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In-Depth Information
and their difference is
⎡
⎤
0
a
12
−
a
21
...
a
1
n
−
a
n
1
⎣
⎦
−
(a
12
−
a
21
)
0
...
a
2
n
−
a
n
2
A
T
A
−
=
.
.
.
.
.
.
.
−
(a
1
n
−
a
n
1
)
−
(a
2
n
−
a
n
2
)
...
0
A
T
It is clear that
A
−
is antisymmetric, and if we divide throughout by 2 we have
2
A
A
T
.
1
Q
=
−
For example:
⎡
⎤
⎡
⎤
a
11
a
12
a
13
a
11
a
21
a
31
⎣
⎦
,
A
T
⎣
⎦
A
=
a
21
a
22
a
23
=
a
12
a
22
a
32
a
31
a
32
a
33
a
13
a
23
a
33
⎡
⎣
⎤
⎦
(a
12
−
(a
13
−
0
a
21
) /
2
a
31
) /
2
(a
21
−
(a
23
−
Q
=
a
12
) /
2
0
a
32
) /
2
(
a
31
−
a
13
)
/
2
(
a
32
−
a
23
)
/
2
0
and if we maintain some symmetry with the subscripts, we have
⎡
⎤
0
(a
12
−
a
21
) /
2
−
(a
31
−
a
13
) /
2
⎣
⎦
Q
=
−
(a
12
−
a
21
) /
2
0
(a
23
−
a
32
) /
2
(a
31
−
a
13
) /
2
−
(a
23
−
a
32
) /
2
0
⎡
⎤
0
q
3
/
2
−
q
2
/
2
⎣
⎦
=
−
q
3
/
2
0
q
1
/
2
q
2
/
2
−
q
1
/
2
0
where
q
1
=
a
32
q
2
=
a
31
−
a
13
q
3
=
a
12
−
a
21
.
a
23
−
Using a real example:
⎡
⎤
⎡
⎤
014
314
426
034
112
446
⎣
⎦
,
A
T
⎣
⎦
A
=
=
⎡
⎤
0
10
101
0
−
⎣
⎦
.
Q
=
−
10
Furthermore, we have already computed
⎡
⎣
⎤
⎦
024
213
436
=
S
