C1,' + 0(a)/(1 + )N'-N+
AG QA' +O(a)/(1+a)o*'-N+'
O(a)/(l + +a)N'-N +
SI' Q + O(a)/( +a)ON-N+'
nA e SA'
m = nmAe SA'
me S'- SA
An equivalent result expressed in terms of the auxiliary matrices P- is
Pj I' + 0(a)/(1 + a)N'-N+1
lE A I' + O(a/( + a)"'-"' m=mA
q(m) -= A
O(ap/(1 + (a)'-(N+-' + I
imE SE -
C P ,' + O(a)/(l+ a)N'-N+'
nAE SA
SA
SA
Eq. 7.9
By retracing the preceding steps by which P', was transformed into PnA compan-
ion results to Eq. 7.7 and Eq. 7.9 can be developed for P(mA)'. The companion to Eq. 7.7
differs only in the elements of row n = mA. Thus, if P(m I n)' denotes the general element
in P(mA)', then
Eq. 7.10
P(m I n)
Vm, ne
S S.'+P(m I n)' =
S'-S '+mI
me S'- SA'
+ {mA + S(A)' SA"
1 me S(nA)'
P(m I n)+-P(n I n)
L nA e SA'- ImA
Further, examination of Eq. 7.10 reveals that the row sum constraint on P is preserved in
the transformation by which P(mA)' is generated (i.e. P(mA)' is a stochastic matrix). Thus,
I P(m)' I'| = 0. A consequence is the Proposition 6.8 counterpart of Eq. 7.9,
Eq. 7.11
SP( nA)' I P I' + O(a)/(l + a)N'-N+1
qa(m)= nE s- +
I { P(nA)'-I' A I'll +0(a)/(l +a)N'-N+
"A e SA
m=m E SA'
me S SA'