Vm, ne S': P(m I n)l, = (M= I i Eq. 9.7
m Ies 2
M1^2
Thus, Pi = = [P(m I n)],,= is a rank one matrix, and therefore all minors of I PI = 1 of
order u 2 2 are identically zero. Eq. 9.4 then reduces to
{| P-| | P-I } = {M(1)- (1)}
= (-1)N'{A-Al + (-1)N {Ao- A =1,
and substituting from Eq. 9.2, 9.5 and 9.7 into this result produces
{|P-I| -|PI,- } ,= = (-1)' {trace(P)- [trace(P)-P(m I m)]}f,=
(=(-)N'- P(m | m) =l Eq. 9.8
= (-1)N M
Employing Eq. 9.8 with Proposition 6.8 yields an explicit result for the a = I limit-
ing value of q(m), i.e.
IP-II -IP5-I|
q(m)l== = (I P-i I P|P- i[ )
Ie S' a=l
-(_)NS_ I M}2ML (M)
MM
E S'(--1)s'-l (M- 2" m sT
(M)
m 2ML'
It is independent of the objective function of the underlying optimization problem
because at pm = 1/2 = ac = 1 mutation completely nullifies the reproduction operator.
Although this trivial case is not of any particular interest on its own, it serves as the
basis for developing the general case 0 < a < 1 in Section 9.3. The essential idea is that P