(Q) =IP- XI
= A.- AN'._ + AN._-22- AN_ 3 Eq. 9.1
+... + (-1)N'-AN'-I + (_)N'A0N'
= Y(-1) N' -uAN'-u
U
where N' = card(S') is the dimension of P and A, is the sum of its order u principal
minors. This result is fundamental to the theory of square matrices and follows from
application of elementary determinant expansion operations to I P- II [Aitk54, MoSt64,
Muir60]. Exactly
(N') N'!
lu =u!(N' -u)!
order u principal minors are summed to produce Au. The values of some of the Au's are
I iP u=N'
A,= trace(P) u=l Eq. 9.2
1 u=0
where the Ao result follows from the convention that the single order zero principal minor
of I X has value 1.
In a fashion exactly analogous to Eq. 9.1, the characteristic polynomial of Pn can be
written as
OIMA) = I P-ii XII
= A.-A._lX + A, 2 2-A ._3N3 Eq. 9.3
+.. +(--1)N'- iA N'+(-1N' A N'
(-l)N'-WuA N'- u
= "(-1) -A:'
U
where Am is the sum of all order u principal minors of I P~I. Thus, the value of each deter-
minant required for expressing q via Propositions 6.7-8 can be written respectively in the
form