) = Eq. 6.1
The following proposition establishes a significant fact concerning the rank of the
matrix (P-I) in Eq. 6.1.
Proposition 6.1: The rank of the matrix (P- 1) in Eq. 6.1 is exactly N'- 1 where
N' = card(S') is the dimension of P.
This result follows from Theorem B4(f). Its significance is that exactly one column
of the system of equations in Eq. 6.1 can be replaced without sacrificing any of the con-
straints which Eq. 6.1 imposes on q,. Proposition 6.2 below concerns such a modification
of the system. The modification consists of replacing any column (e.g. the column
indexed by n E S') of Eq. 6.1 by a column corresponding to the constraint
q,(m) = =1, Eq. 6.2
me S'
thus producing a system of the form
qi(P I = Eq. 6.3
where (P )n is generated by replacing the column of (P- ) indexed by n e S' with the
vector 1 whose components all have the value 1, and where J is the row vector contain-
ing 1 in column n and O's elsewhere.