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 determinantAn=|φm|nis

a.homogeneous of degreenin its elements and

b.homogeneous of degreen(n−1)in the suffixes of its elements.

Proof. Each of then! terms in the expansion ofAnis of the form

±φ1+k 1 − 2 φ2+k 2 − 2 ···φn+kn− 2 ,

where{kr}

n

1 is a permutation of{r}

n

1. The number of factors in each term

isn, which proves (a). The sum of the suffixes in each term is

n

∑

r=1

(r+kr−2)=2

n

∑

r=1

r− 2 n

=n(n−1),

which is independent of the choice of{kr}
n
1
, that is, the sum is the same

for each term, which proves (b).

Exercise.Prove thatA

(n)

ij is homogeneous of degree (n−1) in its elements

and homogeneous of degree (n

2

−n+2−i−j) in the suffixes of its elements.

Prove also that the scaled cofactorA

ij

nis homogeneous of degree (−1) in

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

∑

m=1

φm+r− 2 A

ms

n =δrs,^1 ≤r, s≤n. (4.8.16)

Exercise.Prove that, in addition to the sum formula,

a.

n

∑

m=1

φm+n− 1 A

(n)

im

=−A

(n+1)

i,n+1

, 1 ≤i≤n,

b.

n

∑

m=1

φm+nA

(n)

im

=A

(n+1)

1 n

,

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.