E.2 Quaternions in the GOE
465
If we next take the determinant of the 2N×2N matrix, ¯
Q N , and remember that
we can subtract multiples of one row of a determinant from another row without
changing the value of the determinant, then Det[ ¯
Q N ] takes the form
Det[ ¯
Q N ] = Det[ ¯
J + ¯
], where J =
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
¯
J 1
¯
0
¯
J 2
¯
0
. . .
¯
J N
¯
0
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
.
(E.57)
If we rearrange rows and columns, Det[ ¯
Q N ] takes the form
Det[ ¯
Q N ] = Det
¯
S N ¯
S N
¯
N ¯
0 N
,
(E.58)
where ¯
0 N is an N ×N matrix with every matrix element equal to zero,
¯
S N =
⎛
⎜
⎜
⎜
⎜
⎝
S N
1,1 (x 1 , x 1 ) S N
1,1 (x 1 , x 2 ) . . . S N
1,1 (x 1 , x N )
S N
1,1 (x 2 , x 1 ) S N
1,1 (x 2 , x 2 ) . . . S N
1,1 (x 2 , x N )
. . .
. . .
. . .
. . .
S N
1,1 (x N , x 1 ) S N
1,1 (x N , x 2 ) . . . S N
1,1 (x N , x N )
⎞
⎟
⎟
⎟
⎟
⎠
,
(E.59)
¯
S N =
⎛
⎜
⎜
⎜
⎜
⎝
0
S N
1,2 (x 1 , x 2 ) . . . S N
1,2 (x 1 , x N )
S N
1,2 (x 2 , x 1 )
0
. . . S N
1,2 (x 2 , x N )
. . .
. . .
. . .
. . .
S N
1,2 (x N , x 1 ) S N
1,2 (x N , x 2 ) . . .
0
⎞
⎟
⎟
⎟
⎟
⎠
,
(E.60)
and
¯
N =
⎛
⎜
⎜
⎜
⎝
0
− N (x 1 , x 2 ) . . . − N (x 1 , x N )
− N (x 2 , x 1 )
0
. . . − N (x 2 , x N )
. . .
. . .
. . .
. . .
− N (x N , x 1 ) − N (x N , x 2 ) . . .
0
⎞
⎟
⎟
⎟
⎠
.
(E.61)
Thus, we find
Det[ ¯
Q N ] = Det[¯ N ]×Det[ ¯
S N ].
(E.62)