Determinants and Their Applications in Mathematical Physics

192 5. Further Determinant Theory

5.4 The Cofactors of the Matsuno Determinant

5.4.1 Introduction

Let

En=|eij|n,

where

eij=

{

1

ci−cj

,j=i

xi,j=i,

(5.4.1)

and where thec’s are distinct but otherwise arbitrary and thex’s are

arbitrary. In some detail,

En=

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

x 1

1

c 1 −c 2

1

c 1 −c 3

···

1

c 1 −cn

1

c 2 −c 1

x 2

1

c 2 −c 3

··· ···

1

c 3 −c 1

1

c 3 −c 2

x 3 ··· ···

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

1

cn−c 1

··· ··· ··· xn

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ n

. (5.4.2)

This determinant is known here as the Matsuno determinant in recognition

of Matsuno’s solutions of the Kadomtsev–Petviashvili (KP) and Benjamin–

Ono (BO) equations (Sections 6.8 and 6.9), where it appears in modified

forms. It is shown below that the first and higher scaled cofactors ofE

satisfy a remarkably rich set of algebraic multiple-sum identities which can

be applied to simplify the analysis in both of Matsuno’s papers.

It is convenient to introduce the symbol†into a double sum to denote

that those terms in which the summation variables are equal are omitted

from the sum. Thus,

∑

r

∑

s

†

urs=

∑

r

∑

s

urs−

∑

r

urr. (5.4.3)

It follows from the partial derivative formulae in the first line of (3.2.4),

(3.6.7), (3.2.16), and (3.2.17) that

∂Epq

∂xi

=Eip,iq,

∂Epr,qs

∂xi

=Eipr,iqs

∂E

pq

∂xi

=−E

pi

E

iq

,

(

E

ii

+

∂

∂xi

)

E

pq

=E

ip,iq

,

(

E

ii

+

∂

∂xi

)

E

pr,qs

=E

ipr,iqs

,