464
E Quaternions
and
¯
β =
− ¯
I 1 , ¯
φ 1 , − ¯
I 2 , ¯
φ 2 , . . ., − ¯
I N , ¯
φ N
.
(E.51)
The N ×N quaternion matrix, ¯
, 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)
⎞
⎟
⎟
⎟
⎠
,
(E.52)
where
¯
e(j, j ) =
0 0
0 0
and ¯
e(i, j ) =
0
0
i − x j ) 0
.
(E.53)
Since each row in Eq. (E.49) is a linear combination of ¯
α and ¯
β, the rows are not
independent. Let us write
⎛
⎜
⎜
⎜
⎝
¯
J 1
¯
J 2
. . .
¯
J N
⎞
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎝
¯
φ T
1 · ¯
α + ¯
p T
1 · ¯
β
¯
φ T
2 · ¯
α + ¯
p T
2 · ¯
β
. . .
¯
φ T
N · ¯
α + ¯
p T
N · ¯
β
⎞
⎟
⎟
⎟
⎠
.
(E.54)
Then the 2N ×2N matrix ¯
Q N (N×N quaternion) takes the form
¯
Q N =
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
¯
J 1
N
n=1 T 1,n · ¯
J n
¯
J 2
N
n=1 T 2,n · ¯
J n
. . .
¯
J N
N
n=1 T N,n · ¯
J n
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
+ ¯
,
(E.55)
where T i,j are matrix elements of the N ×N matrix, ¯
T , which is given by
¯
T =
⎛
⎜
⎜
⎜
⎝
¯
I T
1 , ¯
φ T
1
¯
I T
2 , ¯
φ T
2
. . .
. . .
¯
I T
N , ¯
φ T
N
⎞
⎟
⎟
⎟
⎠
·
⎛
⎜
⎜
⎜
⎝
¯
φ T
1 , ¯
p T
1
¯
φ T
2 , ¯
p T
2
. . .
. . .
¯
φ T
N , ¯
p T
N
⎞
⎟
⎟
⎟
⎠
−1
.
(E.56)