Page 123 - Determinants and Their Applications in Mathematical Physics

108 4. Particular Determinants

4.8.3 Two Kinds of Homogeneity
The definitions of a function which is homogeneous in its variables and of
a function which is homogeneous in the suffixes of its variables are given in
Appendix A.9.
Lemma. The determinant A n = |φ m | n is

a. homogeneous of degree n in its elements and
b. homogeneous of degree n(n − 1) in the suffixes of its elements.
Proof. Each of the n! terms in the expansion of A n is of the form

±φ 1+k 1 −2 φ 2+k 2 −2 ··· φ n+k n −2 ,
where {k r } is a permutation of {r} . The number of factors in each term
n
n
1 1
is n, which proves (a). The sum of the suffixes in each term is
n n

(r + k r − 2)=2 r − 2n
r=1 r=1
= n(n − 1),
which is independent of the choice of {k r } , that is, the sum is the same
n
1
for each term, which proves (b).
(n)
Exercise. Prove that A is homogeneous of degree (n−1) in its elements
ij
2
and homogeneous of degree (n −n+2−i−j) in the suffixes of its elements.
Prove also that the scaled cofactor A ij is homogeneous of degree (−1) in
n
its elements and homogeneous of degree (2 − i − j) in the suffixes of its
elements.
4.8.4 The Sum Formula
The sum formula for general determinants is given in Section 3.2.4. The
sum formula for Hankelians can be expressed in the form
n

φ m+r−2 A ms = δ rs , 1 ≤ r, s ≤ n. (4.8.16)
n
m=1
Exercise. Prove that, in addition to the sum formula,
n
(n) (n+1)
a. φ m+n−1 A = −A , 1 ≤ i ≤ n,
im i,n+1
m=1
n
(n) (n+1)
b. φ m+n A = A ,
im 1n
m=1
where the cofactors are unscaled. Show also that there exist further sums
of a similar nature which can be expressed as cofactors of determinants of
orders (n + 2) and above.