264
Appendix A: On the Independence of Certain Random Variables
write the eigenvalues λ A and λ B next to each other. For illustrative purposes we
chose p = 2 when fitting with A and q = 4 when fitting with B. Next to the eigenvalues we show the corresponding χ
2
p and χ
2
q , expressed in the new variables r k and
r
k , the components of r and r
, respectively
λ A = (0, 0, 1, 1, 1, 1, 1, . . . ) such that χ
2
p =
N
k= p+1
r
2
k ,
λ B = (0, 0, 0, 0, 1, 1, 1, . . . ) such that χ
2
q =
N
k=q+1
r
2
k .
(A.10)
Note that r k = r
k where the eigenvalues of P A and P B are unity, such that the difference between χ
2
p and χ
2
q is given by χ
2
p − χ
2
q =
q
k= p+1 r
2
k = r
2
3 + r
2
4 , whereas
χ
2
q =
N
k=q+1 r
2
k = r
2
5 + · · · + r
2
N . This shows that the random variables that make
up the numerator and denominator of (7.37) are independent.
Appendix A: On the Independence of Certain Random Variables
write the eigenvalues λ A and λ B next to each other. For illustrative purposes we
chose p = 2 when fitting with A and q = 4 when fitting with B. Next to the eigenvalues we show the corresponding χ
2
p and χ
2
q , expressed in the new variables r k and
r
k , the components of r and r
, respectively
λ A = (0, 0, 1, 1, 1, 1, 1, . . . ) such that χ
2
p =
N
k= p+1
r
2
k ,
λ B = (0, 0, 0, 0, 1, 1, 1, . . . ) such that χ
2
q =
N
k=q+1
r
2
k .
(A.10)
Note that r k = r
k where the eigenvalues of P A and P B are unity, such that the difference between χ
2
p and χ
2
q is given by χ
2
p − χ
2
q =
q
k= p+1 r
2
k = r
2
3 + r
2
4 , whereas
χ
2
q =
N
k=q+1 r
2
k = r
2
5 + · · · + r
2
N . This shows that the random variables that make
up the numerator and denominator of (7.37) are independent.