Page 105 - Determinants and Their Applications in Mathematical Physics

90 4. Particular Determinants

When t 0 =1, t r =0, r> 0, T 2n =1, E n = O n , A n = diag[1 0 0 ... 0].
1
Hence, P n = , Q n = 1, and the sign of T 2n is positive, which proves part
2
(b) of the theorem.
The above theorem is applied in Section 6.10 on the Einstein and Ernst
equations.

Exercise. Prove that
(n) (n) (n+1)
T 12 = T n−1,n = T 1n;1,n+1 .

4.5.3 Skew-Centrosymmetric Determinants

The determinant A n = |a ij | n is said to be skew-centrosymmetric if
a n+1−i,n+1−j = −a ij .
In A 2n+1 , the element at the center, that is, in position (n +1,n + 1), is
necessarily zero, but in A 2n , no element is necessarily zero.

Exercises
1. Prove that A 2n can be expressed as the product of two determinants of
order n which can be written in the form (P + Q)(P − Q) and hence as
the difference between two squares.
2. Prove that A 2n+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 of A 2n+1 is replaced by x,
then A 2n+1 can be expressed in the form x(p + q)(p − q).

4.6 Hessenbergians

4.6.1 Definition and Recurrence Relation
The determinant

H n = |a ij | n ,
where a ij = 0 when i − j> 1 or when j − i> 1 is known as a Hessenberg
determinant or simply a Hessenbergian. If a ij = 0 when i − j> 1, the
Hessenbergian takes the form

a 11 a 12 a 13 ··· a 1,n−1 a 1n

a 21 a 22 a 23 ··· a 2,n−1 a 2n

a 32 a 33 ··· ···
···

a 43 ··· ··· ··· . (4.6.1)
H n =

··· ···
···

a n−1,n−1
a n−1,n

a n,n−1 a nn
n