9.9. The Independence of Certain Quadratic Forms 565SinceQ 2 ≥0, each of the matricesA,A 1 ,andA 2 is positive semidefinite. Because
A^2 =A, we can find an orthogonal matrix Γ such that
Γ′AΓ=[
Ir O
OO]
.If we multiply both members ofA=A 1 +A 2 on the left byΓ′andontherightby
Γ,wehave [
Ir 0
00]
=Γ′A 1 Γ+Γ′A 2 Γ.Now each ofA 1 andA 2 , and hence each ofΓ′A 1 ΓandΓ′A 2 Γis positive semidefi-
nite. Recall that if a real symmetric matrix is positive semidefinite, each element on
the principal diagonal is positive or zero. Moreover, if an element on the principal
diagonal is zero, then all elements in that row and all elements in that column are
zero. ThusΓ′AΓ=Γ′A 1 Γ+Γ′A 2 Γcanbewrittenas
[
Ir 0
00
]
=[
Gr 0
00]
+[
Hr 0
00]. (9.9.13)
SinceA^21 =A 1 ,wehave
(Γ′A 1 Γ)^2 =Γ′A 1 Γ=[
Gr 0
00]
.If we multiply both members of Equation (9.9.13) on the left by the matrixΓ′A 1 Γ,
we see that [
Gr 0
00]
=[
Gr 0
00]
+[
GrHr 0
00]or, equivalently,Γ′A 1 Γ=Γ′A 1 Γ+(Γ′A 1 Γ)(Γ′A 2 Γ). Thus (Γ′A 1 Γ)×(Γ′A 2 Γ)= 0
andA 1 A 2 = 0. In accordance with Theorem 9.9.1,Q 1 andQ 2 are independent.
This independence immediately implies thatQ 2 /σ^2 isχ^2 (r 2 =r−r 1 ). This com-
pletes the proof whenk=2. Fork>2, the proof may be made by induction. We
shall merely indicate how this can be done by usingk=3. TakeA=A 1 +A 2 +A 3 ,
whereA^2 =A,A^21 =A 1 ,A^22 =A 2 ,andA 3 is positive semidefinite. Write
A=A 1 +(A 2 +A 3 )=A 1 +B 1 , say. NowA^2 =A,A^21 =A 1 ,andB 1 is
positive semidefinite. In accordance with the case ofk=2,wehaveA 1 B 1 = 0 ,
so thatB^21 =B 1 .WithB 1 =A 2 +A 3 ,whereB^21 =B 1 ,A^22 =A 2 , it follows
from the case ofk=2thatA 2 A 3 = 0 andA^23 =A 3. If we regroup by writing
A=A 2 +(A 1 +A 3 ), we obtainA 1 A 3 = 0 ,andsoon.
Remark 9.9.3.In our statement of Theorem 9.9.2, we tookX 1 ,X 2 ,...,Xnto
be observations of a random sample from a distribution that isN(0,σ^2 ). We did
this because our proof of Theorem 9.9.1 was restricted to that case. In fact, if
Q′,Q′ 1 ,...,Q′kare quadratic forms in any normal variables (including multivariate
normal variables), ifQ′=Q′ 1 +···+Q′k,ifQ′,Q′ 1 ,...,Q′k− 1 are central or noncentral
chi-square, and ifQ′kis nonnegative, thenQ′ 1 ,...,Q′kare independent andQ′kis
either central or noncentral chi-square.