G.3 Circular Orthogonal Ensemble (COE)
501
The quaternion matrix ¯
Q
(1)
N can be written
¯
Q
(1)
N =
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
¯
†
1 · ¯
α
¯ I
†
1 · ¯
α
¯
†
2 · ¯
α
¯ I
†
2 · ¯
α
. . .
¯
†
N · ¯
α
¯ I
†
N · ¯
α
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
+ ¯
,
(G.50)
where ¯
α is a 2N ×N matrix defined as
¯
α =
¯
1 −¯ p 1 ¯
2 −¯ p 2 . . . ¯
N −¯ p N
.
(G.51)
The N ×N quaternion ¯
is defined as
¯
=
⎛
⎜
⎜
⎜
⎝
¯
e(1, 1) ¯
e ( 1, 2) . . . ¯
e(1, N)
¯
e(2, 1) ¯
e(2, 2) . . . ¯
e(2, N)
. . .
. . .
. . .
. . .
¯
e(N, 1) ¯
e(N, 2) . . . ¯
e(N, N)
⎞
⎟
⎟
⎟
⎠
,
(G.52)
where
¯
e(j, j ) =
0 0
0 0
and ¯
e(i, j ) =
0
0
i − θ j ) 0
.
(G.53)
In the determinant Det[ ¯
Q
(1)
N ], we can subtract multiples of odd-numbered rows
from even-numbered rows without changing the value of the determinant. We then
obtain
Det[ ¯
Q
(1)
N ] = Det[ ¯
J + ¯
], where ¯
J =
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
¯
†
1 · ¯
α
¯
0
¯
†
2 · ¯
α
¯
0
. . .
¯
†
N · ¯
α
¯
0
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
.
(G.54)
If we rearrange rows and columns, Det[ ¯
Q
(1)
N ] takes the form
Det[ ¯
Q
(1)
N ] = Det
¯
S N ¯
S N
¯
N ¯
0 N
,
(G.55)