Page 303 - Determinants and Their Applications in Mathematical Physics

288 6. Applications of Determinants in Mathematical Physics

b rs = ω s−r a rs . (6.10.6)
(n+1) (n+1)
E n = |e rs | n =(−1) A =(−1) A . (6.10.7)
n
n
1,n+1 n+1,1
In some detail,

u 0 ωu 1 −u 2 −ωu 3 ···

ωu 1 u 0 ωu 1 −u 2 ···
2
A n = −u 2 ωu 1 u 0 ωu 1 ··· (ω = −1), (6.10.8)

−ωu 3 −u 2 ωu 1 u 0 ···
..............................

n

u 0 −u 1 u 2 −u 3 ···

u 1 u 0 −u 1 u 2 ···

B n = u 2 u 1 u 0 −u 1 ··· , (6.10.9)

u 3 u 2 u 1 u 0 ···
..........................

n

ωu 1 u 0 ωu 1 −u 2 ···

−u 2 ωu 1 u 0 ωu 1 ···
2
E n = −ωu 3 −u 2 ωu 1 u 0 ··· (ω = −1), (6.10.10)

u 4 −ωu 3 −u 2 ωu 1 ···
..............................

n
(n+1)
A n =(−1) E n+1,1 . (6.10.11)
n
A n is a symmetric Toeplitz determinant (Section 4.5.2) in which t r =
ω u r . All the elements on and below the principal diagonal of B n are
r
positive. Those above the principal diagonal are alternately positive and
negative.
The notation is simplified by omitting the order n from a determinant
(n)
or cofactor where there is no risk of confusion. Thus A n , A , A , etc.,
ij
ij n
may appear as A, A ij , A , etc. Where the order is not equal to n, the
ij
appropriate order is shown explicitly.
A and E, and their simple and scaled cofactors are related by the
following identities:
A 11 = A nn = A n−1 ,
A 1n = A n1 =(−1) n−1 E n−1 ,
E p1 =(−1) n−1 A pn ,
E nq =(−1) n−1 A 1q ,
E n1 =(−1) n−1 A n−1 , (6.10.12)
2
2
A E

n1
= , (6.10.13)
E A 11
p1
2
2
E E E nq = A A A . (6.10.14)
1q
pn
Lemma 6.17.
A = B.