Determinants and Their Applications in Mathematical Physics

4.4 Circulants 83

= exp

[

n

∑

r=1

n− 1

∑

t=1

ω

(r−1)t

xt

]

= exp

[

n− 1

∑

t=1

xt

n

∑

r=1

ω

(r−1)t

]

= exp(0).

The lemma follows.

Theorem.

A=A(H 1 ,H 2 ,H 3 ,...,Hn)=1.

Proof. The definition (4.4.16) implies that

A(H 1 ,H 2 ,H 3 ,...,Hn)=











H 1 H 2 H 3 ··· Hn

Hn H 1 H 2 ··· Hn− 1

Hn− 1 Hn H 1 ··· Hn− 2

··· ··· ··· ··· ···

H 2 H 3 H 4 ··· H 1

     n

=W

− 1

diag

(

E 1 E 2 E 3 ...En

)

W. (4.4.18)

Taking determinants,

A(H 1 ,H 2 ,H 3 ,...,Hn)=

∣

∣

W

− 1

W

∣

∣

n

∏

r=1

Er.

The theorem follows from Lemma 4.19.

Illustrations

Whenn=2,ω= exp(iπ)=−1.

W=

[

11

1 − 1

]

,

W

− 1

=

1

2

W,

Er= exp[(−1)

r− 1

x 1 ],r=1, 2.

Letx 1 →x; then,

E 1 =e

x

,

E 2 =e

−x

,

[

H 1

H 2

]

=

1

2

[

11

1 − 1

][

e

x

e

−x

]

,

H 1 =chx,

H 2 =shx,