K = K n= {n,,n2, ,nk,n)} S'), then
IP s. =IP I u=N'
A,(r)= -K P01 u=k+l Eq. 9.14
nA s'-K
IPI K u=k
Also, since every principal minor of I P2| is either identical to the corresponding principal
minor of I P I or 0 depending upon whether or not it includes row m, and since m e K by
hypothesis, it follows that the A (r)'s corresponding to Eq. 9.14 are
P = -=O0 u=N'
A (r)= s'-K=Ak+I,(r)li -I u=k+l Eq.9.15
n S'-K- {in}
I|FIK u=k
All of the N' k undifferentiated rows in p = I are identical (Eq. 9.7) and conse-
quently all minors of Pi = 1 of order u 2 k + 2 have value 0. Thus, in a fashion exactly
analogous to the derivation of Eq. 9.8 from Eq. 9.4, Eq. 9.13 yields
( P-| -|Pi ) {Ak(r)- A ()} +(-)N-k {Ak() A(r)},=
and substituting from Eq. 9.14 and 9.15 into this result produces
{(IP-I -|IP,-il|) )=,= (-1) '-- I Ak+()--[Ak(r)- IPK )}a=1 Eq. 9.16
= (-1)N'-k-I iPI
from which, by substitution into Eq. 9.11, it follows that
(-1)N''( -k- 1 K-l
C(m, r)= r! iil = 0 Eq. 9.17
0 otherwise
Evaluating C(m,r) thus requires evaluating the quotient of the order k + 1 principal minor
IP1 =1 and r!
The order k + 1 principal minor I PI I is completely determined by the
(k + 1) x (k + 1) sub-array of P given by IP(-w I v)] for w,ve K, where P(w | v)