Determinants and Their Applications in Mathematical Physics

90 4. Particular Determinants

Whent 0 =1,tr=0,r>0,T 2 n=1,En=On,An= diag[1 0 0...0].

Hence,Pn=

1

2

,Qn= 1, and the sign ofT 2 nis positive, which proves part

(b) of the theorem.

The above theorem is applied in Section 6.10 on the Einstein and Ernst

equations.

Exercise.Prove that

T

(n)

12

=T

(n)

n− 1 ,n

=T

(n+1)

1 n;1,n+1

4.5.3 Skew-Centrosymmetric Determinants........

The determinantAn=|aij|nis said to be skew-centrosymmetric if

an+1−i,n+1−j=−aij.

InA 2 n+1, the element at the center, that is, in position (n+1,n+ 1), is

necessarily zero, but inA 2 n, no element is necessarily zero.

Exercises

1.Prove thatA 2 ncan be expressed as the product of two determinants of

ordernwhich can be written in the form (P+Q)(P−Q) and hence as

the difference between two squares.

2.Prove thatA 2 n+1 can be expressed as a determinant containing an

(n+1)×(n+ 1) block of zero elements and is therefore zero.

3.Prove that if the zero element at the center ofA 2 n+1is replaced byx,

thenA 2 n+1can be expressed in the formx(p+q)(p−q).

4.6 Hessenbergians

4.6.1 Definition and Recurrence Relation.........

The determinant

Hn=|aij|n,

whereaij= 0 wheni−j>1 or whenj−i>1 is known as a Hessenberg

determinant or simply a Hessenbergian. Ifaij = 0 wheni−j>1, the

Hessenbergian takes the form

Hn=

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

a 11 a 12 a 13 ··· a 1 ,n− 1 a 1 n

a 21 a 22 a 23 ··· a 2 ,n− 1 a 2 n

a 32 a 33 ··· ··· ···

a 43 ··· ··· ···

··· ··· ···

an− 1 ,n− 1 an− 1 ,n

an,n− 1 ann

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ n

. (4.6.1)