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2
3
2
3
2
3
dX
dY
dZ
X
Y
Z
X
Y
Z
4
5
ᄐ
4
5
new
4
5
old
,
and
2
3
2
3
2
3
dL
dB
dH
L
B
H
L
B
H
4
5
ᄐ
4
5
new
4
5
old
:
In order to derive the inverse matrix J
1
, we decompose J into the product of
two matrices:
J
ᄐ
SH,
ð
7
:
17
Þ
with
2
4
3
5
,
sin L
sin B cos L
cos B cos L
S
ᄐ
cos L
sin B sin L
cos B sin L
0
cos B
sin B
and
2
3
ð
N
þ
H
Þ
cos B
0
0
4
5
:
H
ᄐ
0
M
þ
H 0
0
0
1
According to the invertible matrix theorem, we get:
J
1
H
1
S
1
ᄐ
:
ð
7
:
18
Þ
H is the diagonal matrix and its inverse matrix is:
2
3
1
0
0
4
5
ð
N
þ
H
Þ
cos B
H
1
ᄐ
1
:
ð
7
:
19
Þ
0
1
M
þ
H
0
0
1
S is an orthogonal matrix; hence, we have:
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