Determinants and Their Applications in Mathematical Physics

6.10 The Einstein and Ernst Equations 289

Proof. Multiply therth row ofAbyω
−r
,1≤r≤nand thesth column

byω

s

,1≤s≤n. The effect of these operations is to multiplyAby the

factor 1 and to multiply the elementarsbyω

s−r

. Hence, by (6.10.6),Ais

transformed intoBand the lemma is proved.

UnlikeA, which is real, the cofactors ofAare not all real. An example

is given in the following lemma.

Lemma 6.18.

A 1 n=ω

n− 1

B 1 n (ω

2

=−1).

Proof.

A 1 n=(−1)

n+1

|ers|n− 1 ,

where

ers=ar+1,s

=ω

|r−s+1|

u|r−s+1|

=ar,s− 1

and

B 1 n=(−1)

n+1

|βrs|n− 1 ,

where

βrs=br+1,s

=br,s− 1 ,

that is,

βrs=ω

s−r− 1

ers.

Multiply therth row ofA

(n)

1 n

byω

−r− 1

,1≤r≤n−1 and thesth column

byω

s

,1≤s≤n−1. The effect of these operations is to multiplyA

(n)

1 nby

the factor

ω

−(2+3+···+n)+(1+2+3+···+n−1)

=ω

1 −n

and to multiply the elementersbyω
s−r− 1

. The lemma follows.

BothAandBare persymmetric (Hankel) about their secondary diag-

onals. However,Ais also symmetric about its principal diagonal, whereas

Bis neither symmetric nor skew-symmetric about its principal diagonal.

In the analysis which follows, advantage has been taken of the fact thatA

with its complex elements possesses a higher degree of symmetry thanB

with its real elements. The expected complicated analysis has been avoided

by replacingBand its cofactors byAand its cofactors.