100
Since p is independent of the P(i I m)'s associated with row m (row m of p. is set
to 0 ), it follows that no monomial term containing any of the factors (P(i I m) 1/2 )
appears in the expansion of P II. Further, since I 1 II 1Pj- I = -| P II, no such
monomial terms appear with nonzero coefficient in the expansion of I P-I| -I PE-I1
either. (Due to the constraint Vin S' : Y P(i I n) = 1, the aggregate of such terms
ie S
appearing in | P- I is identically zero). This observation establishes the l| rIl = 0 bound
and permits the following revision of the Eq. 9.10 definition of C(m,r)
(IP-11 -|PE-Il) | l
C(m,r)= r! rII 1 = 0 Eq. 9.11
0 otherwise
The derivative of an order n determinant with respect to a variable x can be written
as the sum of the n determinants generated by differentiating each row (or column) in
turn with respect to x [Aitk54, MoSt64, Muir60]. For example, if
Sal a,2]
A= a,2
then
d|l da,, da12 a, a12
d dx dx + da21 da22
dx
a21 a22 dx dx
If the elements of any row in the given determinant are independent of x, then dif-
ferentiation of that row introduces an all zero row and the value of the corresponding
determinant is zero. In particular, if only one row of the given determinant depends upon
x, then only one nonzero determinant appears in the row-derivative expansion.
Higher order and mixed partial derivatives of an order n determinant can be
expressed similarly, e.g.