Determinants and Their Applications in Mathematical Physics

288 6. Applications of Determinants in Mathematical Physics

brs=ω

s−r

ars. (6.10.6)

En=|ers|n=(−1)

n

A

(n+1)

1 ,n+1

=(−1)

n

A

(n+1)

n+1, 1

. (6.10.7)

In some detail,

An=

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

u 0 ωu 1 −u 2 −ωu 3 ···

ωu 1 u 0 ωu 1 −u 2 ···

−u 2 ωu 1 u 0 ωu 1 ···

−ωu 3 −u 2 ωu 1 u 0 ···

..............................

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ n

(ω

2

=−1), (6.10.8)

Bn=

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

u 0 −u 1 u 2 −u 3 ···

u 1 u 0 −u 1 u 2 ···

u 2 u 1 u 0 −u 1 ···

u 3 u 2 u 1 u 0 ···

..........................

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ n

, (6.10.9)

En=

∣ ∣ ∣ ∣ ∣ ∣ ∣

∣

∣

ωu 1 u 0 ωu 1 −u 2 ···

−u 2 ωu 1 u 0 ωu 1 ···

−ωu 3 −u 2 ωu 1 u 0 ···

u 4 −ωu 3 −u 2 ωu 1 ···

..............................

∣ ∣ ∣ ∣ ∣ ∣ ∣

∣

∣

n

(ω

2

=−1),(6.10.10)

An=(−1)

n

E

(n+1)

n+1, 1

. (6.10.11)

Anis a symmetric Toeplitz determinant (Section 4.5.2) in whichtr=

ω

r

ur. All the elements on and below the principal diagonal of Bn are

positive. Those above the principal diagonal are alternately positive and

negative.

The notation is simplified by omitting the ordernfrom a determinant

or cofactor where there is no risk of confusion. ThusAn,A

(n)

ij

,A

ij

n

, etc.,

may appear asA,Aij,A
ij
, etc. Where the order is not equal ton, the

appropriate order is shown explicitly.

A and E, and their simple and scaled cofactors are related by the

following identities:

A 11 =Ann=An− 1 ,

A 1 n=An 1 =(−1)

n− 1

En− 1 ,

Ep 1 =(−1)

n− 1

Apn,

Enq=(−1)

n− 1

A 1 q,

En 1 =(−1)

n− 1

An− 1 , (6.10.12)

(

A

E

) 2

=

(

E

n 1

A

11

) 2

, (6.10.13)

E

2

E

p 1

E

nq

=A

2

A

pn

A

1 q

. (6.10.14)

Lemma 6.17.

A=B.