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) inM

∗

.

Theorem 5.14. The determinants B

nr

s, where n and r are fixed,

s=1, 2 , 3 ,..., represent identical polynomials of degree(r+2−n)(2n− 2 −r).

Denote their common polynomial byBnr.

Theorem 5.15.

Tr+2−n,r=(−1)

k

Bnr,r≥ 2 n− 2 ,n=1, 2 , 3 ,...

where

k=n+r+

[

1

2

(r+2)

]

.

Both of these theorems have been proved by Fiedler using the theory

ofS-matrices but in order to relate the present notes to Fiedler’s, it is

necessary to change the sign ofx.

Whenr=2n−2, Theorem 5.15 becomes the symmetric identity

Tn, 2 n− 2 =Bn, 2 n− 2 ,

that is
∣ ∣ ∣ ∣ ∣ ∣ ∣
φ 0 ... φn− 1

.

.

.

.

.

.

φn− 1 ... φ 2 n− 2

∣ ∣ ∣ ∣ ∣ ∣ ∣ n

=

∣ ∣ ∣ ∣ ∣ ∣ ∣

α 0 ... αn− 1

.

.

.

.

.

.

αn− 1 ... α 2 n− 2

∣ ∣ ∣ ∣ ∣ ∣ ∣ n

(degree 0)

|φm|n=|αm|n, 0 ≤m≤ 2 n− 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

ofMandM

∗

, that is, by interchangingφm(x)andαmand changing the

sign of eachxwhere 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):

|φ 1 |=

∣

∣

∣

∣

1 −x

α 0 α 1

∣

∣

∣

∣

,

|α 1 |=

∣

∣

∣

∣

1 x

φ 0 φ 1

∣

∣

∣

∣

; (5.6.8)

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

|φ 2 |=−

∣ ∣ ∣ ∣ ∣ ∣

1 − 2 x

1 −xx

2

α 0 α 1 α 2

∣ ∣ ∣ ∣ ∣ ∣

=−

∣ ∣ ∣ ∣ ∣ ∣

1 −x

1 α 0 α 1

−xα 1 α 2

∣ ∣ ∣ ∣ ∣ ∣

(symmetric),