Page 229 - Determinants and Their Applications in Mathematical Physics

214 5. Further Determinant Theory

coaxial in the sense that all their secondary diagonals lie along the same
diagonal parallel to diag(1) in M .
∗
Theorem 5.14. The determinants B , where n and r are fixed,
nr
s
s =1, 2, 3,..., represent identical polynomials of degree (r+2−n)(2n−2−r).
Denote their common polynomial by B nr .
Theorem 5.15.
T r+2−n,r =(−1) B nr , r ≥ 2n − 2, n =1, 2, 3,...
k
where
1
k = n + r + (r +2) .
2
Both of these theorems have been proved by Fiedler using the theory
of S-matrices but in order to relate the present notes to Fiedler’s, it is
necessary to change the sign of x.
When r =2n − 2, Theorem 5.15 becomes the symmetric identity
T n,2n−2 = B n,2n−2 ,
that is

... ...
φ 0 φ n−1 α 0 α n−1

. . . .

. . . . (degree 0)

. . = . .

φ n−1 ... φ 2n−2 α n−1 ... α 2n−2

n n
|φ m | n = |α m | n , 0 ≤ m ≤ 2n − 2,
which is proved by an independent method in Section 4.9 on Hankelians 2.
Theorem 5.16. To each identity, except one, described in Theorems 5.14
and 5.15 there corresponds a dual identity obtained by reversing the role
of M and M , that is, by interchanging φ m (x) and α m and changing the
∗
sign of each x where it occurs explicitly. The exceptional identity is the
symmetric one described above which is its own dual.
The following particular identities illustrate all three theorems. Where
n = 1, the determinants on the left are of unit order and contain a single
element. Each identity is accompanied by its dual.
(n, r)=(1, 1):

1 −x
|φ 1 | = ,

α 0 α 1

1 x
|α 1 | = ; (5.6.8)
φ 0 φ 1

(n, r)=(3, 2):

1 −2x 1 −x

|φ 2 | = − 1 −x x 2 = − 1 α 0 (symmetric),

α 1

α 0 α 1 α 2 −xα 1 α 2