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[
Z
S
]
G
[
T
][
S
][
A
][
Z
R
][
h
]
−
1
=
,
(3
.
46)
where
1
G
=
g
(1
+
s
2
)(1
+
t
2
)(1
+
a
2
)
and
1
1
s
s
1
1
−
t
+
a
0
01
[
T
]
[
S
]
[
A
]
=
,
=
,
=
,
t
1
−
a
0
1
[
Z
R
]
[
h
][
Z
R
]
=
,
[
h
]
=
[
I
]
+
.
−
2
0
The incorporation of [
h
] into matrix equation (3.46) aggravates its indeterminacy.
To get a resolvable system of equations giving the regional strike, we have to reduce
the number of unknowns. To this end we expand [
h
] into the sum of diagonal and
anti-diagonal tensors:
h
xx
h
xx
0
h
xy
h
xy
0
[
h
]
[
h
]
D
[
h
]
A
=
=
+
=
+
,
h
yx
h
yy
h
yy
h
yx
0
0
where
h
xx
0
h
xy
0
[
h
]
D
=
[
h
]
A
=
,
.
h
yy
h
yx
0
0
Using this representation, we simplify the (3.46). On rather cumbersome algebra,
we obtain:
[
h
][
Z
R
]
[
h
][
Z
R
])[
Z
R
]
−
1
[
Z
R
][
h
]
−
1
[
Z
R
]
}
−
1
=
{
[
I
]
+
={
([
I
]
+
}
[
Z
R
]
−
1
[
h
]
D
−
1
[
h
]
A
[
Z
R
])[
Z
R
]
−
1
[
h
]
D
}
−
1
={
([
I
]
+
+
=
+
(3
.
47)
[
h
]
D
[
Z
R
])[
Z
R
]
−
1
[
Z
R
]([
I
]
[
h
]
D
[
Z
R
])
−
1
}
−
1
={
([
I
]
+
=
+
[
Z
R
][
h
]
−
1
=
,
where [
Z
R
] is the transformed regional tensor with the anti-diagonal matrix:
[
Z
R
]
[
h
]
A
[
Z
R
])
−
1
[
Z
R
]([
I
]
=
+
0
0
10
01
0
0
−
1
R
h
xy
1
1
=
+
=
,
h
yx
2
2
−
0
−
0
(3
.
48)
0
R
−
0
R
1
R
2
R
R
=
1
,
=
h
yx
h
xy
2
1
+
1
−
