An alternative way to use permutation random variable to indicate the two stage sampling Considering the two stage sampling problems with no measurement errors, we take an example to try to solve the problems. Assume we have 2N = clusters, the cluster size is denoted by SM . Assume , 1 2M = 2 3M = . So the list of our population is
. ( )11 12 21 22 23y y y y y ′=y
Define and to denote the permutation within
cluster 1 and cluster 2. Define denote the permutation of clusters.
11 121
21 22
U UU U⎛ ⎞
= ⎜ ⎟⎝ ⎠
U11 12 13
2 21 22 2
31 32 33
U U UU U UU U U
⎛ ⎞⎜= ⎜⎜ ⎟⎝ ⎠
U 3⎟⎟
⎟11 12
21 22
V VV V⎛ ⎞
= ⎜⎝ ⎠
V
Define the new random variables *Y2 11
3 12 1*
2 21 2
3 22
00
VV
VV
⎛ ⎞⎜ ⎟⎛ ⎞⎜ ⎟= =⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎝ ⎠
Ι 00 Ι U
Y y VUyΙ 0 U
0 Ι
Find the expectation and variance for *Y
( ) ( )2 11
3 12 1* *| |
2 21 2
3 22
2 111
13 12
2 212
23 22
2 11
3 12
2 21
3 22
00
1 0
10
UV V U V V U V
V
V
VV
E E E E EV
V
VMV
EV
MV
VV
EV
V
⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟ ⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟⎡ ⎤= = ⎢ ⎥⎜ ⎟⎣ ⎦ ⎢ ⎥⎜ ⎟ ⎝ ⎠⎣ ⎦⎢ ⎥⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
⎡ ⎤⎛ ⎞⎛ ⎞⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟=⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎝ ⎠⎣ ⎦
⎛
=
Ι 00 Ι U
Y Y yΙ 0 U
0 Ι
Ι 0J
0 Ιy
Ι 0 J0 Ι
Ι 00 Ι
Ι 00 Ι
1 1
1 1
2 2
2 2
2 2
1 1N N
u uu uu u
N Nu uu u
⎡ ⎤⎛ ⎞ ⎛ ⎞⎞⎢ ⎥⎜ ⎟ ⎜ ⎟
⎜ ⎟⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟ ⎜ ⎟= ⊗ = ⊗⎜ ⎟⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦
1 1 u
( ) ( ) ( )* *| |UV V U V V U VVar E Var Var E⎡ ⎤ ⎡= +⎣ ⎦ ⎣Y Y * ⎤⎦Y
C07bx06 4/17/2007 1
For the second item:
( ) ( )
2 11
3 12*| |
2 21
3 22
U V U V
VV
E EV
V
⎛ ⎞⎜ ⎟⎜ ⎟= =⎜ ⎟⎜ ⎟⎝ ⎠
Ι 00 Ι
Y VUy uΙ 0
0 Ι
( )
( )
12 11
13 12*
| 22 21
23 22
2
1 11 1 11 2 12 2 12 2 12 1 21 1 21 2 22 2 22 2 22
V U V V
V
uV
uV
Var E Var uV
uV
u
Var u V u V u V u V u V u V u V u V u V u V
⎡ ⎤⎛ ⎞⎛ ⎞⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟⎡ ⎤ ⎢ ⎥⎜ ⎟=⎣ ⎦ ⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟⎝ ⎠⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
⎡ ⎤′= ⎢ ⎥⎣ ⎦
Ι 00 Ι
YΙ 0
0 Ι
C07bx06 4/17/2007 2
1 11
1 111 1 2 2 1 2 2 3 1 1 2 2 1 2 3 22 2
2 12
2 121 2 3 2 2 22
2 12
1 21
1 21
2 22
2 22
2 22
1 1 1 1 1 1 111
1 1 11
V
u Vu V
u u u u u u u uu V N N N N N N Nu V
u u u uu V N N N
Varu Vu Vu Vu Vu V
× ×
⎛ ⎞⎜ ⎟
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟ ′ ′− − −⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ −⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎜ ⎟
⎛ ⎞ ⎛⎜ ⎟ ′− −⎜ ⎟ ⎜⎜ ⎟ ⎝ ⎠ ⎝⎜ ⎟= =⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
J 1 1 J 1 1
1 1 3 3 1 2 2 2 2 3 32
1 1 2 2 1 2 2 3 1 1 2 2 1 2 22 2
1 2 3 2 2 2 3 3 1 2 3 2 2 22 2
1 1 1 11
1 1 1 1 1 1 111
1 1 1 1 1 1 111
u u u uN N N N
u u u u u u u uN N N N N N N
u u u u u u u uN N N N N N N
× ×
× ×
×
⎞ ⎛ ⎞ ⎛ ⎞′ −⎟ ⎜ ⎟ ⎜ ⎟−⎠ ⎝ ⎠ ⎝ ⎠⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞′ ′− − −⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞′ ′− − −⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜−⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝
J 1 1 J
J 1 1 J 1 1
1 1 J 1 1 3 3
1 1 2 2 1 1 2 2 1 2 2 3 1 1 2 22 2
1 2 3 2 1 1 3 3 1 1 3 3 1 2 3 22 2 2
1 1 1 1 1 111
1 1 1 1 11
u u u u u u u uN N N N N N
u u u u u u u uN N N N N
×
× × ×
× ×
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎟
⎠⎣ ⎦
⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛′− − − −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜−⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝⎛ ⎞ ⎛ ⎞′ ′− − − − −⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠=
J
J J 1 1 J
1 1 J J 1 1
1 1 2 2 1 2 2 3 1 1 2 2 1 1 2 22 2
1 2 3 2 2 22
1 1 1 1 1 1 111 11 1 1 1 11
1 1
N
u u u u u u u uN N N N N N N
u u u uN N N N N
× × ×
⎛⎜⎝
⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ′− − − − −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟− −⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞′− − − −⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟− −⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠
J 1 1 J J
1 1 J3 3 1 2 3 22
2 22 2
1 1
2 22 2
1 1
1
1 1 1 1 11
1 1 1 1 11
111
N N
Ms Ms Ms Mss s
N N
Ms Ms Ms Mss s
u u uN
u uN N N N N
u uN N N N N
N
×
= =
= =
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢
′−⎢⎣⎡ ⎤⎛ ⎞⎛ ⎞′ ′− − −⎜ ⎟⎜ ⎟⎢ ⎥−⎝ ⎠⎝ ⎠⎢ ⎥=⎢ ⎥⎛ ⎞⎛ ⎞ ′ ′− − −⎢ ⎥⎜ ⎟⎜ ⎟−⎝ ⎠⎝ ⎠⎣ ⎦
−−=
⊕ ⊕
⊕ ⊕
1 1
J uu J uu
J uu J uu
22
1
22
1
22
1
1 11 1
11 11
1 11 11 1
1 11
N
Ms Mss
N
Ms Mss
N
N Ms Mss
uN N
N
N N N uN N N
N NN u
N N N
=
=
=
⎛ ⎞⎜ ⎟ ⎛ ⎞′⊗ −⎜ ⎟ ⎜ ⎟
⎝ ⎠⎜ ⎟−⎜ ⎟−⎝ ⎠⎛ ⎞− −⎜ ⎟ ⎛ ⎞′= ⊗ −⎜ ⎟ ⎜ ⎟− ⎝ ⎠⎜ ⎟− −⎜ ⎟⎝ ⎠
⎛ ⎞′= ⊗ −⎜ ⎟− ⎝ ⎠
⊕
⊕
⊕
J uu
J uu
P J uu
For the first item:
( ) ( ) ( ) ( )( )*| | |V U V V U V V U VE Var E Var E Var vec⎡ ⎤ ′⎡ ⎤= = ⊗⎡ ⎤⎣ ⎦⎣ ⎦⎣ ⎦Y VUy y V U
C07bx06 4/17/2007 3
( ) ( ) ( ) ( ) ( )| |U V U VVar vec Var vec′ ′⊗ = ⊗⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦y V U y V U y V′⊗
For ( )|U VVar vec⎡ ⎤⎣ ⎦U
( )
111121
112122
211221231
212222232
213223233
000
00000
00
00
UU
UU
vec UUU
UUU
UUU
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟
= ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
U
( )
1 1
1 1
2 2
2 2
|
1 1 1
1 1 1
2 2 2 2 2
2 2 2 2 2
2
1 1 10 0 0 0 0 0 0 01
0 0 0 0 0 0 0 0 0 01 1 10 0 0 0 0 0 0 0
10 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0 0
1 1 1 1 10 0 0 0 0 0 01 1
0 0 0 0 0 0 0 0 0 01 1 1 1 10 0 0 0 0 0 0
1 10 0 0 0 0 0 0 0 0 0
10 0 0 0 0
U V
M M
M M
M M
M M
Var vec
M M M
M M M
M M M M M
M M M M M
M
=⎡ ⎤⎣ ⎦
−−
−−
− −− −
− −− −
−
U
P P
P P
P P
P P
2
2
M
M
P
P
2 22 2 2 2
1 1 1 10 01 1M MM M M M
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟
−⎜ ⎟− −⎝ ⎠P P
2MP
C07bx06 4/17/2007 4
Define 1 11 1
1 0
0 0
MM⎛ ⎞⎜ ⎟ = ∆⎜ ⎟⎜ ⎟⎝ ⎠
P and 1 1
1 1 2
1 1 01
0 0
MM M⎛ ⎞−⎜ ⎟− = ∆⎜ ⎟⎜ ⎟⎝ ⎠
P
Define 2 22 1
1 0
0 0
MM⎛ ⎞⎜ ⎟ = ∆⎜ ⎟⎜ ⎟⎝ ⎠
P and 2 2
2 2 2
1 1 01
0 0
MM M⎛ ⎞−⎜ ⎟− = ∆⎜ ⎟⎜ ⎟⎝ ⎠
P
Then
( )
1 11 21 12 1
2 2 2| 1 2 2
2 2 21 1 22 2 22 2 1
0 0 00 0 0
0 00 00 0
U VVar vec
⎛ ⎞∆ ∆⎜ ⎟∆ ∆⎜ ⎟⎜ ⎟=⎡ ⎤ ∆ ∆ ∆⎣ ⎦ ⎜ ⎟
∆ ∆ ∆⎜ ⎟⎜ ⎟∆ ∆ ∆⎝ ⎠
U
( ) ( ) ( ) ( ) ( )
( )
( )
| |
1 1111 2
1 1122 1
2 2 211 12 21 22 23 211 2 2
2 2 2221 1 2
2 2 2232 2 1
1 12 2
11 12 21 22 23
0 0 00 0 0
0 00 00 0
0 0 0
U V U VVar vec Var vec
yy
y y y y y yyy
y y y y y
′ ′ ′⊗ = ⊗ ⊗⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦′⎛ ⎞∆ ∆ ⎛ ⎞
⎜ ⎟⎜ ⎟′∆ ∆⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟′= ∆ ∆ ∆⎜ ⎟⎜ ⎟′∆ ∆ ∆⎜ ⎟⎜ ⎟
⎜ ⎟⎜ ⎟ ′∆ ∆ ∆ ⎝ ⎠⎝ ⎠
∆ ∆
=
y V U y V U y V
VV
V V V V V VVV
V V V V V
( )
111 1
122 12 2 2
211 2 22 2 2
222 1 22 2 2
232 2 1
1 1 1 1 2 2 2 2 2 2 2 2 211 1 12 2 11 2 12 1 21 1 22 2 23 2 21 2 22 1 23 2 21 2 22 2 23 1
0 0 00 00 00 0
yyyyy
y y y y y y y y y y y y y
′⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟′∆ ∆⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟′∆ ∆ ∆⎜ ⎟⎜ ⎟′∆ ∆ ∆⎜ ⎟⎜ ⎟
⎜ ⎟⎜ ⎟ ′∆ ∆ ∆ ⎝ ⎠⎝ ⎠′
= ∆ + ∆ ∆ + ∆ ∆ + ∆ + ∆ ∆ + ∆ + ∆ ∆ + ∆ + ∆
VVVVV
V V V V V V V V V V V V V
( ) ( ) ( )( ) ( )
11
12
21
22
23
1 1 1 1 2 2 211 1 12 2 11 11 2 12 1 12 21 1 22 2 23 2 21
2 2 2 2 2 221 2 22 1 23 2 22 21 2 22 2 23 1 23
1 1 111 1 11 12 2 11 11 2 12
yyyyy
y y y y y y y y y y
y y y y y y y y
y y y y y y
⎛ ⎞⎜ ⎟′⎜ ⎟⎜ ⎟′⎜ ⎟′⎜ ⎟⎜ ⎟′⎝ ⎠
′ ′ ′= ∆ + ∆ + ∆ + ∆ + ∆ + ∆ + ∆
′ ′+ ∆ + ∆ + ∆ + ∆ + ∆ + ∆
= ∆ + ∆ + ∆
VVVVV
V V V V V V V V V V
V V V V V V V V
V ( )( )
112 1 12
2 2 2 2 2 2 2 2 221 21 1 22 21 2 23 21 2 21 2 22 22 22 1 23 22 2 21 23 2 22 23 2 23 23 1
y y
y y y y y y y y y y y y y y y y y y
′+ ∆ +
′∆ + ∆ + ∆ + ∆ + ∆ + ∆ + ∆ + ∆ + ∆
V
V V
For
C07bx06 4/17/2007 5
( )
1 1 1
1
1 1 1 111 1 11 12 2 11 11 2 12 12 1 12
11 11 12 12 12 111 1 1 1
21
1 1 1 10 0 21
0 0 0 0 0 0
0
0 0
M M M
M s
y y y y y y y y
y y y y y yM M M M
σ
′∆ + ∆ + ∆ + ∆
⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛−⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ′−= + +⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜⎜ ⎟ ⎜ ⎟ ⎜⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝⎣ ⎦⎛ ⎞
′= ⎜ ⎟⎜ ⎟⎝ ⎠
V V
P P PV V
PV V
0⎞⎟⎟⎟⎠
For ( )
[ ] ( )( )
[ ] ( )
2 2 2 2 2 2 2 2 221 21 1 22 21 2 23 21 2 21 2 22 22 22 1 23 22 2 21 23 2 22 23 2 23 23 1
2 221 21 22 22 23 23 1 22 21 23 21 23 22 2
221 21 22 22 23 23 1 22 21 23 21 23 22
2
2
121
y y y y y y y y y y y y y y y y y y
y y y y y y y y y y y y
y y y y y y y y y y y yM
′∆ + ∆ + ∆ + ∆ + ∆ + ∆ + ∆ + ∆ + ∆
′= + + ∆ + + + ∆
⎛= + + ∆ + + + −⎜ −⎝
V V
V V
V
2
21
22
0 00 M sσ
⎛ ⎞⎞′∆⎜ ⎟⎟⎜ ⎟⎠⎝ ⎠
⎛ ⎞′= ⎜ ⎟
⎝ ⎠
V
V VP
( ) ( ) ( ) ( ) ( )
1
2
1
2
| |
21
22
21
22
0 0000 0
0
0
U V U V
M s
M s
M s
M s
Var vec Var vec
σσ
σ
σ
′ ′⊗ = ⊗⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦⎛ ⎞ ⎛ ⎞
′ ′= +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠⎛ ⎞
′= ⎜ ⎟⎜ ⎟⎝ ⎠
′⊗y V U y V U y V
PV V V V
P
PV V
P
C07bx06 4/17/2007 6
( ) ( )( )( ) ( ) ( )( )
1
2
1
2
1
2
|
|
2 11 2 11213 12 3 12
22 21 2 212
3 22 3 22
21 11
22 12
21 21
0
0
V U V
V U V
M sV
M s
M s
M sV
M s
M
E Var vec
E Var vec
V VV V
EV V
V V
V
VE
V
σ
σ
σ
σ
σ
σ
′⊗⎡ ⎤⎣ ⎦
′ ′= ⊗ ⊗⎡ ⎤⎣ ⎦
⎛ ⎞′⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟= ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠
=
y V U
y V U y V
Ι 0 Ι 0P0 Ι 0 Ι
Ι 0 Ι 0P0 Ι 0 Ι
P 0
0 P
P 0
0 P
1 1
2 2
1 1
2 2
1
2 11 2 21
3 12 3 22
22 22
2 21 11 11 1 11 21
2 22 12 12 2 12 22
2 21 21 11 1 21 21
2 22 22 12 2 22 22
1
0 0
0 0
0 0
0 0
s
M s M s
M s M sV
M s M s
M s M s
M s
V VV V
V
V V V V
V V V VE
V V V V
V V V V
σ σ
σ σ
σ σ
σ σ
σ
⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟⎛ ⎞⎢ ⎥⎜ ⎟⎜ ⎟
⎝ ⎠⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦⎡ ⎤⎢ ⎥⎢ ⎥
= ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
=
Ι 0 Ι 00 Ι 0 Ι
P P
P P
P P
P P
P
2
1
2
2
22
21
22
2
1
1 0 0 0
10 0 0
10 0 0
10 0 0
1s
M s
M s
M s
N
N M ss
N
N
N
N
N
σ
σ
σ
σ=
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
⎛ ⎞= ⊗⎜ ⎟⎝ ⎠⊕
P
P
P
I P
Add two items together:
( ) ( ) ( )* * *| |
2 22
1 1
1 1 11 s
UV V U V V U V
N N
N Ms Ms N Ms s
Var E Var Var E
N uN N N N sσ
= =
⎡ ⎤ ⎡ ⎤= +⎣ ⎦ ⎣ ⎦⎛ ⎞ ⎛′= ⊗ − + ⊗⎜ ⎟ ⎜− ⎝ ⎠ ⎝⊕ ⊕
Y Y Y
P J uu I P ⎞⎟⎠
Where ( )2
2 1
1
sM
st st
ss
y u
Mσ =
−=
−
∑
C07bx06 4/17/2007 7
( ) ( ) ( )* * *| |
2 22
1 1
1 1 11 s
UV V U V V U V
N N
N Ms Ms N Ms s
Var E Var Var E
N uN N N N sσ
= =
⎡ ⎤ ⎡ ⎤= +⎣ ⎦ ⎣ ⎦⎛ ⎞ ⎛′= ⊗ − + ⊗⎜ ⎟ ⎜− ⎝ ⎠ ⎝⊕ ⊕
Y Y Y
P J uu I P ⎞⎟⎠
Partition to sample and remainder:
( ) ( )1
0 0s
N
I n n Ms ns Sn N ns Sn n
− × −= ×
⎛ ⎞⎛ ⎞⎡ ⎤= ⊗⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟⎣ ⎦⎝ ⎠⎝ ⎠⊕K I I n is the sampling cluster numbers.
( )( )( )
( ) ( )
( ) ( ) ( )
10 0
0
s ss s s
N
n M n S n N nM n ns S n nII
N n n N n N n
− − × −− ×= − ×
− × − × −⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦ ⎣ ⎦
⎛ ⎞⎛ ⎞⎡ ⎤⊗⎜ ⎟⎜ ⎟⎢ ⎥⎣ ⎦⎝ ⎠= ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
⊕I IK
I
Target is to predicting PSU mean:
1 2
*
1 2
1 1 ...j j M MP eM M
⎡ ⎤⎛ ⎞′ ′ ′= ⊗⎢ ⎥⎜ ⎟
⎝ ⎠⎣ ⎦1 1 Y
*
**
I I
II II
⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟
⎝ ⎠ ⎝ ⎠
K YY
K Y
Find the expectation value of *
**
I I
II II
⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟
⎝ ⎠ ⎝ ⎠
K YY
K Y
( ) ( )
( )1
2 2
**
*
1 1
2 2
1 1
2
1
1 1
I IIUV UV N
II IIII
n
n
IN
M nII
M n
E EN
uu
uN Nu
−
−
⎛ ⎞ ⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞⎜ ⎟= ⊗ =⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
K KYY 1
K KY
11
K1 u 1K
1
⊗u
C07bx06 4/17/2007 8
( )
( ) ( )( ) ( )
**
*
,
,
* *
* *
var var
var var
I IIUV UV
II IIII
I I II
I II II
I UV I I UV II
II UV I II UV II
Var Var′⎛ ⎞ ⎛ ⎞ ⎛ ⎞
=⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠⎛ ⎞
= ⎜ ⎟′⎝ ⎠⎛ ⎞′ ′⎜ ⎟=⎜ ⎟′ ′⎝ ⎠
K KYY
K KY
V VV V
K Y K K Y K
K Y K K Y K
( )*varI I UV ′=V K Y K I is given by
( )*
2 22
1 1
var
1 1 11 s
I I UV I
N N
I N Ms Ms N M ss s
N uN N N N
σ= =
′= =
⎛ ⎞⎛ ⎞ ⎛I
⎞′ ′= ⊗ − + ⊗⎜ ⎟⎜ ⎟ ⎜− ⎝ ⎠ ⎝⎝ ⎠⊕ ⊕
V K Y K
K P J uu I P ⎟⎠
K
For the second item:
( ) ( ) ( ) ( )
2
1
2
1 1 1
2
1
1
10 0 0 0
s
s s s
s
s
N
I N M s Is
N N N
n n Ms ns N M s n n Ms nsSn N n Sn N ns s sSn n Sn n
Nn
n n ss s
N
N
M
σ
σ
σ
=
− −× − × −= = =× ×
=
⎛ ⎞ ′⊗⎜ ⎟⎝ ⎠
′⎛ ⎞ ⎛⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎡ ⎤ ⎛ ⎞ ⎡ ⎤= ⊗ ⊗ ⊗⎜ ⎟ ⎜⎜ ⎟ ⎜ ⎟⎜ ⎟⎢ ⎥⎢ ⎥ ⎢ ⎥⎜ ⎟ ⎜⎣ ⎦ ⎝ ⎠ ⎣ ⎦⎝ ⎠ ⎣ ⎦ ⎝ ⎠⎝ ⎠ ⎝⎛ ⎞⎛ ⎞
= ⊗ −⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
⊕
⊕ ⊕ ⊕
⊕
K I P K
I I I P I I
JI I
⎞⎟⎟⎠
( ) ( )
( ) ( )
22
1
22
1 1
1
1 11
1 10 01
0 0
1
s
s
N
I N Ms Ms Is
N N
n n Ms ns N Ms MsSn N ns sSn n
N
n n Ms ns Sn N ns Sn n
nn
N uN N N
N uN N N
NN N
=
− × −= =×
− × −= ×
⎛ ⎞⎛ ⎞′ ′⊗ −⎜ ⎟⎜ ⎟− ⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞ ⎛⎡ ⎤ ⎛ ′= ⊗ ⊗ −⎜ ⎟⎜ ⎟ ⎜ ⎜ ⎟⎢ ⎥⎜ ⎟ −⎣ ⎦ ⎝⎝ ⎠ ⎝⎝ ⎠
′⎛ ⎞⎛ ⎞⎡ ⎤⊗⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟⎣ ⎦⎝ ⎠⎝ ⎠
⎛= −− ⎝
⊕
⊕ ⊕
⊕
K P J uu K
I I P J uu
I I
JI
⎞⎞⎟⎠⎠
22
1
1 1N
Ms ns n ns
uN N=
⎛ ⎞⎛ ⎞⎞ ′⊗ −⎜ ⎟⎜ ⎟ ⎜ ⎟⎠ ⎝ ⎠⎝ ⎠
⊕ J u u
Where ( )1 1 1 11 1n n nu u′ ′=uAdding them together,
2 22
1 1
1 11
s
s
N Nn n
I n n s n Ms ns n ns ss
N uM N N N N
σ= =
⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞⎛ ⎞ ′= ⊗ − + − ⊗ −⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ − ⎝ ⎠ ⎝ ⎠⎝ ⎠⎝ ⎠⎝ ⎠⊕ ⊕
J JV I I I J u u
Find the inverse 1
I−V
C07bx06 4/17/2007 9
2 22
1 1
2 2 22 2
1 1 1
1 11
1 1 1 11 1
s
s
s
s
N Nn n
I n n s n Ms ns n ns ss
N N Nn n
n n s n Ms ns n n Ms ns ns s ss
N uM N N N N
N Nu uM N N N N N N N
σ
σ
= =
= = =
⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞⎛ ⎞ ′= ⊗ − + − ⊗ −⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ − ⎝ ⎠ ⎝ ⎠⎝ ⎠⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞′ ′= ⊗ − + ⊗ − + − ⊗ −⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ − − ⎝ ⎠⎝ ⎠⎝ ⎠⎝ ⎠
⊕ ⊕
⊕ ⊕ ⊕
J JV I I I J u u
J JI I I J u u J u u
( )
( )
2 2 22
1 1 1
2 2
1
1 1 1 11 1 1
1 1 11 1
s
s
s
s
n
N N Nn n
n n s n Ms ns n n n Ms ns ns s ss
Nn
n n s Ms ns n n ns s
Nu uM N N N N N N
uM N N N N
σ
σ
= = =
=
⎛ ⎞⎜ ⎟⎝ ⎠
⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛⎛ ⎞nN⎞′ ′= ⊗ − + ⊗ − ⊗ + − ⊗ −⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜⎜ ⎟ − − − ⎝ ⎠⎝ ⎠ ⎝⎝ ⎠⎝ ⎠
⎛ ⎞⎡ ⎤⎛ ⎞′= ⊗ − + − − ⊗⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟− − −⎢ ⎥⎝ ⎠⎣ ⎦⎝ ⎠
⊕ ⊕ ⊕
⊕
J JI I I J I u u J u
JI I J u u J
⎟⎠
u
22
1
1 11
N
Ms ns n ns
uN N=
⎛ ⎞⎛ ⎞ ′−⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⊕ J u u
For a matrix , . I n n= ⊗ − ⊗V I A J B ( ) 11 1 1I n n n −− − −= ⊗ − ⊗ −V I A J A B B A
With the definitions of and B , A
( )2 2 2
21 1
22 2
21 1
1 1 1 1 11 1
1 1 1 1 11 1 1
s
s
s
s
N Nn
Ms ns n n n s Ms ns n ns s s
N NnMs
ns n n n s Ms nss s s
n n u uN N N M N N N
nu n uN N N N M N N
σ
σ
= =
= =
⎛ ⎞⎡ ⎤⎛ ⎞⎛ ⎞⎛ ⎞1
′ ′− = − − − + −⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟− −⎝ ⎠⎝ ⎠ ⎢ ⎥⎝ ⎠⎣ ⎦⎝ ⎠⎡ ⎤⎛ ⎞⎛ ⎞ ⎛ ⎞ ′= − − − + +⎢ ⎥⎜ ⎟⎜ ⎟ ⎜ ⎟− − −⎝ ⎠ ⎝ ⎠ ⎢ ⎥⎝ ⎠⎣ ⎦
⊕ ⊕
⊕ ⊕
JB A J u u I J u u
JJ u u I J
( )
−
( )
( )
22 2
21
2 2
1
1
1 1 1 11 1 1
1 11 11 1
s
s
s
s
n n
NnMs
ns n s Ms ns n n n ns s
Nn
Ms ns n s n ns s
N
nu u nN N M N N N N N
n nuN N M N N N
σ
σ
=
=
′−
⎛ ⎞⎡ ⎤ 11
⎡ ⎤⎛ ⎞⎛ ⎞ ⎛ ⎞′ ′= − − + + −⎜ ⎟⎢ ⎥ ⎢ ⎥⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟− − −⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠ −⎣ ⎦⎣ ⎦⎝ ⎠⎛ ⎞ ⎡⎛ ⎞⎛ ⎞⎛ ⎞ ⎛ ⎞ ′= − − − − + −⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟− −⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠⎝ ⎠
⊕
⊕
u u
JJ I J u u u u
JJ I u u
⎤⎢ ⎥⎣ ⎦
. In order to determine the inverse, we need to evaluate 1−A and ( where ) 1n −−B A
( )2 2
1
1 11 1
s
s
Nn
n s Ms ns ns s
uM N N N
σ=
⎛ ⎞⎡ ⎤⎛ ⎞n′= − + −⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟− −⎢ ⎥⎝ ⎠⎣ ⎦⎝ ⎠
⊕J
A I J u u and
( ) ( )2 2
1
1 11 11 1
s
s
Nn
Ms ns n s n ns s
n nn uN N M N N N
σ=
⎛ ⎞ ⎡ ⎤⎛ ⎞⎛ ⎞⎛ ⎞ ⎛ ⎞ ′− = − − − − + −⎜ ⎟ ⎢ ⎥⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟− −⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎣ ⎦⎝ ⎠⊕
JB A J I u u .
We invert these matrices using the results of pattern matrices where if is a
square non-singular matrix, and n n×R
1n×s and
1n×u are column vectors. Then
. In our example for ( ) ( ) 11 1 1 11−− − − −′ ′+ = − +R su R u R s R su R 1−′ ′= +A R su ,
2 2
1
11
s
s
Nn
n s Ms nss s
uM N
σ=
⎡ ⎤⎛ ⎞= − +⎢ ⎥⎜ ⎟ −⎢ ⎥⎝ ⎠⎣ ⎦⊕
JR I J , 1
nN= −s u and 1
1 nN=
−u u .
First, in order to get , we need to find the inverse of each diagonal block. 1−R
C07bx06 4/17/2007 10
2 2
2 22
11
1
s
s
s
nn s Ms
s
s Msn s ns
s
uM N
uM N
σ
σσ
⎛ ⎞− +⎜ ⎟ −⎝ ⎠
⎛ ⎞= − −⎜ ⎟−⎝ ⎠
JI J
I J
ns
Since we know, ( ) 12
1A A A
ba ba a Aab
−− = +−
I J I J A
So
( )
2 2
12 22
2 2 222 2
111
1
s
s Ms
ss Msn s ns ns ns
s s s Mss s s
s
uM Nu
M N unM N
σσσ
σ σσ σ
−⎛ ⎞
−⎜ ⎟⎡ ⎤ −⎛ ⎞ ⎝ ⎠− − = +⎢ ⎥⎜ ⎟− ⎛ ⎞⎝ ⎠⎣ ⎦ − −⎜ ⎟−⎝ ⎠
I J I J
( )
2 2
12 2 21 22 2
11
1
s MsN
sns nss
s s Mss s s
s
uM N
unM N
σ
σ σσ σ
−
=
⎡ ⎤⎛ ⎞−⎢ ⎥⎜ ⎟−⎝ ⎠⎢ ⎥= ⊕ +
⎢ ⎥⎛ ⎞− −⎢ ⎥⎜ ⎟−⎢ ⎥⎝ ⎠⎣ ⎦
R I J and
( ) ( )
( ) ( )
2 2
12 2 21 22 2
2 22 2
2
2 2 221 2 2
11 11
1
111
1
s MsN
sn ns nss
s s Mss s s
s
s Mss sN
ss s
s s s Mss s s
s
uM N
N N unM N
u u nM Nu n
N N unM N
σ
σ σσ σ
σ
σ σσ σ
−
=
=
⎛ ⎞⎡ ⎤⎛ ⎞−⎜ ⎟⎢ ⎥⎜ ⎟−⎜ ⎝ ⎠⎢ ⎥′ ′= − ⊕ +⎜ ⎟⎢ ⎥− ⎛ ⎞
⎜ ⎟− −⎢ ⎥⎜ ⎟⎜ ⎟−⎢ ⎥⎝ ⎠⎣ ⎦⎝ ⎠⎛ ⎡ ⎤⎛ ⎞
−⎢ ⎥⎜ ⎟−⎝ ⎠⎢ ⎥= − +⎢ ⎥− ⎛ ⎞
− −⎢ ⎥⎜ ⎟−⎢ ⎥⎝ ⎠⎣ ⎦
∑
u R s u I J u
( )
n⎟
( )
2
2 21 2
11
1
Ns s
s s Mss s
s
u nN N un
M Nσσ=
⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟= − ⎜ ⎟⎜ ⎟− ⎛ ⎞⎜ ⎟− −⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟−⎝ ⎠⎝ ⎠⎝ ⎠
∑
C07bx06 4/17/2007 11
( ) ( )
( ) ( )( )
( )
( ) ( )
21
2 21 2
2
2 21 2
2 22 2
11 11
1
1 11
1
111
1
Ns s
s s Mss s
s
Ns s
s s Mss s
s
s Mss s s s
s
s
u nN N un
M N
u nNN N un
M N
uN n uM N
N N
σσ
σσ
σσ
σ
−
=
=
⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟′+ = − ⎜ ⎟⎜ ⎟− ⎛ ⎞⎜ ⎟− −⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟−⎝ ⎠⎝ ⎠⎝ ⎠
⎛ ⎞⎡ ⎤⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥= − −⎜ ⎟⎢ ⎥− ⎛ ⎞⎜ ⎟− −⎢ ⎥⎜ ⎟⎜ ⎟−⎢ ⎥⎝ ⎠⎣ ⎦⎝ ⎠
⎡ ⎤⎛ ⎞− − − −⎢ ⎥⎜ ⎟−⎝ ⎠⎣ ⎦=
−
∑
∑
u R s
( )
n
2 21 2
2
21 2
1
11
11
N
s s Mss
s
ssN
s
s s s Mss
s
unM N
nM
N n n uM N
σ
σ
σ
=
=
⎛ ⎞⎡ ⎤⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥⎛ ⎞⎜ ⎟⎢ ⎥− −⎜ ⎟⎜ ⎟−⎢ ⎥⎝ ⎠⎣ ⎦⎝ ⎠
⎛ ⎞⎡ ⎤⎛ ⎞−⎜ ⎟⎢ ⎥⎜ ⎟
⎜ ⎝ ⎠ ⎟⎢ ⎥= ⎜ ⎟⎢ ⎥⎛ ⎞⎜ ⎟− +⎢ ⎥⎜ ⎟⎜ ⎟−⎢ ⎥⎝ ⎠⎣ ⎦⎝ ⎠
∑
∑
( ) 11
2
21 2
11
11
11
ssN
s
s s s Mss
s
nM
N n n uM N
σ
σ
−−
=
′+ =⎛ ⎞⎡ ⎤⎛ ⎞
−⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎝ ⎠⎢ ⎥⎜ ⎟⎢ ⎥⎛ ⎞
− +⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟−⎢ ⎥⎝ ⎠⎣ ⎦⎝ ⎠
∑
u R s
⎟
And another part is
C07bx06 4/17/2007 12
( ) ( )
( )
( )
2 2
12 2 21 22 2
2 2
2212
12
11 11
1
11 11
11
1 11
1
s sN
sn nI nss
s s ss s s
s
s ssN
s sn nss
ss s ss
s
N
n ss
s
uM N
N N unM N
u nM N u
N N n n uM N
N N n
σ
σ σσ σ
σ
σσ
σ
−
=
=
=
⎡ ⎤⎛ ⎞−⎢ ⎥⎜ ⎟−⎝ ⎠⎢ ⎥′ ′= − ⊕ +
⎢ ⎥− ⎛ ⎞− −⎢ ⎥⎜ ⎟−⎢ ⎥⎝ ⎠⎣ ⎦
⎡ ⎤⎛ ⎞−⎢ ⎥⎜ ⎟−⎝ ⎠⎢ ⎥ ′= − ⊕ +
⎢ ⎥− ⎛ ⎞− +⎢ ⎥⎜ ⎟ −⎢ ⎥⎝ ⎠⎣ ⎦
= − ⊕−
−
su R u u I J
u 1
u
ns
2
1
s nss s
s
un u
M N
⎡ ⎤⎢ ⎥⎢ ⎥ ′⎢ ⎥⎛ ⎞
+⎢ ⎥⎜ ⎟ −⎢ ⎥⎝ ⎠⎣ ⎦
1
C07bx06 4/17/2007 13
So finally, And ( ) ( )
( )
( )
11 1 1 1 1
11 1 1
2 2
2 2 21 22 2 2
21 2
1
1
11 1
11 1
11
s MsN
sns nss
s s Ms ss s s s
s s
s s s Mss
s
uM N
u nnM N M
N n n uM N
σ
σ σσ σ σ
σ
−− − − − −
−− − −
=
=
′ ′ ′+ = − +
⎡ ⎤′ ′= − +⎢ ⎥⎣ ⎦
⎡ ⎤⎛ ⎞−⎢ ⎥⎜ ⎟−⎝ ⎠⎢ ⎥= ⊕ + −
⎢ ⎥⎛ ⎞ ⎡ ⎤⎛ ⎞− −⎢ ⎥ −⎜ ⎟ ⎢ ⎥⎜ ⎟−⎢ ⎥⎝ ⎠⎣ ⎦ ⎝ ⎠⎢ ⎥⎢ ⎥⎛ ⎞
− +⎢ ⎥⎜ ⎟ −⎢ ⎥⎝ ⎠⎣ ⎦
R su R u R s R su R
R I u R s su R
I J I( )
( )
212
2 2
2 2 21 22 2
1 11
11
11
1
N
n s nsss s s
sNs
s MsN
sns nss
s s Mss s s
s
uN N n n u
M N
uM N
unM N
σ
σ
σ σσ σ
=
=
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎡ ⎤⎢ ⎥⎢ ⎥⎛ ⎞⎢ ⎥⎢ ⎥ ′− ⊕⎜ ⎟⎢ ⎥⎜ ⎟ ⎢ ⎥−⎛ ⎞ ⎛ ⎞⎝ ⎠⎢ ⎥− +⎢ ⎥⎜ ⎟ ⎜ ⎟⎢ ⎥−⎢ ⎥⎝ ⎠⎣ ⎦⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟
⎝ ⎠⎣ ⎦
⎛ ⎞⎡ ⎤⎛ ⎞−⎜ ⎟⎢ ⎥⎜ ⎟−⎜ ⎝ ⎠ ⎟⎢ ⎥= ⊕ +⎜ ⎟⎢ ⎥⎛ ⎞
⎜ ⎟− −⎢ ⎥⎜ ⎟⎜ −⎢ ⎥⎝ ⎠⎣ ⎦⎝ ⎠
∑
u 1
I J
( )
( )
2122
21 2
2 2
21 22
1 1
11 11
11
11
N
n s nsss s ss
ssNss
s s s Mss
s
s MsN
snss
ss s s
un n unM NM
Nn n uM N
uM N
n
σσ
σ
σ
σσ σ
=
=
=
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ ′+ ⊕⎢ ⎥⎢ ⎥⎛ ⎞ ⎛ ⎞⎡ ⎤⎛ ⎞⎢ ⎥− +⎢ ⎥−⎜ ⎟ ⎜ ⎟⎢ ⎥⎜ ⎟⎟ ⎢ ⎥−⎢ ⎥⎝ ⎠⎣ ⎦⎜ ⎝ ⎠ ⎟⎢ ⎥−⎢ ⎥⎜ ⎟⎢ ⎥⎛ ⎞⎢ ⎥− +⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟−⎢ ⎥⎝ ⎠⎣ ⎦⎝ ⎠⎣ ⎦
⎛ ⎞−⎜ ⎟−⎝ ⎠= ⊕ +
−
∑
I u 1
I
( )
2
2 2 212 22
21 2
1
111 11
11
Ns
ns nsss Ms s s ss
ssNs ss
s s s Mss
s
uu n n un
M N M NMN
n n uM N
σ σσ
σ
=
=
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎛ ⎞⎡ ⎤ ⎡ ⎤⎢ ⎥⎜ ⎟⎢ ⎥ ⎢ ⎥⎢ ⎥⎜ ⎟⎢ ⎥ ⎢ ⎥+ ⊕⎢ ⎥⎜ ⎟⎢ ⎥ ⎢ ⎥⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎡ ⎤⎛ ⎞⎢⎜ ⎟− − +⎢ ⎥ ⎢ ⎥−⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟ ⎢− −⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦⎝ ⎠ ⎜ ⎝ ⎠ ⎟⎢ ⎥−⎢ ⎜ ⎟⎢ ⎥⎛ ⎞⎢ − +⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎜ ⎟−⎢ ⎥⎝ ⎠⎣ ⎦⎝ ⎠⎣ ⎦
∑
J I J
( )( )
2 2
2 22 21 122 2 2
21 2
11 1 1
111
11
s MsN N
sns ns nss s
s ss Ms ss s s sN
s s
s s s Mss
s
uM N
u nnM N M
Nn n uM N
σ
σ σσσ σ σ
σ
= =
=
⎥⎥⎥⎥⎥
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎡ ⎤⎛ ⎞ ⎢ ⎥−⎢ ⎥⎜ ⎟− ⎢ ⎥⎝ ⎠⎢ ⎥= ⊕ + + ⊕⎢ ⎥⎢ ⎥⎛ ⎞ ⎛ ⎞⎡ ⎤⎛ ⎞⎢ ⎥− −⎢ ⎥ −⎜ ⎟ ⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥−⎢ ⎥⎝ ⎠⎣ ⎦ ⎜ ⎝ ⎠ ⎟⎢ ⎥−⎢ ⎥⎜ ⎟⎢ ⎥⎛ ⎞⎢ ⎥− +⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟−⎢ ⎥⎝ ⎠⎣ ⎦⎝ ⎠⎣ ⎦
∑
I J I( )
2 2
2 222 2
1
1
s MsN
sns s
s Mss s s
s
uM N
unM N
σ
σσ σ=
⎛ ⎞⎛⎡ ⎤⎛ ⎞−⎜ ⎟⎜⎢ ⎥⎜ ⎟−⎜ ⎝ ⎠ ⎟⎜⎢ ⎥+ ⊕⎜ ⎟⎜⎢ ⎥⎛ ⎞
⎜ ⎟⎜− −⎢ ⎥⎜ ⎟⎜ ⎟⎜−⎢ ⎥⎝ ⎠⎣ ⎦⎝ ⎠⎝
J
For the second item:
C07bx06 4/17/2007 14
( )( )
2 2
2 2 21 122 22
21 2
11 1
1 11
11
s MsN N
sns nss s
s s Msss s ssN
ss
s s s Mss
s
uM N
un nM NM
Nn n uM N
σ
σ σσ σσ
σ
= =
=
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥ ⎛ ⎞⎡ ⎤⎛ ⎞⎢ ⎥ −⎜ ⎟⎢ ⎥⎜ ⎟−⎢ ⎥ ⎜ ⎝ ⎠ ⎟⎢ ⎥⊕ + ⊕⎢ ⎥ ⎜ ⎟⎢ ⎥⎛ ⎞ ⎛ ⎞⎡ ⎤⎛ ⎞⎢ ⎥ ⎜ ⎟− −⎢ ⎥−⎜ ⎟ ⎜ ⎟⎢ ⎥⎜ ⎟ ⎜ ⎟⎢ ⎥ −⎢ ⎥⎝ ⎠⎣ ⎦⎜ ⎝ ⎠ ⎟ ⎝ ⎠⎢ ⎥−⎢ ⎥⎜ ⎟⎢ ⎥⎛ ⎞⎢ ⎥− +⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟−⎢ ⎥⎝ ⎠⎣ ⎦⎝ ⎠⎣ ⎦
∑
I J
( )
2
22
2
2 2122
21 2
11
1 1
11 11
11
sns
s s ss
s
Ns
ss s s ss
ssNss
s s s Mss
s
un n uM N
u un n unM NM
Nn n uM N
σ
σσσ
σ
=
=
⎛ ⎞⎡ ⎤⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥⎛ ⎞⎜ ⎟− +⎢ ⎥⎜ ⎟⎜ ⎟−⎢ ⎥⎝ ⎠⎣ ⎦⎝ ⎠
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
= ⊕ +⎢ ⎥⎛ ⎞ ⎛ ⎞⎡ ⎤⎛ ⎞⎢ ⎥ − +−⎜ ⎟ ⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥ −⎝ ⎠⎜ ⎝ ⎠ ⎟⎢ ⎥−⎢ ⎥⎜ ⎟⎢ ⎥⎛ ⎞⎢ ⎥− +⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟−⎢ ⎥⎝ ⎠⎣ ⎦⎝ ⎠⎣ ⎦
∑
J
( )
( )
2 2
2
2 2 222 2 2
2
21 2
1
11 1
1
11
11
s Ms
ss sns
s s s s Mss s s s
s s
ssN
s
s s s Mss
s
uM Nn
n n u unM N M N
nM
Nn n uM N
σ
σσ σ σ
σ
σ=
⎛ ⎞⎛ ⎞⎡ ⎤ ⎛ ⎞⎜ ⎟−⎜ ⎟⎢ ⎥ ⎜ ⎟−⎜ ⎟⎜ ⎝⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟− + − −⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟− −⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦⎝ ⎠⎝ ⎠
⎡
=⎛ ⎞⎡ ⎤⎛ ⎞
−⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎝ ⎠ ⎟⎢ ⎥− ⎜ ⎟⎢ ⎥⎛ ⎞
− +⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟−⎢ ⎥⎝ ⎠⎣ ⎦⎝ ⎠
∑
J⎠ ⎟
2
21 22 1
1
Ns
nsss s s
ss
u
n n uM N
σ=
⎤⎢ ⎥⎢ ⎥⎢ ⎥ ⎛ ⎞
⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥⊕⎜ ⎟⎢ ⎥⎡ ⎤⎛ ⎞⎜ ⎟⎢ ⎥ − +⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥ −⎝ ⎠⎣ ⎦⎝ ⎠⎢ ⎥
⎢ ⎥⎢ ⎥⎣ ⎦
J
Hence,
( ) ( )
( )( )
111 1 1 1
2 2
2 2 21 22 2 2
21 2
1
11 1
111
11
s MsN
sns nss
s s Ms ss s s sN
s s
s s s Mss
s
uM N
u nnM N M
Nn n uM N
σ
σ σσ σ σ
σ
−−− − − −
=
=
⎡ ⎤′ ′ ′= + = − +⎢ ⎥⎣ ⎦⎡⎢⎢
⎡ ⎤⎛ ⎞−⎢ ⎥⎜ ⎟−⎝ ⎠⎢ ⎥= ⊕ + +
⎢ ⎥⎛ ⎞ ⎛ ⎞⎡ ⎤⎛ ⎞− −⎢ ⎥ −⎜ ⎟ ⎜ ⎟⎢ ⎥⎜ ⎟−⎢ ⎥⎝ ⎠⎣ ⎦ ⎜ ⎝ ⎠ ⎟⎢ ⎥− ⎜ ⎟⎢ ⎥⎛ ⎞− +⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟−⎢ ⎥⎝ ⎠⎣ ⎦⎝ ⎠⎣
∑
A R su R I u R s su R
I J
( )
2
21 22
2 2
2 2 212 2
22
11
11 1 1
1 111
11
Ns
nsss s s
ss
s MsN
snss
s s s s Ms ss s
s s
s s s Mss
s
u
n n uM N
uM N
n n u nM N M
Nn n uM N
σ
σ
σ σσ σ
σ
=
=
⎤⎥⎥
⎢ ⎥ ⎛ ⎞⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥⊕⎜ ⎟⎢ ⎥⎡ ⎤⎛ ⎞⎜ ⎟⎢ ⎥ − +⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥ −⎝ ⎠⎣ ⎦⎝ ⎠⎢ ⎥
⎢ ⎥⎢ ⎥
⎦
⎛ ⎞−⎜ ⎟−⎝ ⎠= ⊕ +
⎛ ⎞ ⎡ ⎤⎛ ⎞− + −⎜ ⎟ ⎢ ⎥⎜ ⎟−⎝ ⎠ ⎝ ⎠⎢ ⎥−⎢ ⎥⎛ ⎞
− +⎢ ⎥⎜ ⎟ −⎢ ⎥⎝ ⎠⎣ ⎦
J
I +2
222
1
11
sns
s s ssN
s
u
n n uM N
σ
=
⎡ ⎤⎛ ⎞⎡ ⎤⎢ ⎥⎜ ⎟⎢ ⎥⎢ ⎥⎜ ⎟⎢ ⎥⎢ ⎥⎜ ⎟⎢ ⎥ ⎡ ⎤⎢ ⎥⎜ ⎟⎢ ⎥ ⎢ ⎥⎢ ⎥⎜ ⎟⎢ ⎥ ⎢ ⎥⎢ ⎥⎜ ⎟⎢ ⎥ ⎢ ⎥⎛ ⎞ ⎡ ⎤⎢ ⎥⎛ ⎞⎜ ⎟⎢ ⎥ ⎢ ⎥⎜ ⎟ − +⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥ ⎢ ⎥−⎜ ⎟ ⎝ ⎠⎣ ⎦⎣ ⎦⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦⎝ ⎠⎣ ⎦
∑
J
C07bx06 4/17/2007 15
Next, determine the inverse ( 1n )−−B A where
( ) ( )2 2
1
1 11 11 1
s
s
Nn
Ms ns n s n ns s
n nn uN N M N N N
σ=
⎛ ⎞ ⎡ ⎤⎛ ⎞⎛ ⎞⎛ ⎞ ⎛ ⎞ ′− = − − − − + −⎜ ⎟ ⎢ ⎥⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟− −⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎣ ⎦⎝ ⎠⊕
JB A J I u u .
( ) ( ) 11 1 1 11−− − − −′ ′+ = − +R su R u R s R su R 1−′ .
2 2
1
1 11
s
s
Nn
Ms ns n ss s
n uN N M
σ=
⎛ ⎞⎛ ⎞⎛ ⎞⎛ ⎞= − − − −⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟−⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠⊕
JR J I , 1
nN=s u and 1 1
1 nn
N N⎛ ⎞= −⎜ ⎟− ⎝ ⎠
u u .
First, in order to get , we need to find the inverse of each diagonal block. 1−R
2 2
22 2
1 11
1 11
s
s
s
nMs ns n s
s
ss n Ms
s
n uN N M
n uN N M ns
σ
σσ
⎛ ⎞⎛ ⎞⎛ ⎞− − − −⎜ ⎟⎜ ⎟⎜ ⎟−⎝ ⎠⎝ ⎠ ⎝ ⎠⎡ ⎤⎛ ⎞⎛ ⎞= − − − −⎢ ⎥⎜ ⎟⎜ ⎟−⎝ ⎠⎝ ⎠⎣ ⎦
JJ I
I J
Since we know, ( ) 12
1A A A
ba ba a Aab
−− = +−
I J I J A
So
( ) ( )
122 2
22
2 222 2 2
1 11
1 111
1 11
s
s s
ss n s ns
s
sMs
sn n
s ss s s Ms
s
n uN N M
n uN N M
nn uN N M
σσ
σ
σ σσ σ
−⎛ ⎞⎡ ⎤⎛ ⎞⎛ ⎞− − − −⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟⎜ ⎟−⎝ ⎠⎝ ⎠⎣ ⎦⎝ ⎠
⎡ ⎤⎛ ⎞⎛ ⎞− −⎜ ⎟⎜ ⎟⎢ ⎥−⎝ ⎠⎝ ⎠⎣ ⎦= +− ⎡ ⎤⎛ ⎞⎛ ⎞− − − − −⎜ ⎟⎜ ⎟⎢ ⎥−⎝ ⎠⎝ ⎠⎣ ⎦
I J
I J
( ) ( )
22
12 21 22 2 2
1 111
1 11
s s
sMsN
sn ns
s ss s s Ms
s
n uN N M
nn uN N M
σ
σ σσ σ
−
=
⎡ ⎤⎡ ⎤⎛ ⎞⎛ ⎞− −⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥−⎝ ⎠⎝ ⎠⎣ ⎦⎢ ⎥= ⊕ +⎢ ⎥− ⎡ ⎤⎛ ⎞⎛ ⎞− − − − −⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥−⎝ ⎠⎝ ⎠⎢ ⎥⎣ ⎦⎣ ⎦
R I J and
C07bx06 4/17/2007 16
( ) ( ) ( )
( )
22
12 21 22 2 2
22
2
2
1 111 11
1 1 11
1 111 1
1
s s
sMsN
sn n n ns
s ss s s Ms
s
sMs
sMs s
s
n uN N Mn
N N N nn uN N M
n uN N Mu nn
N N N
σ
σ σσ σ
σ
σ
−
=
⎡ ⎤⎡ ⎤⎛ ⎞⎛ ⎞− −⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥−⎝ ⎠⎝ ⎠⎛ ⎞ ⎣ ⎦⎢ ⎥′ ′= − ⊕ +⎜ ⎟ ⎢ ⎥− − ⎡ ⎤⎝ ⎠ ⎛ ⎞⎛ ⎞− − − − −⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥−⎝ ⎠⎝ ⎠⎢ ⎥⎣ ⎦⎣ ⎦
⎡⎛ ⎞⎛ ⎞− −⎜ ⎟⎜ ⎟−⎝ ⎠⎝ ⎠⎛ ⎞= − +⎜ ⎟− −⎝ ⎠
u R s u I J u
( ) ( )
2 2
221 2 2 21 11
Ms sN
s ss s s Ms
s
u n
nn uN N M
σσ σ=
⎡ ⎤⎤⎢ ⎥⎢ ⎥
⎣ ⎦⎢ ⎥⎢ ⎥⎡ ⎤⎛ ⎞⎛ ⎞− − − − −⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥−⎝ ⎠⎝ ⎠⎢ ⎥⎣ ⎦⎣ ⎦
∑
( ) ( ) ( )
( )( )
( )
22 2 2
21
2 221 2 2 2
2
22
1 1111 1 1
1 1 11
11 1
1
sMs Ms sN
sMs s
s s ss s s Ms
s
Ms s
s s Ms
n u u nN N Mu nn
N N N nn uN N M
N N u nN nN N N n N nn u
N N
σ
σ σσ σ
σ
−
=
⎡ ⎤⎡ ⎤⎛ ⎞⎛ ⎞− −⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥−⎝ ⎠⎝ ⎠⎛ ⎞ ⎣ ⎦⎢ ⎥′+ = + − +⎜ ⎟ ⎢ ⎥− − ⎡ ⎤⎝ ⎠ ⎛ ⎞⎛ ⎞− − − − −⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥−⎝ ⎠⎝ ⎠⎢ ⎥⎣ ⎦⎣ ⎦
−−= +
− − −⎛ ⎞⎛ ⎞− − ⎜ ⎟⎜ ⎟−⎝ ⎠⎝ ⎠
∑u R s
( )
( ) ( ) ( )
( ) ( )
21 2
22 2 2
2 22 2
111
1 11
N
s s
s
ss s Ms Ms s
s
s ss s Ms
s
M
N nN N n u N n u nN N MN n
N N N nN n n uN N M
σ
σσ
σσ
=
=
⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎡ ⎤
−⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦⎣ ⎦
⎡ ⎤⎡ ⎤⎡ ⎤⎡ ⎤−⎛ ⎞⎛ ⎞− − − − + −⎢ ⎥⎢ ⎥⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥−⎝ ⎠⎝ ⎠− ⎢ ⎥⎣ ⎦⎢ ⎥⎣ ⎦= ⎢ ⎥⎢ ⎥− ⎡ ⎤⎡ ⎤−⎛ ⎞⎛ ⎞⎢ ⎥⎢ ⎥− − − −⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎢ ⎥−⎢ ⎥⎝ ⎠⎝ ⎠⎣ ⎦⎣ ⎦⎣ ⎦⎣ ⎦
∑
1
2
1 2 2
11
111
N
ssN
s
s ss s Ms
s
nM
N n N n n uM N N
σ
σ=
⎡ ⎤⎛ ⎞− +⎢ ⎥⎜ ⎟
⎢ ⎥⎝ ⎠= ⎢ ⎥⎡ ⎤⎛ ⎞ −⎛ ⎞⎛ ⎞⎢ ⎥− + −⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟−⎢ ⎥⎝ ⎠⎝ ⎠⎝ ⎠⎣ ⎦⎣ ⎦
∑
∑
C07bx06 4/17/2007 17
( ) ( ) ( )
( )
22
12 21 22 2 2
21
1 111 11
1 1 11
1 111 1
1
s s
sMsN
sn n n ns
s ss s s Ms
s
Ns s
n ss
n uN N Mn
N N N nn uN N M
nN Nu nn
N N N
σ
σ σσ σ
σ
−
=
=
⎡ ⎤⎡ ⎤⎛ ⎞⎛ ⎞− −⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎡ ⎤ −⎝ ⎠⎝ ⎠⎛ ⎞ ⎣ ⎦⎢ ⎥′ ′= − ⊕ +⎢ ⎥⎜ ⎟ ⎢ ⎥− − ⎡ ⎤⎝ ⎠ ⎛ ⎞⎛ ⎞⎣ ⎦ − − − − −⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥−⎝ ⎠⎝ ⎠⎢ ⎥⎣ ⎦⎣ ⎦
⎛ ⎞⎛ ⎞−⎜ ⎟⎜ ⎟−⎝ ⎠⎝ ⎠⎛ ⎞= − ⊕ +⎜ ⎟− −⎝ ⎠
su R u u I J
u( ) ( )
( ) ( )
22 2
222 2 2
22
2 212 2
1 11
1 111 1 1
1 1 11
sMs s s
sns
ss s s Ms
s
sMs sN
ss sn s
s s ss s Ms
s
u u nM
nn uN N M
n u nN N Mu nn
N N N nnn uN N M
σ
σσ σ
σ
σ σσ=
⎛ ⎞⎡ ⎤−⎜ ⎟⎢ ⎥
⎣ ⎦⎜ ⎟ ′⎜ ⎟⎡ ⎤⎛ ⎞⎛ ⎞− − − − −⎜ ⎟⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟−⎝ ⎠⎝ ⎠⎣ ⎦⎝ ⎠
⎡ ⎤⎛ ⎞⎛ ⎞− −⎜ ⎟⎜ ⎟⎢ ⎥−⎝ ⎠⎝ ⎠⎛ ⎞ ⎣ ⎦= − ⊕ +⎜ ⎟− − ⎡⎝ ⎠ ⎛ ⎞⎛ ⎞− − − −⎜ ⎟⎜ ⎟⎢ −⎝ ⎠⎝ ⎠⎣
1
u
( ) ( )21
2 2
1 11 1 1
1
ns
Ns s
n nsss s
s s Mss
u nnN N N nnn u
N N Mσσ
=
⎛ ⎞⎜ ⎟⎜ ⎟ ′⎜ ⎟⎤⎜ ⎟⎥⎜ ⎟⎦⎝ ⎠
⎛ ⎞⎜ ⎟
⎛ ⎞ ⎜ ⎟ ′= − ⊕⎜ ⎟ ⎜ ⎟− ⎡ ⎤⎝ ⎠ ⎛ ⎞⎛ ⎞− − − −⎜ ⎟⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟−⎝ ⎠⎝ ⎠⎣ ⎦⎝ ⎠
1
u 1
C07bx06 4/17/2007 18
( ) ( )
( ) ( )
11 1 1 1 1
22
2 21 22 2 2
2
2
1
1 111
1 11
1
11
111
s s
sMsN
sn ns
s ss s s Ms
s
ss
s
ss
s
n uN N M
nn uN N M
nM
N n N nM N N
σ
σ σσ σ
σ
σ
−− − − − −
=
′ ′ ′+ = − +
⎡ ⎤⎡ ⎤⎛ ⎞⎛ ⎞− −⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥−⎝ ⎠⎝ ⎠⎣ ⎦⎢ ⎥= ⊕ +⎢ ⎥− ⎡ ⎤⎛ ⎞⎛ ⎞− − − − −⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥−⎝ ⎠⎝ ⎠⎢ ⎥⎣ ⎦⎣ ⎦
−⎛ ⎞− +⎜ ⎟⎝ ⎠
⎛ ⎞ −⎛ ⎞⎛− + −⎜ ⎟ ⎜ ⎟−⎝ ⎠⎝⎝ ⎠
R su R u R s R su R
I J
( ) ( )
( ) ( )
1 2
22
2 21 22 2 2
2
2 12 2
1 111
1 11
1 1 11
s s
N
ss Ms
sMsN
sn ns
s ss s s Ms
s
NMs s
s
s s Ms
n u
n uN N M
nn uN N M
u nN nN N nn u
N N
σ
σ σσ σ
σ
=
=
=
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎡ ⎤⎞⎢ ⎥⎢ ⎥⎜ ⎟⎢ ⎥⎠⎣ ⎦⎣ ⎦
⎡ ⎤⎡ ⎤⎛ ⎞⎛ ⎞− −⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥−⎝ ⎠⎝ ⎠⎣ ⎦⎢ ⎥⊕ +⎢ ⎥− ⎡ ⎤⎛ ⎞⎛ ⎞− − − − −⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥−⎝ ⎠⎝ ⎠⎢ ⎥⎣ ⎦⎣ ⎦
−⊕
− ⎛ ⎞⎛ ⎞− − −⎜ ⎟⎜ ⎟−⎝ ⎠⎝ ⎠
∑
I J
2 sns s
s
nMσ
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎡ ⎤
−⎜ ⎟⎢ ⎥⎜ ⎟⎣ ⎦⎝ ⎠
J
for the second item:
C07bx06 4/17/2007 19
( )
( )
2
2
1 2 2
22
2
2212 2
1
11
111
1 111
1 11
ssN
s
s ss s Ms
s
sMsN
sMs s
sss s
s s Mss
N nN N
nM
N n N n n uM N N
n uN N Mu n
nnn uN N M
σ
σ
σ
σσσ
=
=
−−
⎡ ⎤⎛ ⎞− +⎢ ⎥⎜ ⎟
⎢ ⎥⎝ ⎠⎢ ⎥⎡ ⎤⎛ ⎞ −⎛ ⎞⎛ ⎞⎢ ⎥− + −⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟−⎢ ⎥⎝ ⎠⎝ ⎠⎝ ⎠⎣ ⎦⎣ ⎦
⎛ ⎞ ⎛ ⎞⎛ ⎞− −⎜ ⎟ ⎜ ⎟⎜ ⎟−⎝ ⎠⎝ ⎠⎜ ⎟⊕ +⎜ ⎟ −⎡ ⎤⎛ ⎞⎛ ⎞− − − −⎜ ⎟⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟−⎝ ⎠⎝ ⎠⎣ ⎦⎝ ⎠
∑
( ) ( )
( )
( )
222 2 2
2 2
12 2
1 2 2
1 11
1
11
111
s
s
ns
s s s Mss
NMs s
ss
s s sNs
s ss s Ms
s
n
nn uN N M
N nN N u n
nn
MN n N n n u
M N N
σσ σ
σ σ
σ
=
=
⎡ ⎤⎛ ⎞⎡ ⎤⎢ ⎥⎜ ⎟⎢ ⎥
⎣ ⎦⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟⎡ ⎤⎛ ⎞⎛ ⎞− − − − −⎢ ⎥⎜ ⎟⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟−⎝ ⎠⎝ ⎠⎢ ⎥⎣ ⎦⎝ ⎠⎣ ⎦
−−
= ⊕⎡ ⎤⎛ ⎞
− +⎢ ⎥ − −⎜ ⎟⎢ ⎥⎝ ⎠⎢ ⎥⎡ ⎤⎛ ⎞ −⎛ ⎞⎛ ⎞⎢ ⎥− + −⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟−⎢ ⎥⎝ ⎠⎝ ⎠⎝ ⎠⎣ ⎦⎣ ⎦
∑
J
2221 1
1
sn
s sMs
s
nn uN N M
σ
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞⎡ ⎤⎜ ⎟⎛ ⎞⎛ ⎞− −⎜ ⎟⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟−⎝ ⎠⎝ ⎠⎣ ⎦⎝ ⎠⎝ ⎠
J
so finally,
C07bx06 4/17/2007 20
( ) ( )
( ) ( )
( )
11 1 1 1 1
22
2 21 22 2 2
2
2
1
1 111
1 11
1
1
111
s s
sMsN
sn ns
s ss s s Ms
s
ss
s
ss
s
n uN N M
nn uN N M
N nN N
nM
n NM N
σ
σ σσ σ
σ
σ
−− − − − −
=
′ ′ ′+ = − +
⎡ ⎤⎡ ⎤⎛ ⎞⎛ ⎞− −⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥−⎝ ⎠⎝ ⎠⎣ ⎦⎢ ⎥= ⊕ +⎢ ⎥− ⎡ ⎤⎛ ⎞⎛ ⎞− − − − −⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥−⎝ ⎠⎝ ⎠⎢ ⎥⎣ ⎦⎣ ⎦
−−
−⎛ ⎞− +⎜ ⎟⎝ ⎠
⎛ ⎞ −⎛ ⎞− + −⎜ ⎟ ⎜ ⎟−⎝ ⎠⎝ ⎠
R su R u R s R su R
I J
( )
( ) ( )
2
21 22 2
1 2
22
222 2
1
1 11
1 111
1 11
s
s
NMs s
nss s
s s MsNs
ss Ms
sMs
sn
ss s s
N
s
u n
nnn uN N M
n n uN
n uN N M
nnN N
σσ
σ
σσ σ
=
=
=
⎛ ⎞⎜ ⎟⎜ ⎟
⊕ ⎜ ⎟⎡ ⎤ ⎛ ⎞⎡ ⎤⎜ ⎟⎛ ⎞⎛ ⎞⎢ ⎥ − − − −⎜ ⎟⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟−⎝ ⎠⎝ ⎠⎢ ⎥ ⎣ ⎦⎝ ⎠⎝ ⎠⎢ ⎥⎡ ⎤⎛ ⎞⎢ ⎥⎢ ⎥⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎣ ⎦
⎡ ⎤⎛ ⎞⎛ ⎞− −⎜ ⎟⎜ ⎟⎢ ⎥−⎝ ⎠⎝ ⎠⎣ ⎦+− ⎛ ⎞⎛ ⎞− − − −⎜ ⎟⎜−⎝ ⎠⎝
= ⊕
∑
J
I
( )
( )
22
2
222 2 2
1 2 2
1
11 11
111
sns
Mss
Ms s
s s ss s s MsN
s s
s ss s Ms
s
uM
N nN N u n
n nnn uM N N M
n N n n uM N N
σ
σσ σ
σ=
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎡ ⎤
−⎢ ⎥⎟⎢ ⎥⎠⎢ ⎥⎣ ⎦⎣ ⎦⎛ ⎞− ⎜ ⎟⎜ ⎟−
− ⎜ ⎟⎡ ⎤ ⎛ ⎞⎛ ⎞ ⎡ ⎤⎜ ⎟⎛ ⎞⎛ ⎞− +⎢ ⎥ − − − −⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟−⎝ ⎠⎝ ⎠⎢ ⎥⎝ ⎠ ⎣ ⎦⎝ ⎠⎝ ⎠⎢ ⎥⎡ ⎤⎛ ⎞ −⎛ ⎞⎛ ⎞⎢ ⎥− + −⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟−⎢ ⎥⎝ ⎠⎝ ⎠⎝ ⎠⎣ ⎦⎣ ⎦
∑
J
Jsn
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
For a matrix , I n n= ⊗ − ⊗V I A J B ( ) 11 1 1I n n n −− − −= ⊗ − ⊗ −V I A J A B B A
( ) ( )
( )
111 1 1 1
2 2
2 2 212 2
21 2
1
11 1 1
1 111
11
s MsN
snss
s s s s Ms ss sN
s s
s s s Mss
s
uM N
n n u nM N M
Nn n uM N
σ
σ σσ σ
σ
−−− − − −
=
=
⎡ ⎤′ ′ ′= + = − +⎢ ⎥⎣ ⎦
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎛ ⎞ ⎢ ⎥−⎜ ⎟− ⎢ ⎥⎝ ⎠= ⊕ + ⎢ ⎥⎛ ⎞ ⎛ ⎞⎡ ⎤⎛ ⎞⎢− + −⎜ ⎟ ⎜ ⎟⎢ ⎥⎜ ⎟⎢−⎝ ⎠ ⎜ ⎝ ⎠ ⎟⎢ ⎥−⎢ ⎜ ⎟⎢ ⎥⎛ ⎞⎢ − +⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎜ ⎟−⎢ ⎥⎝ ⎠⎣ ⎦⎝ ⎠⎣ ⎦
∑
A R su R I u R s su R
I +2
222 1
1
sns
s s ss
s
u
n n uM N
σ
⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟⎡ ⎤⎢ ⎥⎜ ⎟⎢ ⎥⎢ ⎥⎜ ⎟⎢ ⎥⎢ ⎥⎜ ⎟⎢ ⎥
⎡ ⎤⎢ ⎥⎛ ⎞⎜ ⎟⎥ ⎢ ⎥− +⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟⎥ ⎢ ⎥−⎝ ⎠⎣ ⎦⎣ ⎦⎢ ⎥⎜ ⎟⎥⎢ ⎥⎜ ⎟⎥⎢ ⎥⎜ ⎟⎥⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
J
Simplification of this formula:
C07bx06 4/17/2007 21
Define 2
2 11
s s Mss s
s
n n ukM N
σ⎛ ⎞
= − +⎜ ⎟ −⎝ ⎠and 2 1 s
s ss
nvM
σ⎛ ⎞
= −⎜ ⎟⎝ ⎠
, so
( ) ( )2
21 12
11 1
11
ssN N
s s
s s ss s Mss
s
nM vN N
kn n uM N
σ
σ= =
⎛ ⎞⎡ ⎤⎛ ⎞−⎜ ⎟⎢ ⎥⎜ ⎟ ⎛ ⎞⎜ ⎝ ⎠ ⎟⎢ ⎥− = ⎜ ⎟⎜ ⎟⎢ ⎥⎛ ⎞ ⎝ ⎠⎜ ⎟− +⎢ ⎥⎜ ⎟⎜ ⎟−⎢ ⎥⎝ ⎠⎣ ⎦⎝ ⎠
∑ ∑−
And
( )
2 2
21
2 212
1
11
1
s MsN
s sns nsNs
s s s ss
s s
uM N u
k vk Nk
σ
σ σ−
=
=
⎡ ⎤⎛ ⎞⎛ ⎞−⎢ ⎥⎜ ⎟⎜ ⎟−⎝ ⎠⎢ ⎥⎜ ⎟= ⊕ +⎢ ⎥⎜ ⎟⎛ ⎞
−⎢ ⎥⎜ ⎟⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎝ ⎠⎣ ⎦∑
A I + Jand 2
21
1 1 11
N
Ms ns n ns
uN N N=
⎛ ⎞⎛ ⎞ ′= −⎜ ⎟⎜ ⎟−⎝ ⎠⎝ ⎠⊕B J u u
( )
( )
2 2
21 2
2 2 21 12
1
2 2
2
2 212
11 1 1 11
1
11 11
1
s Ms
NNs s
ns ns Ms ns n nNs ss s s ss
s s
s Ms
Ns s
nsss s s s
ss s
uM N u
uN Nk Nv
k Nk
uM N u
N k vk N
k
σ
σ σ
σ
σ σ
−
= =
=
=
⎡ ⎤⎛ ⎞⎛ ⎞−⎢ ⎥⎜ ⎟⎜ ⎟− ⎛ ⎞⎛ ⎞⎢ ⎥⎝ ⎠⎜ ⎟ ′= ⊕ + −⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟ −⎛ ⎞ ⎝ ⎠⎝ ⎠⎢ ⎥⎜ ⎟− ⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎝ ⎠⎣ ⎦
⎛ ⎞−⎜ ⎟−⎛ ⎞ ⎝ ⎠= ⊕ +⎜ ⎟−⎝ ⎠
−
⊕∑
A B I + J J u u
I +( )
2 2
22
2 2 212
1 1
2 2
21
11 1 1
1
11
s Ms
Ns s
ns Ms ns ns ns n nN Nss s s s
ss s
s
NMs Ms s
ss
uM N u
uN k Nv
k Nk
u u nN NN
σ
σ σ
σ
σ
=
= =
=
⎛ ⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ −⎢ ⎥ ⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟−⎜ ⎢ ⎥ ⎢ ⎥⎝ ⎠⎜ ⎟ ⎜ ⎟ ′− ⊕ +⎜ ⎢ ⎥ ⎢ ⎥⎜ ⎟ ⎜ ⎟⎛ ⎞ ⎛ ⎞⎜ ⎢ ⎥ ⎢ ⎥⎜ ⎟ ⎜ ⎟−⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦⎝
⎛ ⎞= ⊕ +⎜ ⎟−⎝ ⎠
∑ ∑J J I + J u u
( ) ( )
2 2 2 2
2 2 22
2 2 2 212 2
1 1
1 11
1 1
Ms s Ms
Ns ss Ms s
ns Ms s nsN Nss s s s ss s
s ss ss s
u uM N M Nu u u
u nk N kv v
k N k Nk k
σ
σ σ σ=
= =
⎛ ⎡ ⎤⎛ ⎞ ⎡⎡ ⎤ ⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ − −⎢ ⎥⎜ ⎟ ⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟ ⎜ ⎟− −⎜ ⎢ ⎥⎜ ⎟ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎜ ⎟⎢ ⎥ − ⊕ +⎜ ⎢ ⎥⎜ ⎟ ⎢ ⎥⎜ ⎟⎢ ⎥⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎢ ⎥⎜ ⎟⎢ ⎥− −⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟⎢ ⎥ ⎢ ⎥⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎝ ⎠⎝ ⎠ ⎣⎣ ⎦⎝∑ ∑
+ J +
⎤
⎦
J
( ) ( )
2 2 2 2
2 2 2 2 22
2 2 2 2 212 2
1 1
1 11 11
1 1
s Ms s Ms
Ns sMs Ms s s Ms s
Ms sN Nss s s s s ss s
s ss ss s
u uM N M Nu u n u u u
u nN NN k N kv v
k N k Nk k
σ σ
σ σ σ σ=
= =
⎞⎟⎟⎟
⎜ ⎟⎜ ⎟
⎠
⎛ ⎡ ⎤⎡ ⎤ ⎛ ⎞⎛ ⎞ ⎛ ⎞− −⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟ ⎜ ⎟− −⎛ ⎞ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎜ ⎟⎢ ⎥= ⊕ + − +⎜ ⎟ ⎢ ⎥⎜ ⎟⎢ ⎥− ⎛ ⎞ ⎛ ⎞⎝ ⎠ ⎢ ⎥⎜ ⎟⎢ ⎥− −⎜ ⎟ ⎜ ⎟⎜ ⎟⎢ ⎥⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎝ ⎠⎣ ⎦
∑ ∑+ +
( )
( )
2 2
2 2 2
2 2 2 212
1
2 2
21
1
1
1
11
ns
s Ms
NsMs Ms s s
nsNss s s s
ss s
NMs s s
Nss s
ss s
uM Nu u n u
N N k vk N
k
u n uN k v
k Nk
σ
σ σ=
=
=
=
⎛ ⎞⎡ ⎤⎞⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟⎜ ⎟⎢ ⎥
⎝ ⎠⎣ ⎦⎝ ⎠⎛ ⎞⎡ ⎤⎛ ⎞
−⎜ ⎟⎢ ⎥⎜ ⎟−⎜ ⎟⎝ ⎠⎢ ⎥= ⊕ +⎜ ⎟⎢ ⎥⎛ ⎞⎜ ⎟⎢ ⎥− ⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎝ ⎠⎛ ⎞⎜ ⎟⎜ ⎟= ⊕ ⎜ ⎛ ⎞⎜ − ⎜ ⎟⎜ ⎝ ⎠⎝ ⎠
∑
∑
J
+ J
+ ns⎟⎟⎟
J
next simplify
C07bx06 4/17/2007 22
( )
( ) ( )
( )
22
2 222 2 2
1
1
2
2 2
1 111
1 11
1
1
111
s s
sMs
sn n
s ss s s Ms
s
N
s
ss
s
ss s Ms
s
n uN N M
nn uN N M
N nnN N
nM
n N n n uM N N
σ
σ σσ σ
σ
σ
−
=
⎡ ⎤⎡ ⎤⎛ ⎞⎛ ⎞− −⎢ ⎥⎢ ⎥⎜ ⎟⎜ ⎟−⎝ ⎠⎝ ⎠⎣ ⎦⎢ ⎥+⎢ ⎥− ⎡ ⎤⎛ ⎞⎛ ⎞⎢ ⎥− − − − −⎢ ⎥⎜ ⎟⎜ ⎟−⎝ ⎠⎝ ⎠⎢ ⎥⎣ ⎦⎣ ⎦
−− = ⊕−
−⎡ ⎛ ⎞
− +⎢ ⎜ ⎟⎢ ⎝ ⎠⎢ ⎡ ⎤⎛ ⎞ −⎛ ⎞⎛ ⎞− + −⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟−⎝ ⎠⎝ ⎠⎝ ⎠⎣ ⎦⎣
I J
B A
( )
2
222 2
1
1 11
s
Ms sn
s ss s MsN
s
s
u n
nnn uN N M
σσ
=
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
⎛ ⎞⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥⎤ ⎛ ⎞⎜ ⎟⎡ ⎤⎢ ⎥⎛ ⎞⎛ ⎞⎥ − − − −⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟−⎝ ⎠⎝ ⎠⎥ ⎣ ⎦⎝ ⎠⎝ ⎠⎢ ⎥⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎦⎣ ⎦
∑
J
Define 2 21 1 11s s Ms
s s
N nl uM n N N
σ⎛ ⎞ −⎛ ⎞⎛ ⎞= − −⎜ ⎟ ⎜ ⎟⎜ ⎟−⎝ ⎠⎝ ⎠⎝ ⎠
and 21 1s s
s s
wM n
σ⎛ ⎞
= −⎜ ⎟⎝ ⎠
( )( )
( )
22
21
2 21
1
22
2 21
1 111
1
1 111
1
s s s
s
sMsN
s Msn n Ns
s s s s ss
s s
sMsN
sns
s s s s s
s
n uN N M uN nn
l n n lwN N
l
n uN N M N n
l n wN N
l
σ
σ σ
σ
σ σ
−
=
=
=
⎡ ⎤⎡ ⎤⎛ ⎞⎛ ⎞− −⎢ ⎥⎢ ⎥⎜ ⎟⎜ ⎟− ⎛ ⎞⎝ ⎠⎝ ⎠ −⎣ ⎦⎢ ⎥− = ⊕ + − ⎜ ⎟⎢ ⎥− − ⎡ ⎤ ⎝ ⎠⎢ ⎥− ⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦
⎡ ⎤⎛ ⎞⎛ ⎞− −⎢ ⎥⎜ ⎟⎜ ⎟−⎝ ⎠⎝ ⎠ −⎣ ⎦= ⊕ + −− − ⎡ ⎤
− ⎢ ⎥⎣ ⎦
∑B A I J J
I
2 ns
2
2
1
s
MsnN
s s
s
un l
=
⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥
⎛ ⎞⎢ ⎥⎢ ⎥⎜ ⎟⎢ ⎥⎢ ⎥⎝ ⎠⎢ ⎥⎢ ⎥
⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦∑
J
C07bx06 4/17/2007 23
( )( ) ( )
22
2 2 211
2 2 2 21 1
1 1
1
1 1111
1 1s s
sMsN N
sMs s s Msns n nN Ns s
s s s s s s ss ss
s ss s
s
n uN N Mu n u N n un
N k l n n lv wk N N Nk l
σ
σ σ−−
= =
= =
=
⎡ ⎤⎛ ⎞ ⎡ ⎤⎡ ⎤⎛ ⎞⎛ ⎞− −⎢ ⎥⎜ ⎟ ⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥− ⎛ ⎞−⎝ ⎠⎝ ⎠⎣ ⎦⎢ ⎥⎜ ⎟ ⎢ ⎥− = ⊕ ⊕ + − ⎜ ⎟⎢ ⎥⎜ ⎟ ⎢ ⎥− −⎛ ⎞ ⎡ ⎤ ⎝ ⎠− −⎢ ⎥⎜ ⎟ ⎢ ⎥⎜ ⎟ ⎢ ⎥⎜ ⎟ ⎢ ⎥⎢ ⎥⎝ ⎠ ⎣ ⎦⎝ ⎠ ⎣ ⎦⎣ ⎦
= ⊕
∑ ∑A B B A + J I J
( ) ( ) ( )
22
2 2 2 2 2
2 2 2 2 2
1 1 1
1 1111 1
1 1 1
sMsN
sMs s s Ms s s s MsN N N
s s s s s s s ss s ss s
s s ss s s
n uN N Mu n u u n n u N n u
N k N k l n n lv v wk N k N N Nk k l
σ
σ σ
= = =
⎛ ⎞ ⎛ ⎞ ⎡ ⎡ ⎤⎛ ⎞⎛ ⎞− −⎜ ⎟ ⎜ ⎟ ⎢ ⎜ ⎟⎜ ⎟⎢ ⎥−⎛ ⎞ ⎛ ⎞−⎝ ⎠⎝ ⎠⎣ ⎦⎜ ⎟ ⎜ ⎟ ⎢+ −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎢− −⎛ ⎞ ⎛ ⎞ ⎡ ⎤⎝ ⎠ ⎝ ⎠− − −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎣ ⎦⎝ ⎠ ⎝ ⎠ ⎣∑ ∑ ∑
+ +
( ) ( ) ( )
22
2 2 2 2
2 2 2 2 21
1 1 1
1 11 11 11 1
1 1 1
sMsN
sMs s s Ms s sN N Ns
s s s s s ss s ss s
s s ss s s
n nuN N M N Nu n u N n u n u
N k l lv w vk N N N k Nk l k
σ
σ σ σ=
= = =
⎛ ⎞⎤⎜ ⎥⎜ ⎥⎜ ⎥⎜ ⎢ ⎥⎜ ⎢ ⎥⎦⎝ ⎠
⎡ ⎤⎛ ⎞⎛ ⎞ ⎛ ⎞⎛− − −⎜ ⎟⎜ ⎟ ⎜ ⎟⎜⎢ ⎥− −⎛ ⎞−⎝ ⎠⎝ ⎠ ⎝ ⎠⎝⎣ ⎦= ⊕ + − +⎜ ⎟− − −⎛ ⎞ ⎡ ⎤ ⎛ ⎞⎝ ⎠− − −⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎣ ⎦ ⎝ ⎠∑ ∑ ∑
+
( )
( )
( )
22 2
21
1 1
1 1 11 1
s
Ns sMs s s
N Nss s s s s
s ss ss s
N n n uu n uN k n l v wk N l N N
k l
σ
=
= =
⎛ ⎡⎜ ⎢
⎣⎜⎜ −⎜⎜⎝
⎛ ⎞⎛ ⎞⎜ ⎟⎜
−⎜ ⎟⎜= ⊕ + −⎜ ⎟⎜⎛ ⎞ ⎡ ⎤− −⎜ ⎟⎜⎜ ⎟ ⎢ ⎥⎜ ⎟⎜⎝ ⎠ ⎣ ⎦⎝ ⎠⎝
∑ ∑ns
⎛ ⎞⎜ ⎟⎟⎜ ⎟⎟⎜ ⎟⎟⎜ ⎟⎟⎟⎜ ⎟⎠⎝ ⎠
J
( )
( )
( )
( )
( )
11 1 1
2 2
2
2 212
1
22 2
21
1
11
1
1 1 11 1
I n n
s MsN
s sn ns nsNs
s s s ss
s s
Ns sMs s s
n Nss s s s
s ss s
n
uM N u
k vk Nk
N n n uu n uN k n l vk N l N
k
σ
σ σ
−− − −
=
=
=
=
= ⊗ − ⊗ −
⎡ ⎤⎛ ⎞⎛ ⎞−⎢ ⎥⎜ ⎟⎜ ⎟−⎝ ⎠⎢ ⎥⎜ ⎟= ⊗ ⊕ +⎢ ⎥⎜ ⎟⎛ ⎞
−⎢ ⎥⎜ ⎟⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎝ ⎠⎣ ⎦
⎛ ⎞⎜ ⎟
−⎜ ⎟− ⊗ ⊕ + −⎜ ⎟⎛ ⎞− −⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
∑
∑
V I A J A B B A
I I + J
J
1
nsNs
s s
wNl=
⎛ ⎞⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎡ ⎤⎜ ⎟⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟⎜ ⎟⎣ ⎦⎝ ⎠⎝ ⎠⎝ ⎠
∑J
Our target is ( )1 2
*
1 2
1 1 ... 0j j M M N nP eM M −
⎡ ⎤⎛ ⎞′ ′ ′ ′= ⊗⎢ ⎥⎜ ⎟
⎝ ⎠⎣ ⎦1 1 Y where j n<=
C07bx06 4/17/2007 24
( ) ( )
( ) 1
2 2
**
*
1 1
2 2
1 1
2
1
1
1
1
1 1
1
1
1
I IIUV UV N
II IIII
n
n
IM nN
IIM n
N
n nss
N
n Ms nss
N
N n Mss
E EN
uu
uN N u
N
N
N
−
−
=
−=
−=
⎛ ⎞ ⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞
= ⊗ = ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
⎡ ⎤⎛ ⎞⊗ ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
⎡ ⎤⎛ ⎞= ⊗ ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
⎛ ⎞⊗ ⎜ ⎟⎝ ⎠
⊕
⊕
⊕
K KYY 1
K KY
11
K 11 uK 1
u
1 1
1 1
1 1
⊗u
1
2
1
*
1
1
1
1
1
N
N
n nss
N
n Ms nss
N
N n Mss
uu
u
N
N
N
=
−=
−=
⎛ ⎞⎜ ⎟⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟
⎡ ⎤ ⎜ ⎟⎜ ⎟⎝ ⎠⎢ ⎥⎜ ⎟⎣ ⎦⎝ ⎠⎛ ⎞⎡ ⎤⎛ ⎞⊗⎜ ⎟⎜ ⎟⎢ ⎥
⎝ ⎠⎣ ⎦⎜ ⎟⎜ ⎟⎡ ⎤⎛ ⎞⎜ ⎟= ⊗ ⎜ ⎟⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦⎜ ⎟
⎡ ⎤⎛ ⎞⎜ ⎟⊗ ⎜ ⎟⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦⎝ ⎠
⊕
⊕
⊕
1 1
1 1 u
1 1
( )1 2
*
1 2 *1 2
* *1 2 1 2
1 2 1 2
1 1 ...
1 1 1 1... ... 0
Ij j n n
II
j n n I j M n M n N n
P eM M
e eM M M M− − −
⎡ ⎤⎛ ⎞⎛ ⎞′ ′ ′= ⊗⎢ ⎥⎜ ⎟⎜ ⎟
⎝ ⎠ ⎝ ⎠⎣ ⎦⎛ ⎞⎛ ⎞ ⎛ ⎞
′ ′ ′ ′ ′ ′ ′= ⊗ + ⊗⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠
Y1 1
Y
1 1 Y 1 1 IIY
Hence,
1 21 2
1 1 ...I j n ng eM M
⎛ ⎞′ ′ ′ ′= ⊗ ⎜ ⎟
⎝ ⎠1 1 , ( )1 21 2
1 2
1 1 ... 0II j M n M n N ng eM M− − −
⎛ ⎞⎛ ⎞′ ′ ′ ′ ′= ⊗⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
1 1
1
1 N
I n nsN =
⎡ ⎤⎛= ⊗⎜⎢ ⎥⎝ ⎠⎣ ⎦⊕X 1 1 s
⎞⎟ and 1
1
1
1
N
n Ms nss
II N
N n Mss
N
N
−=
−=
⎛ ⎞⎡ ⎤⎛ ⎞⊗⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎜ ⎟= ⎜ ⎟⎡ ⎤⎛ ⎞⎜ ⎟⊗ ⎜ ⎟⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦⎝ ⎠
⊕
⊕
1 1X
1 1.
C07bx06 4/17/2007 25
( ) ( )1 1* 1 1 1 1 1 1
,ˆ I I I I I I I I I II iII I I I I I II iI
− −− − − − − −⎡ ⎤′ ′ ′ ′= − +⎢ ⎥⎣ ⎦
a V V X X V X X V V g V X X V X X g I
( )( )
( )( )
11 1 12
1 1
11 12 2
1 1 1
1
1 1
N N
I I I n ns n n n nss s
N N N
n ns n n ns n ns n n nss s s
nN
nN N
−− − −
= =
−− −
= = =
⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞′ ′ ′= ⊗ ⊗ − ⊗ − ⊗⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞′ ′ ′ ′= ⊗ ⊗ ⊗ − ⊗ ⊗ − ⊗⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣
⊕ ⊕
⊕ ⊕ ⊕
X V X 1 1 I A J A B B A 1 1
1 1 I A 1 1 1 1 J A B B A 1 11
N
s=⊕
( )( )2
11 12 2
1 1 1 1
N N N N
ns ns ns nss s s s
n n nN N
−− −
= = = =
⎤⎢ ⎥
⎦⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞′ ′= − −⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⊕ ⊕ ⊕ ⊕1 A 1 1 A B B A 1
for
( )
1
1 1
2 2
2
2 211 12
1
2 2
2
2 212
11
1
1
N N
ns nss s
s MsN NN
s sns ns ns nsNss ss s s s
ss s
s MsN
ss s
ss s s
s
uM N u
k vk Nk
uM Nn u
kk N
σ
σ σ
σ
σ σ
−
= =
== =
=
=
⎛ ⎞ ⎛ ⎞′⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
⎡ ⎤⎛ ⎞⎛ ⎞−⎢ ⎥⎜ ⎟⎜ ⎟−⎛ ⎞ ⎛ ⎞⎝ ⎠⎢ ⎥⎜ ⎟′= ⊕ +⎜ ⎟ ⎜ ⎟⎢ ⎥⎜ ⎟⎛ ⎞⎝ ⎠ ⎝ ⎠−⎢ ⎥⎜ ⎟⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎝ ⎠⎣ ⎦
⎛ ⎞−⎜ ⎟−⎝ ⎠= ⊕ +
−
⊕ ⊕
⊕ ⊕∑
1 A 1
1 I + J
+( )
1
( )
2
1
2
1
1
1
1
sNs
s s
Ns s s
Nss s
ss s
nvk
n n uk vk N
k
=
=
=
⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟⎛ ⎞⎢ ⎥⎜ ⎟⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎝ ⎠⎣ ⎦⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟= ⊕ ⎢ ⎥⎜ ⎟⎛ ⎞
−⎢ ⎥⎜ ⎟⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎝ ⎠⎣ ⎦
∑
∑1 +
C07bx06 4/17/2007 26
( )( )
( )
( )
( )
11
1 1
22 2
211 1
1 1
1 1 11 1
N N
ns nss s
N NNs sMs s s
ns ns nsN Nss ss s s s ss s
s ss s
n
N n n uu n uN k n l v wk N l N N
k l
−−
= =
== =
= =
⎛ ⎞ ⎛ ⎞′ −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
⎛ ⎞⎛ ⎞⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟
−⎛ ⎞ ⎛⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟′= ⊕ + −⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞ ⎡ ⎤⎝ ⎠ ⎝⎜ ⎟⎜ ⎟− −⎜ ⎟⎜ ⎟⎜ ⎟ ⎢ ⎥⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠ ⎣ ⎦⎝ ⎠⎝ ⎠⎝ ⎠⎝ ⎠
⊕ ⊕
⊕ ⊕∑ ∑
1 A B B A 1
1 J 1
( )
( )
( )
( )
( )
( )
22 22
21
1 1
22 2
21
1
1 1 11 1
1 11 1
Ns sMs s s
sN Nss s s s s
s ss ss s
Ns sMs s s s
Nss s s
s ss s
N n n uu n u nN k n l v wk N l N N
k l
N n n uu n n uN k l vk N l N
k
=
= =
=
=
⎞⎜ ⎟
⎠
⎛ ⎞⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟
−⎜ ⎟⎜ ⎟⎜ ⎟= ⊕ + −⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞ ⎡ ⎤⎜ ⎟− −⎜ ⎟⎜ ⎟⎜ ⎟ ⎢ ⎥⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠ ⎣ ⎦⎝ ⎠⎝ ⎠⎝ ⎠
⎛ ⎞⎜ ⎟
−⎜ ⎟= ⊕ + −⎜ ⎟⎛ ⎞− −⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
∑ ∑
∑1
Ns
s s
wNl=
⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎡ ⎤⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟⎣ ⎦⎝ ⎠⎝ ⎠
∑
C07bx06 4/17/2007 27
( )( )
( ) ( )
211 1 1
2 21 1 1 1
2 2 2 2
2 2 21 1
1 1
1 1
N N N N
I I I ns ns ns nss s s s
N Ns s s Ms s s s
N Ns ss s ss s
s ss ss s
n n nN N
n n n u n u n n uN k N N k lv vk N k N
k k
−− − −
= = = =
= =
= =
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞′ ′ ′= − −⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟= ⊕ − ⊕⎢ ⎥⎜ ⎟⎛ ⎞ ⎛
− −⎢ ⎥⎜ ⎟⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠ ⎝⎝ ⎠⎣ ⎦
⊕ ⊕ ⊕ ⊕
∑ ∑
X V X 1 A 1 1 A B B A 1
1 + ( )
( )
( ) ( )
( )
2
1
22 2 2
2 21
1 1
1 11
1 11 1
s sN
ss
s s
Ns ss s s Ms s s s
N Nss s ss s
s s ss ss s
N n n uwl N Nl
N n n un n n u nu n n uN k N k lv vk N k N l
k k
=
=
= =
⎛ ⎞⎛ ⎞⎛⎜ ⎟⎜ ⎟⎜
−⎜ ⎟⎜ ⎟⎜+ −⎜ ⎟⎜ ⎟⎜⎞ ⎡⎜ ⎟−⎜ ⎟⎜⎜ ⎟
⎞⎟⎟⎟⎤⎟⎢ ⎥⎜ ⎟⎜⎜ ⎟⎠ ⎣⎝ ⎠⎝⎝ ⎠
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟
−⎜ ⎟ ⎜ ⎟= ⊕ − + −⎜ ⎟ ⎜ ⎟⎛ ⎞ ⎛ ⎞− −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠
∑
∑ ∑1 +
( )
⎟⎦ ⎠
( )
( )
( )
1
22 2
2 21
1 1
1
1 11 1
Ns
s s
Ns ss s s Ms
N Nss ss s
s ss ss s
wN Nl
N n n un n n u nuN k N lv wk N l N N
k l
=
=
= =
⎡ ⎤⎛ ⎞⎛ ⎞⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟⎡ ⎤⎢ ⎥⎜ ⎟−⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎣ ⎦⎝ ⎠⎝ ⎠⎣ ⎦⎡ ⎤⎛ ⎞⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟⎜ ⎟ ⎜ ⎟
−⎢ ⎥⎜ ⎟⎜ ⎟ ⎜ ⎟= ⊕ − −⎢ ⎥⎜ ⎟⎜ ⎟ ⎜ ⎟⎛ ⎞ ⎡ ⎤⎢ ⎥⎜ ⎟− −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠ ⎣ ⎦⎝ ⎠ ⎝ ⎠⎝ ⎠⎣ ⎦
=
∑
∑ ∑1 +
( )
( )
( )
22 2 2
2 2 21
1 1
11 1
Ns ss s s Ms Ms
N Nss s ss s
s ss ss s
N n n un n n u nu nuN k N l N lv wk N l N N
k l
=
= =
⎡ ⎤⎛ ⎞⎛ ⎞⎢ ⎥⎜ ⎟⎜ ⎟
−⎢ ⎥⎜ ⎟⎜ ⎟⊕ − +⎢ ⎥⎜ ⎟⎜ ⎟⎛ ⎞ ⎡ ⎤− −⎢ ⎥⎜ ⎟⎜ ⎟⎜ ⎟ ⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠ ⎣ ⎦⎝ ⎠⎝ ⎠⎣ ⎦
∑ ∑1 +
( )( )
( )
( )
( )
1
22 2 211
2 2 21
1 1
2
1
2
1
11 1
1
1
Ns ss s s Ms Ms
I I I N Nss s ss s
s ss ss s
N
s
s s sN
s ss
s s
N n n un n n u nu nuN k N l N lv wk N l N N
k l
Nn
n n uk vk N
k
−
−−
=
= =
=
=
⎛ ⎞⎡ ⎤⎛ ⎞⎛ ⎞⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟
−⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟′ = ⊕ − +⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟⎛ ⎞ ⎡ ⎤⎜ ⎟− −⎢ ⎥⎜ ⎟⎜ ⎟⎜ ⎟ ⎢ ⎥⎜ ⎟⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠ ⎣ ⎦⎝ ⎠⎝ ⎠⎣ ⎦⎝ ⎠
= ⊕
⎛ ⎞− ⎜ ⎟
⎝ ⎠
∑ ∑
∑
X V X 1 +
1 + ( )
( )
22 2
2 2
1
11
s sMs MsN
s s ss
s s
N n n unu nuN l N l wl N N
l=
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎛ ⎞⎛ ⎞⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥−⎜ ⎟⎜ ⎟− +⎢ ⎥⎜ ⎟⎜ ⎟⎡ ⎤⎢ ⎥−⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎣ ⎦⎝ ⎠⎝ ⎠⎣ ⎦
∑
C07bx06 4/17/2007 28
( )( )
( )
( )
1
11 1
1
11 1
1 1
2 2
2
2 211 2
1
1
1
11 1
1
I I
N
n ns n ns
N N
n ns n nss s
s MsN N
s sn ns ns Nss s s s s
ss s
nN
n nN N
uM N u
N k vk Nk
σ
σ σ
−
−− −
=
−− −
= =
==
=
′
⎡ ⎤⎛ ⎞′ ′= ⊗ ⊗ − ⊗ −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎛ ⎞⎛ ⎞ ⎛ ⎞′ ′ ′ ′= ⊗ − ⊗ −⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠
⎛ ⎞−⎜ ⎟−⎛ ⎞ ⎝ ⎠′ ′= ⊗ ⊕ +⎜ ⎟ ⎛ ⎞⎝ ⎠ − ⎜
⎝
⊕
⊕ ⊕
⊕∑
X V
1 1 I A J A B B A
1 1 A 1 1 A B B A
1 1 I +
( )
( )
( )
22 2
211
1 1
1 1 11 1
ns
N Ns sMs s s
n ns nsN Nss s s s s ss s
s ss s
N n n un u n uN N k n l v wk N l N N
k l
==
= =
⎛ ⎞⎡ ⎤⎛ ⎞⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟⎟⎜ ⎟⎜ ⎟⎢ ⎥⎠⎝ ⎠⎣ ⎦⎝ ⎠
⎛ ⎞⎛ ⎞⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟
−⎛ ⎞⎜ ⎜ ⎟⎜ ⎟⎜ ⎟′ ′− ⊗ ⊕ + −⎜ ⎟⎜ ⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞ ⎡ ⎤⎝ ⎠⎜ ⎜ ⎟− −⎜ ⎟⎜ ⎟⎜ ⎟ ⎢ ⎥⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎝ ⎠ ⎣ ⎦⎝ ⎠⎝ ⎠⎝ ⎠⎝ ⎠
⊕∑ ∑
J
1 1 J
( )
( )
( )
( )
2
1
1
22 2
21
1 1
1 1
1
1 1 11 1
Ns s
n nsNss s
ss s
Ns sMs s s
n N Nss s s s
s ss ss s
n uN k vk N
k
N n n unu n uN N k l v wk N l N N
k l
=
=
=
= =
⎛ ⎞⎜ ⎟⎜ ⎟⎟⎜ ⎟⎟⎜ ⎟⎟
⎟⎜ ⎟⎝ ⎠⎛ ⎞⎡ ⎤⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥′ ′= ⊗ ⊕⎜ ⎟⎢ ⎥⎛ ⎞⎜ ⎟−⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎝ ⎠
⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟
−⎜ ⎟⎜ ⎟′− ⊗ ⊕ + −⎜ ⎟⎜ ⎟⎛ ⎞ ⎡ ⎤− −⎜ ⎟⎜ ⎟⎜ ⎟ ⎢ ⎥⎜ ⎟⎜⎝ ⎠ ⎣ ⎦⎝ ⎠⎝ ⎠
∑
∑ ∑
1 1 + 1
1
( ) ( )
( )
( )
22 2 2
21 1
1 1 1
1 1 1 11 1 1
ns
N Ns ss s Ms s s
n nsN N Ns ss s ss s s
s s ss s ss s s
N n n un u nu n uN k N k lv v wk N k N l N N
k k l
= =
= = =
⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟ ′⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
⎛ ⎞⎡ ⎤ ⎛ ⎞⎛⎜ ⎟ ⎜ ⎟⎜⎢ ⎥ −⎜ ⎟ ⎜ ⎟⎜⎢ ⎥′ ′= ⊗ ⊕ − ⊕ + −⎜ ⎟ ⎜ ⎟⎢ ⎥⎛ ⎞ ⎛ ⎞ ⎡ ⎤⎜ ⎟− − −⎜ ⎟⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎣ ⎦⎣ ⎦ ⎝ ⎠⎝⎝ ⎠
∑ ∑ ∑
1
1 1 + 1
( ) ( )
( )
( )
22 2 2
21
1 1 1
1 1 1 11 1 1
Ns ss s Ms s s
n N N Nss s ss s s
s s ss s ss s s
N n n un u nu n uN k N k lv v wk N k N l N N
k k l
=
= = =
⎡ ⎛ ⎛ ⎞⎢ ⎜ ⎜ ⎟⎢ ⎜ ⎜ ⎟⎢ ⎜ ⎜ ⎜ ⎟⎢ ⎜ ⎜ ⎜ ⎟⎜ ⎟⎜⎜⎢ ⎠⎝⎝⎣
⎛ ⎞ ⎛ ⎞⎛⎜ ⎟ ⎜ ⎟⎜
−⎜ ⎟ ⎜ ⎟⎜′= ⊗ ⊕ − + −⎜ ⎟ ⎜ ⎟⎛ ⎞ ⎛ ⎞ ⎡ ⎤− − −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎣ ⎦⎝ ⎠ ⎝ ⎠⎝
∑ ∑ ∑1 1 +
( )
( )
( )
22 2
21
1 1
1 1 11 1
ns
Ns ss s Ms
n nsN Nss s ss s
s ss ss s
N n n un u nuN k N k lv wk N l N N
k l
=
= =
⎛ ⎞⎡ ⎤⎞⎜ ⎟⎢ ⎥⎟⎜ ⎟⎢ ⎥⎟ ′⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟⎜ ⎟⎢ ⎥⎠⎣ ⎦⎝ ⎠⎛ ⎞⎡ ⎤⎡ ⎤⎛ ⎞ ⎛ ⎞⎜ ⎟⎢ ⎥⎢ ⎥⎜ ⎟ ⎜ ⎟
−⎜ ⎟⎢ ⎥⎢ ⎥⎜ ⎟ ⎜ ⎟′ ′= ⊗ ⊕ − −⎜ ⎟⎢ ⎥⎢ ⎥⎜ ⎟ ⎜ ⎟⎛ ⎞ ⎡ ⎤⎜ ⎟⎢ ⎥− −⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎢ ⎥⎢ ⎥⎝ ⎠ ⎣ ⎦⎝ ⎠ ⎝ ⎠⎣ ⎦⎣ ⎦⎝ ⎠∑ ∑
1
1 1 + 1
⎟
C07bx06 4/17/2007 29
( ) ( )1 1* 1 1 1 1 1 1
,ˆ I I I I I I I I I II iII I I I I I II iII
− −− − − − − −⎡ ⎤′ ′ ′ ′= − +⎢ ⎥⎣ ⎦
a V V X X V X X V V g V X X V X X g
( )
( )
( )
( )
11 1 1
22 2
21
1 1
2
1 1 11 1
1
I I I I I I I
Ns ss s Ms
n nsN Nss s ss s
s ss ss s
N n n un u nuN k N k lv wk N l N N
k l
Nn
−− − −
=
= =
′ ′
⎡ ⎤⎛ ⎞⎡ ⎤⎡ ⎤⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟⎢ ⎥⎢ ⎥⎜ ⎟ ⎜ ⎟
−⎢ ⎥⎜ ⎟⎢ ⎥⎢ ⎥⎜ ⎟ ⎜ ⎟= ⊗ ⊕ − −⎢ ⎥⎜ ⎟⎢ ⎥⎢ ⎥⎜ ⎟ ⎜ ⎟⎛ ⎞ ⎡ ⎤⎢ ⎥⎜ ⎟⎢ ⎥− −⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟⎢ ⎥⎢ ⎥⎢ ⎥⎝ ⎠ ⎣ ⎦⎝ ⎠ ⎝ ⎠⎣ ⎦⎣ ⎦⎝ ⎠⎣ ⎦
⊗
∑ ∑
V X X V X X V
1 1 + 1
( )
( )
( )
( )
1
22 2 2
2 2
1 1
2
1
1
1
11 1
1 1
1
N
s
s ss s s Ms MsN N
s s ss ss s
s ss s
Ns s M
n Nsss
ss s
N n n un n u nu nuk N l N lv wk N l N N
k l
n u nuN kvk N
k
=
= =
=
=
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⊕ ⎢ ⎥⎛ ⎞⎛ ⎞⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥−⎜ ⎟⎜ ⎟− +⎢ ⎥⎜ ⎟⎜ ⎟⎛ ⎞ ⎡ ⎤⎢ ⎥− −⎜ ⎟⎜ ⎟⎜ ⎟ ⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠ ⎣ ⎦⎝ ⎠⎝ ⎠⎣ ⎦
⎛ ⎞⎜ ⎟⎜ ⎟′ ⊗ ⊕ −⎜ ⎟⎛ ⎞
−⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
∑ ∑
∑
1 +
1 1 + ( )
( )
( )
( )
( )
22
2
1
2
22 2
21
1
11
1 11
1 11 1
s ssnsN
s s ss
s s
n n
Ns ss s Ms
Nss s ss s
s ss s
N n n uN k l wl N N
l
NN n N
N n n un u nuk N k lv wk N l N N
k l
=
=
=
⎛ ⎞⎡ ⎤⎡ ⎤⎛ ⎞⎜ ⎟⎢ ⎥⎢ ⎥⎜ ⎟
−⎜ ⎟⎢ ⎥⎢ ⎥⎜ ⎟ ′−⎜ ⎟⎢ ⎥⎢ ⎥⎜ ⎟⎡ ⎤⎜ ⎟⎢ ⎥−⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎢ ⎥⎣ ⎦⎝ ⎠⎣ ⎦⎣ ⎦⎝ ⎠⎛ ⎞
′= ⊗⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟
−⎜ ⎟⊕ − −⎜ ⎟⎛ ⎞− −⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
∑
∑
1
1 1
1 +
( )
( )
( )
1
1
22 2 2
2 2
1 1
1
11 1
nsN
s s
N
s
s ss s s Ms MsN N
s s ss ss s
s ss s
N n n un n u nu nuk N l N lv wk N l N N
k l
=
=
= =
⎛ ⎞⎡ ⎤⎡ ⎤⎛ ⎞⎜ ⎟⎢ ⎥⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎢ ⎥⎜ ⎟⎡ ⎤⎜ ⎟⎢ ⎥⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎢ ⎥⎣ ⎦⎝ ⎠⎣ ⎦⎣ ⎦⎝ ⎠⎡ ⎤⎢⎢⎢⎢⎢⊕ ⎢ ⎛ ⎞⎛ ⎞⎢ ⎜ ⎟⎜ ⎟⎢ −⎜ ⎟⎜ ⎟− +⎢ ⎜ ⎟⎜ ⎟⎛ ⎞ ⎡ ⎤⎢ − −⎜ ⎟⎜ ⎟⎜ ⎟ ⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎝ ⎠ ⎣ ⎦⎝ ⎠⎝ ⎠⎣
∑
∑ ∑
1
1 +
( )
( )
( )
22 2
21
1 1
1 11 1
Ns ss s Ms
nsN Nss s ss s
s ss ss s
N n n un u nuk N k lv wk N l N N
k l
=
= =
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎦⎢⎢⎛ ⎞⎡ ⎤⎡ ⎤⎛ ⎞ ⎛ ⎞⎢⎜ ⎟⎢ ⎥⎢ ⎥⎜ ⎟ ⎜ ⎟⎢ −⎜ ⎟⎢ ⎥⎢ ⎥⎜ ⎟ ⎜ ⎟ ′⊕ − −⎢⎜ ⎟⎢ ⎥⎢ ⎥⎜ ⎟ ⎜ ⎟⎛ ⎞ ⎡ ⎤⎢⎜ ⎟⎢ ⎥− −⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎢ ⎥⎜ ⎟ ⎜ ⎟⎢⎜ ⎟⎢ ⎥⎢ ⎥⎝ ⎠ ⎣ ⎦⎝ ⎠ ⎝ ⎠⎣ ⎦⎣ ⎦⎝ ⎠⎣ ⎦
∑ ∑1 + 1
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
C07bx06 4/17/2007 30
21 1 11n nN
N n N n⎛ ⎞
′ =⎜ ⎟⎝ ⎠
1 1 nJ
( )
( )
( )
( )
22 2
21
1 1
1
2
1
1 1 11 1
1
1
Ns ss s Ms
nsN Nss ss s
s ss ss s
N
s
s s sN
s ss
s s
N n n un u nuk N lv wk N l N N
k l
n n uk vk N
k
=
= =
=
=
⎛ ⎞⎡ ⎤⎡ ⎤⎛ ⎞ ⎛⎜ ⎟⎢ ⎥⎢ ⎥⎜ ⎟ ⎜
−⎜ ⎟⎢ ⎥⎢ ⎥⎜ ⎟ ⎜⊕ − −⎜ ⎟⎢ ⎥⎢ ⎥⎜ ⎟ ⎜⎛ ⎞ ⎡ ⎤⎜ ⎟⎢ ⎥− −⎢ ⎥⎜ ⎟ ⎜⎜ ⎟ ⎢ ⎥⎜ ⎟ ⎜⎜ ⎟⎢ ⎥⎢ ⎥⎝ ⎠ ⎣ ⎦⎝ ⎠ ⎝⎣ ⎦⎣ ⎦⎝ ⎠
⊕⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞
−⎜ ⎟⎜ ⎟⎜ ⎝ ⎠⎝ ⎠
∑ ∑
∑
1 + 1
1 +
⎞⎟⎟⎟⎟⎟⎠
( )
( )
( )
( )
( )
22 2
2 2
1
22 2
21
1 1
11
1 11 1
s sMs MsN
s s ss
s s
Ns ss s Ms
N Nss s ss s
s ss ss s
N n n unu nuN l N l wl N N
l
N n n un u nuk N k lv wk N l N N
k l
=
=
= =
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎛ ⎞⎢ ⎥⎜ ⎟⎢ ⎥−⎜ ⎟− +⎢ ⎥⎜ ⎟⎡ ⎤⎢ ⎥−⎜ ⎟⎢ ⎥⎟⎜ ⎟⎢ ⎥⎣ ⎦⎝ ⎠⎣ ⎦
⎡⎛ ⎞ ⎛⎢⎜ ⎟ ⎜
−⎜ ⎟ ⎜⊕ − −⎜ ⎟ ⎜⎛ ⎞ ⎡ ⎤− −⎜ ⎟ ⎜⎜ ⎟ ⎢ ⎥⎜ ⎟ ⎜⎝ ⎠ ⎣ ⎦⎝ ⎠ ⎝⎣
∑
∑ ∑1 +
⎞⎟⎟⎟⎟⎟⎠
( )
( )
( )
22 2
21
1 1
1 1 11 1
ns
Ns ss s Ms
N Nss s ss s
s ss ss s
N n n un u nun k N lv wk N l N N
k l
=
= =
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎛ ⎞⎡ ⎤⎤⎢ ⎥⎜ ⎟⎢ ⎥⎥⎢ ⎥⎜ ⎟⎢ ⎥⎢ ⎥ ′⎢ ⎥⎜ ⎟⎢ ⎥⎢ ⎥⎢ ⎥⎜ ⎟⎢ ⎥⎢ ⎥⎢ ⎥⎜ ⎟⎢ ⎥⎢ ⎥⎦⎣ ⎦⎝ ⎠⎣ ⎦
⎛ ⎞ ⎛⎜ ⎟ ⎜
−⎜ ⎟ ⎜= ⊕ − −⎜ ⎟ ⎜⎛ ⎞ ⎡ ⎤− −⎜ ⎟ ⎜⎜ ⎟ ⎢ ⎥⎜ ⎟ ⎜⎝ ⎠ ⎣ ⎦⎝ ⎠ ⎝
∑ ∑
1
1 + ns
⎡ ⎤⎢ ⎥⎟⎢ ⎥⎟⎢ ⎥⎟⎢ ⎥⎟⎟⎢ ⎥⎣ ⎦
J
⎞
⎠
( )
( )
( )
( )
11 1 1
22 2
21
1 1
1 1 1 11 1
I I I I I I I
Ns ss s Ms
n nsN Nss s ss s
s ss ss s
N n n un u nun n k N lv wk N l N N
k l
−− − −
=
= =
′ ′
⎡ ⎤⎡ ⎤⎛ ⎞ ⎛⎢ ⎥⎢ ⎥⎜ ⎟ ⎜
−⎢ ⎥⎢ ⎥⎜ ⎟ ⎜= ⊗ ⊕ − −⎢ ⎥⎢ ⎥⎜ ⎟ ⎜⎛ ⎞ ⎡ ⎤⎢ ⎥− −⎢ ⎥⎜ ⎟ ⎜⎜ ⎟ ⎢ ⎥⎜ ⎟ ⎜⎢ ⎥⎢ ⎥⎝ ⎠ ⎣ ⎦⎝ ⎠ ⎝⎣ ⎦⎣ ⎦
∑ ∑
V X X V X X V
J 1 +
⎞⎟⎟⎟⎟⎟⎠
J
Now we want to evaluate ( ) 11 1 1
I I I I I I I
−− − − 1
I−′ ′−V V X X V X X V where
C07bx06 4/17/2007 31
( )
( )
( )
( )
11 1 1 1
2 2
2
2 212
1
22 2
21
1
11
1
1 1 11
I I I I I I I I
s MsN
s sn ns nsNs
s s s ss
s s
Ns sMs s s
n Nss s s s
ss s
uM N u
k vk Nk
N n n uu n uN k n l vk N l
k
σ
σ σ
−− − − −
=
=
=
=
′ ′−
⎡ ⎤⎛ ⎞⎛ ⎞−⎢ ⎥⎜ ⎟⎜ ⎟−⎝ ⎠⎢ ⎥⎜ ⎟= ⊗ ⊕ +⎢ ⎥⎜ ⎟⎛ ⎞
−⎢ ⎥⎜ ⎟⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎝ ⎠⎣ ⎦
⎛ ⎞⎜ ⎟
−⎜ ⎟− ⊗ ⊕ + −⎜ ⎟⎛ ⎞−⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
∑
∑
V V X X V X X V
I I + J
J( )
( )
( )
( )
1
22 2
21
1 1
1
1 1 1 11 1
nsNs
ss s
Ns ss s Ms
n nsN Nss s ss s
s ss ss s
wN Nl
N n n un u nun n k N lv wk N l N N
k l
=
=
= =
⎛ ⎞⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎡ ⎤⎜ ⎟⎜ ⎟−⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟⎜ ⎟⎣ ⎦⎝ ⎠⎝ ⎠⎝ ⎠⎡ ⎡ ⎤⎛ ⎞ ⎛ ⎞⎢ ⎢ ⎥⎜ ⎟ ⎜ ⎟
−⎢ ⎢ ⎥⎜ ⎟ ⎜ ⎟− ⊗ ⊕ − −⎢ ⎢ ⎥⎜ ⎟ ⎜ ⎟⎛ ⎞ ⎡ ⎤− −⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎢ ⎥⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎣ ⎦⎝ ⎠ ⎝ ⎠⎣ ⎦⎣
∑
∑ ∑
J
J 1 + J
( ) ( )
2 2
2 2
2 21 12
1 1
11 1 11 1
s MsN N
s s s sn ns ns nN Ns s
s s s s ss ss s
s ss s
uM N u n u
k nv vk N k Nk k
σ
σ σ= =
= =
⎤⎥⎥⎥
⎢ ⎥⎢ ⎥⎦⎡ ⎤⎛ ⎞ ⎛⎛ ⎞
−⎢ ⎥⎜ ⎟ ⎜⎜ ⎟−⎝ ⎠⎢ ⎥⎜ ⎟ ⎜= ⊗ ⊕ + − ⊗ ⊕ +⎢ ⎥⎜ ⎟ ⎜⎛ ⎞ ⎛ ⎞− −⎢ ⎥⎜ ⎟ ⎜⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜⎢ ⎥⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝⎣ ⎦
∑ ∑I I + J J
( )*, varI II iII I UV II iII′=V g K Y K g
1nsn k
⎞⎟⎟⎟⎟⎟⎠
J
First,
( )( )
( )( )( )
( ) ( )
( ) ( ) ( ) [
1 2
1 2
11 2
1 2 1
1 21 2
0 01 1 ... 00
1 1 ...
s ss s s
N
n M n S n N nM n ns S n nII II j M n M n N n
N SnN n n N n N n
j M n M n
g eM M
eM M
− − × −− ×= − ×− − −
× −− × − × −⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦ ⎣ ⎦
− −
⎛ ⎞⎛ ⎞⎡ ⎤⊗⎜ ⎟⎜ ⎟⎛ ⎞⎛ ⎞ ⎢ ⎥⎣ ⎦′ ′ ′ ′ ′ ⎝ ⎠= ⊗ ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠ ⎜ ⎟⎝ ⎠ ⎜ ⎟⎝ ⎠
⎡ ⎛ ⎞′ ′ ′= ⊗ ⎜ ⎟
⎝ ⎠⎣
⊕I IK 1 1
I
1 1( )
( )( )( )
( )
( )1 1 2 2
11
1 1 1 2 1 11 2 1
0 0
1 1 10 0 0 0
s ss s s
N N
N
n M n N nM n ns S n nS n
j n M n n M n n M nN N nN n
eM M M
− −− ×= − ×× −
× − × − × − × −
×
⎛ ⎞⎤ ⎛ ⎞⎡ ⎤⎜ ⎟′⊗⎢ ⎥ ⎜ ⎟⎢ ⎥⎜ ⎟⎣ ⎦⎝ ⎠⎦⎝ ⎠⎛ ⎞⎡ ⎤⎛ ⎞
′ ′ ′ ′ ′ ′ ′ ′⎜ ⎟= ⊗⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎝ ⎠
⊕I I
1 1 1
C07bx06 4/17/2007 32
( )
( )
( )
1 1 2 2
1
1
2
2
1 1 1 2 1 11 2 1
1
11
1
22
1
1 1 10 0 0 0
0
1
0
1
0
1
0
N N
s
s
II iII iII II
j n M n n M n n M nN N nN n
n
M n
n
M nj
n
M nss
N n
eM M M
M
Me
M
× − × − × − × −
×
×
−
×
−
×
−
−
′′ ′=
′⎛ ⎞⎡ ⎤⎛ ⎞′ ′ ′ ′ ′ ′ ′ ′⎜ ⎟= ⊗⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎝ ⎠
⎡ ⎡ ⎤⎢ ⎢ ⎥⎢ ⎢ ⎥⎢ ⎢ ⎥
⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⊗⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
⎣
K g g K
1 1 1
1
1
1
⎤⎥⎥⎥
⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥
⎦
( )
( ) ( )
( )
1
1
*,
1
12 212
1 1 1
var
0
11 1 10 0
1
0
s s
I II iII I UV II iII
n
N N N j M n
n n Ms ns N Ms Ms N M sSn N ns s sSn n
N n
n
eN MuN N N N
σ
×
−
− × −= = =×
−
′=
⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟⊗⎛ ⎞⎛ ⎞ ⎛ ⎞⎡ ⎤ ⎛ ⎞ ⎛ ⎞ ⎢ ⎥⎜ ⎟′= ⊗ ⊗ − + ⊗⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎢ ⎥⎜ ⎟ ⎢ ⎥⎜ ⎟−⎣ ⎦ ⎝ ⎠ ⎝ ⎠ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎢ ⎥⎝ ⎠⎢ ⎥⎢ ⎥⎣ ⎦
= ⊗
⊕ ⊕ ⊕
V g K Y K g
1I I P J uu I P
I ( ) ( )
( )
( ) ( )
1
1
1
1212
1 1
1 1
0
11 10 0
1
0
10 0
s
s s
n
N N j M n
n Ms ns N Ms MsSn N ns sSn n
N n
N N
n n Ms ns N MSn N ns sSn n
eN MuN N N
N
×
−
− × −= =×
−
− × −= =×
⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟⊗⎛ ⎞⎛ ⎞⎡ ⎤ ⎛ ⎞ ⎢ ⎥⎜ ⎟′⊗ −⎜ ⎟⎜ ⎟ ⎜ ⎟⎢ ⎥⎜ ⎟ ⎢ ⎥⎜ ⎟−⎣ ⎦ ⎝ ⎠ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎢ ⎥⎝ ⎠⎢ ⎥⎢ ⎥⎣ ⎦
⎛ ⎞⎛ ⎞⎡ ⎤+ ⊗ ⊗⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟⎣ ⎦⎝ ⎠⎝ ⎠
⊕ ⊕
⊕ ⊕
1I P J uu
I I I P
( )
1
1
1
121
0
1
0
n
j M n
s
N n
eMσ
×
−
−
⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟⊗⎛ ⎞ ⎢ ⎥⎜ ⎟
⎜ ⎟ ⎢ ⎥⎜ ⎟⎝ ⎠ ⎜ ⎟⎢ ⎥⎝ ⎠⎢ ⎥⎢ ⎥⎣ ⎦
1
For first item:
C07bx06 4/17/2007 33
( ) ( )
( )
1
1
1
1212
1 1
2
21
0
11 10 0
1
0
11
s
n
N N j M n
n n Ms ns N Ms MsSn N ns sSn n
N n
Nn n N n Ms
n ns Ms ns
eN MuN N N
N uN N N N N
×
−
− × −= =×
−
−
=
⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟⊗⎛ ⎞⎛ ⎞⎡ ⎤ ⎛ ⎞ ⎢ ⎥⎜ ⎟′⊗ ⊗ −⎜ ⎟⎜ ⎟ ⎜ ⎟⎢ ⎥⎜ ⎟ ⎢ ⎥⎜ ⎟−⎣ ⎦ ⎝ ⎠ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎢ ⎥⎝ ⎠⎢ ⎥⎢ ⎥⎣ ⎦
⎡ ′ ⎛ ⎞⎛ ⎞ ′ ′= − − ⊗ −⎢ ⎜ ⎟⎜ ⎟− ⎝ ⎠ ⎝ ⎠⎣
⊕ ⊕
⊕
1I I P J uu
J 1 1I 1 1 u u
( )
1
1
1
1
1
11
1
2
121 1
0
1
0
0
1 11
1
n
j M n
N n
n
Nn Ms
n ns Ms n j M ns
nn j
s
eM
N u eN N N N M
N eN N
×
−
−
×
−=
⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟⊗⎤ ⎢ ⎥⎜ ⎟⎥ ⎢ ⎥⎜ ⎟⎜ ⎟⎦ ⎢ ⎥⎝ ⎠⎢ ⎥⎢ ⎥⎣ ⎦
⎛ ⎞⎡ ⎤⎛ ⎞⎜ ⎟⎢ ⎥⎜ ⎟⎡ ⎤⎛ ⎞⎛ ⎞ ⎜ ⎟⎢ ⎥⎜ ⎟′ ′= − ⊗ − ⊗⎢ ⎥⎜ ⎟⎜ ⎟ ⎜ ⎟⎢ ⎥⎜ ⎟− ⎝ ⎠ ⎝ ⎠⎣ ⎦⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎝ ⎠
⎡ ⎤⎛ ⎞= − ⊗⎜ ⎟⎢ ⎥− ⎝ ⎠⎣ ⎦
⊕
1
JI 1 1 u u 1
JI
( )
1
1
1
2
121 1
21 1 1
11
22
1
0
1 1
11
n
NMs
ns Ms n M n
Mn
Ns s sn
j nMs s ssns s
s
uN N M
u M nN M
u M nN e u M nN N N MN M
×
−=
=
⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎛ ⎞⎜ ⎟⎜ ⎟′ ′−⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
⎛ ⎞⎛ ⎞−⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟ −⎛ ⎞⎡ ⎤⎛ ⎞ ⎜ ⎟⎜ ⎟= − ⊗ −− ⎜ ⎟⎜ ⎟⎢ ⎥ ⎜ ⎟− ⎜ ⎟⎝ ⎠⎣ ⎦ ⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
⊕
∑
1 1 u u 1
1
1 u1
For the second item:
C07bx06 4/17/2007 34
( ) ( )
( )
( ) ( )
1
1
1
1
1
121
1 1
1
121
1 1
0
110 0
0
0
11 0 0
s s
s s
n
N N j M n
n n Ms ns N M sSn N ns sSn n
N n
n
N N j M n
n n Ms ns M ss s
eM
N
eM
N
σ
σ
×
−
− × −= =×
−
×
−
−= =
⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟⊗⎛ ⎞⎛ ⎞⎡ ⎤ ⎛ ⎞ ⎢ ⎥⎜ ⎟⊗ ⊗⎜ ⎟⎜ ⎟ ⎜ ⎟⎢ ⎥⎜ ⎟ ⎢ ⎥⎜ ⎟⎣ ⎦ ⎝ ⎠ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎢ ⎥⎝ ⎠⎢ ⎥⎢ ⎥⎣ ⎦
⎛⎜⎜⊗⎡ ⎤⎡ ⎤⎛ ⎞ ⎜= ⊗⎢ ⎥⎜ ⎟⎢ ⎥
⎝ ⎠⎣ ⎦⎣ ⎦ ⎝
⊕ ⊕
⊕ ⊕
1I I I P
1I I P
( )
( ) ( )
( )
( )
1
1
1
121
1
2
1
0
0
11 10 0
0
1 10
s
s
N n
n
N j M n
n s n Ms ns ns Mss s
N n
N
n s n Ms ns ns Mss s
eM
N M
N M
σ
σ
−
×
−
−=
−
−=
⎡ ⎤⎞⎢ ⎥⎟⎢ ⎥⎟⎢ ⎥⎟⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎠⎢ ⎥⎢ ⎥⎣ ⎦
⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟⊗⎡ ⎤⎛ ⎞⎡ ⎤ ⎢ ⎥⎜ ⎟′= ⊗ −⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟ ⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎣ ⎦⎝ ⎠⎣ ⎦ ⎢ ⎥⎝ ⎠⎢ ⎥⎢ ⎥⎣ ⎦
⎡ ⎤⎛ ⎞⎡ ⎤′= ⊗ −⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎥⎣ ⎦⎝ ⎠⎣ ⎦
⊕
⊕
1I I 1 1
I I 1 1
( )
( )
( )
1
1
1
1
1
11
1
21
1 1
21 1 1
121
2
2
0
1
0
1 1 10
1
s
n
j M n
n
N
j s n Ms ns ns Ms M ns s
n
j s s sns
s
eM
eN M M
M nM
e M nNM
σ
σ
σ
×
−
×
− −=
⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⊗⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
⎡ ⎤⎛ ⎞⎛ ⎞⎢ ⎥⎜ ⎟⎜ ⎟⎛ ⎞⎡ ⎤⎢ ⎥⎜ ⎟⎜ ⎟′= ⊗ −⎜ ⎟⎢ ⎥⎢ ⎥⎜ ⎟⎜ ⎟⎜ ⎟⎣ ⎦⎝ ⎠⎢ ⎥⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎝ ⎠⎣ ⎦
⎛ ⎞−−⎜ ⎟
⎜ ⎟⎜= ⊗ −⎜−⎜⎜⎝ ⎠
⊕
1
I 1 1 1
1
1
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎟⎢ ⎥⎟⎢ ⎥⎟⎢ ⎥⎟⎣ ⎦
Add these two items together.
C07bx06 4/17/2007 35
( )
( )
( )
( )
*,
221 1 11 1 1
1 121 1
2 22
1 2
var
1 11
I II iII I UV II iII
Mn n
Ns s sn
j n jMs s s s s ssns s ns
s s
M nu M nN M M
u M nN e eu M n M nN N N M N
N M M
σ
σ=
′=
⎡⎛ ⎞ ⎛ ⎞−⎛ ⎞−−⎢⎜ ⎟ ⎜ ⎟⎜ ⎟
⎢⎜ ⎟ ⎜ ⎟⎜ ⎟ −⎛ ⎞⎡ ⎤⎛ ⎞ ⎢⎜ ⎟ ⎜ ⎟⎜ ⎟= − ⊗ − + ⊗− −⎜ ⎟⎜ ⎟⎢ ⎥ ⎢⎜ ⎟ ⎜ ⎟− ⎜ ⎟ −⎝ ⎠⎣ ⎦ ⎝ ⎠ ⎢⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎣
∑
V g K Y K g
1 1
1 u1 1
( ) ( )
2 21 1 1 1 1 1
1 11 1
2 22 2
1 1
1 11
M Mn n
N Ns s s s s sn
j nMs s s Ms s ss sns nss s
s s
u M n u M nN M N M
u M n u M nN e u M n u M nN N M N N M
N M N M= =
⎤⎥⎥⎥⎥⎥
⎢ ⎥⎦
⎛ ⎞ ⎛⎛ ⎞ ⎛ ⎞− −⎜ ⎟ ⎜⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜⎜ ⎟ ⎜ ⎟− −⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜⎜ ⎟ ⎜ ⎟= ⊗ − − ⊗ −− −⎜ ⎟ ⎜⎜ ⎟ ⎜− ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎜ ⎟ ⎜⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝
∑ ∑
1 1
1u1 1
( )( )
( )( )
2 1 11 11 1
11
22
1
1111
1 11 1
Ns s s
nns s
Nj s s s s s
s ns nss s s
u M nM n uuN N MN M
e M n u M nu uN M N N M
=
=
⎛ ⎞−⎛ ⎞−⎛ ⎞ ⎜ ⎟⎜ ⎟⎜ ⎟ −− ⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟= ⊗ −− −⎛ ⎞⎜⎜ ⎟ ⎜ ⎟− ⎜ −⎜ ⎟ ⎝ ⎠⎜⎜ ⎟ ⎜⎝ ⎠ ⎝ ⎠
∑
∑
11
1 1
( )
( )
( )
( )( )
21 1 1
121
2
2
21 1 1
11
22
1
2 1 11
1
1
1
1
1 11 1
n
s s sns
s
Mn
Ns s sn
nMs s ssns s
s
s s s
j
M nM N
M nM N
u M nN M
u M nu M nN N MN M
u M nM nuN M N N M
e
σ
σ
=
⎛ ⎞⎛ ⎞−⎜ ⎟−⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟+ −⎜ ⎟⎟ ⎜ ⎟−⎜ ⎟⎟ ⎜ ⎟⎜ ⎟⎟ ⎜ ⎟⎟⎜ ⎟⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞−⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟ −⎛ ⎞⎜ ⎟⎜ ⎟− ⊗ −− ⎜ ⎟⎜ ⎟⎜ ⎟ ⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
−−−
− −
= ⊗
∑
1
1
1
1 u1
( )
( )( ) ( )
( )
21 1 1
1 121 1
22
21
21 1 1
1 1211
2
2
1
1 1 11 1
1
1
N
ns s
Ns s s s s ss s
s s nsss s s
Ns s sM
ns s
ns sMs s s
s
M nu
M N
u M n M nM nu uN M N N M M N
u M nu M n uN M N M
u Mu M nNN M N
σ
σ
=
=
=
⎛ ⎞⎡ ⎤−⎛ ⎞−⎜ ⎟⎢ ⎥⎜ ⎟
⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦⎜ ⎟⎡ ⎤− −⎛ ⎞−⎜ ⎟− −⎢ ⎥⎜ ⎟⎜ ⎟− −⎢ ⎥⎝ ⎠⎣ ⎦⎜ ⎟⎜ ⎟⎝ ⎠
⎡ ⎤−⎛ ⎞−−⎢ ⎥⎜ ⎟
⎢ ⎥⎝ ⎠⎣ ⎦− ⊗ −−
−
∑
∑
∑
1
1
1
1( )
1
Ns
s nss s
nu
M=
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎜ ⎟⎜ ⎟⎝ ⎠
∑ 1
C07bx06 4/17/2007 36
Refer ( ) ( )1 1* 1 1 1 1 1 1
,ˆ I I I I I I I I I II iII I I I I I II iII
− −− − − − − −⎡ ⎤′ ′ ′ ′= − +⎢ ⎥⎣ ⎦
a V V X X V X X V V g V X X V X X g
( )
( )( ) ( )
( )( ) ( )
*,
21 1 12 1 1
1 1 211 1
22
21
21 1 1
var
1 11 1
1 11 1
I II iII I UV II iII
Ns s s
ns s
Nj s s s s s ss s
11
1s s n
ss s s
M
n
u M n M nM nu uN M N N M M N
e u M n M nM nu uN M N N M M N
u M nN M
N
σ
σ
=
=
′=
⎛ ⎞⎡ ⎤− −⎛ ⎞−− −⎜ ⎟⎢ ⎥⎜ ⎟− −⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦
⎜ ⎟= ⊗ ⎡ ⎤− −⎛ ⎞−⎜ ⎟− −⎢ ⎥⎜ ⎟⎜ ⎟− −⎢ ⎥⎝ ⎠⎣ ⎦⎜ ⎟⎜ ⎟⎝ ⎠
−
− ⊗
∑
∑
V g K Y K g
1
1
1
s
( )
( )
1 1211
2
21
1
1
Ns s s
ns s
Ns s sMs s s
s nsss s
u M nu
N M
u M nu M n uN M N M
=
=
⎛ ⎞⎡ ⎤−⎛ ⎞−⎜ ⎟⎢ ⎥⎜ ⎟
⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦⎜ ⎟⎛ ⎞−⎛ ⎞−⎜ ⎟−⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎜ ⎟⎜ ⎟⎝ ⎠
∑
∑
1
1
( )
( ) ( )
11 1 1 1
2 2
2 2
2 21 12
1 1
11 1 11 1
I I I I I I I I
s MsN N
s s s sn ns ns nN Ns s
s s s s ss ss s
s ss s
uM N u n u
k nv vk N k Nk k
σ
σ σ
−− − − −
= =
= =
′ ′−
⎡ ⎤⎛ ⎞ ⎛⎛ ⎞−⎢ ⎥⎜ ⎟ ⎜⎜ ⎟−⎝ ⎠⎢ ⎥⎜ ⎟ ⎜= ⊗ ⊕ + − ⊗ ⊕ +⎢ ⎥⎜ ⎟ ⎜⎛ ⎞ ⎛ ⎞
− −⎢ ⎥⎜ ⎟ ⎜⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜⎢ ⎥⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝⎣ ⎦∑ ∑
V V X X V X X V
I I + J J 1nsn k
⎞⎟⎟⎟⎟⎟⎠
J
C07bx06 4/17/2007 37
( )
( ) ( )
11 1 1 1
,
2 2
2 2
2 21 12
1 1
11 1 111 1
I I I I I I I I I II iII
s MsN N
s s s sn ns ns n nN Ns s
s s s s ss ss s
s ss s
uM N u n u
k n n kv vk N k Nk k
σ
σ σ
−− − − −
= =
= =
⎡ ⎤′ ′−⎢ ⎥⎣ ⎦
⎡ ⎤⎛ ⎞ ⎛ ⎞⎛ ⎞−⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟−⎝ ⎠⎢ ⎥⎜ ⎟ ⎜ ⎟= ⊗ ⊕ + − ⊗ ⊕ +⎢ ⎥⎜ ⎟ ⎜ ⎟⎛ ⎞ ⎛ ⎞
− −⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠⎣ ⎦∑ ∑
V V X X V X X V V g
I I + J J J
( )( ) ( )
( )( ) ( )
2 21 1 12 1 1 1 1 1
1 1 12 211 1 1
22
21
1 1 1 11 1
1 1 11 1
s
Ns s s M
ns s
nNj s s s s s ss ss s ns
ss s s
u M n M nM n u M nu uN M N N M M N N M N
e u M n M nM n Nu uN M N N M M N
σ
σ
=
=
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
⎛ ⎞⎡ ⎤− −⎛ ⎞− −− − −⎜ ⎟⎢ ⎥⎜ ⎟− −⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦
⎜ ⎟⊗ − ⊗⎡ ⎤− −⎛ ⎞−⎜ ⎟− −⎢ ⎥⎜ ⎟⎜ ⎟− −⎢ ⎥⎝ ⎠⎣ ⎦⎜ ⎟⎜ ⎟⎝ ⎠
∑
∑
1
11
( )
2
2
2 2
2
2 212
1
1
11
1
Ms s s
s
s MsN
s sn ns nsNs
s s s ss
s s
u M nN M N
uM N u
k vk Nk
σ
σ σ=
=
⎡ ⎛ ⎡⎢ ⎜ ⎢
⎢⎢ ⎜ ⎣⎢ ⎜
⎛ −⎢ ⎜ −⎜⎢ ⎜⎜⎝⎢ ⎜⎜⎢ ⎝⎣
⎡ ⎛ ⎞⎛ ⎞−⎢ ⎜ ⎟⎜ ⎟−⎝ ⎠⎢ ⎜ ⎟= ⊗ ⊕ +⎢ ⎜ ⎟⎛ ⎞
−⎢ ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎣∑
I I + J( )
(
( )(
2 1 11
11
2
1
1 11 1
1 11 1
Ns
s
Nj ss ss
ss
n
uM nuN M N N
e uM nuN M N N
=
=
⎛ ⎛ ⎡ ⎛−⎜ −⎛ ⎞ ⎜⎤ ⎢ ⎜− −⎜⎜ ⎟ ⎢⎜ ⎝⎥ ⎣⎜⎜ ⎟ ⎜⎥ ⊗ ⎡ ⎛−⎜⎜ ⎟ ⎜⎥ −⎢ ⎜⎜⎜ ⎟ ⎜⎥ − −⎢ ⎝⎣⎜ ⎟⎜ ⎜⎢ ⎥⎦⎝ ⎠ ⎜⎜ ⎝⎝
− ⊗
∑
∑
I( )
( )
( )
21 1 12 2
2112
22 21
2 21
1
1
111
1
Ns s sM
s Ms s sN
s s n Nns nsN s s sMs s sss s s s
s ss ss s
u M nu M nu N M N M
M N uu M nu M nk Nvk N N M N Mk
σ
σ σ
=
=
==
⎡ −⎛ ⎞−−⎛ ⎞⎡ ⎤ ⎢⎛ ⎞ ⎜⎛ ⎞
⎜ ⎟ ⎢− ⎝ ⎠⎢ ⎥⎜ ⎟ ⎣⎜ ⎟−⎜ ⎟⎝ ⎠⎢ ⎥⎜ ⎟⊕ + ⊗ ⎛ −⎛ ⎞−⎜ ⎟⎢ ⎥⎜ ⎟⎛ ⎞ − ⎜⎜ ⎟−⎢ ⎥⎜ ⎟⎜ ⎟⎜ ⎟ ⎝ ⎠⎝⎜ ⎟⎢ ⎥⎝ ⎠⎝ ⎠⎣ ⎦⎝ ⎠
∑
∑∑
1I + J
( )
( )( ) (2
12 1 11 1
112
21
1
1 11 1
1 11 11 1
Ns s s
s sN
s sn ns jN ss
s ss ss
s s
u M nM nu uN M N N M
n u e Mn n kv uk N Nk
σ
=
=
=
⎛ ⎛⎜ ⎜⎜ ⎜⎜ ⎜⎜ ⎜
⎜⎜ ⎜⎜⎜ ⎜
⎜⎜ ⎝⎝
⎡ −⎛ ⎞−− −⎛ ⎞ ⎢⎛ ⎞ ⎜ ⎟− −⎢⎜ ⎟ ⎝ ⎠⎜ ⎟ ⎣
⎜ ⎟⎜ ⎟− ⊗ ⊕ + ⊗ −⎜ ⎟⎜ ⎟⎛ ⎞⎜ ⎟−⎜ ⎟⎜ ⎟ −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎝ ⎠
∑
∑J J
( )( ) (
( )
( )
2
1
21 1 1
212
1
1
11
1
1 111
Ns s s ss
sss s
s s sM
sN
s s nn nsNs
s sss
s s
u M nn uM N N M
u M nu M nN M N M
n un n k Nvk N
k
σ
=
=
=
⎛ ⎛⎜ ⎜⎜ ⎜⎜ ⎜
⎡ −⎛ ⎞⎜ ⎜ − −⎢ ⎜ ⎟⎜ ⎜ −⎢ ⎝ ⎠⎣⎜ ⎜⎜⎜ ⎝⎝
−⎛−−⎛ ⎞⎛ ⎞ ⎜
⎜ ⎟ ⎝⎜ ⎟⎜ ⎟⎜ ⎟+ ⊗ ⊕ + ⊗⎜ ⎟⎜ ⎟⎛ ⎞⎜ ⎟−⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎝ ⎠
∑
∑
1J J ( )
1 11
2
21
1
N
ns
Ns s sMs s s
s nsss s
u
u M nu M n uN M N M
=
=
⎛ ⎞⎛ ⎞⎡ ⎤⎞⎜ ⎟⎜ ⎟⎢ ⎥⎟⎜ ⎟⎢ ⎥⎜ ⎟⎠⎣ ⎦⎜ ⎟⎜ ⎟
⎛ ⎞−⎛ ⎞−⎜ ⎟⎜ ⎟−⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
∑
∑
1
1
C07bx06 4/17/2007 38
For first item:
( )
( )(
( )
2 1 12 2 1112
2 2 212
1
1 11 1
111 1
1 1 1
Ns
s Ms sN
s sn ns ns jN s ss
s s s s ss s
s s
u MM nuu N M N N MM N u e M nk v uk N N M N Nk
σ
σ σ
=
=
=
⎡ ⎛−−⎛ ⎞⎡ ⎤ ⎢⎛ ⎞ ⎜⎛ ⎞ − −⎜ ⎟ ⎢− ⎝⎢ ⎥⎜ ⎟ ⎣⎜ ⎟−⎜ ⎟⎝ ⎠⎢ ⎥⎜ ⎟⊗ ⊕ + ⊗ −⎜ ⎟⎢ ⎥⎜ ⎟⎛ ⎞ −⎜ ⎟−⎢ ⎥⎜ ⎟⎜ ⎟ − −⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎝ ⎠⎣ ⎦⎝ ⎠
∑
∑I I + J (
( ) ( )( ) ( )
1
222
21
1
1 1 1 1 11 1
1
Ns
s
Ns s s s s ss s s
j s s nsNss s s ss
ss s
u MM
u M n M nu n M ne u uk N M N N M M Nvk N
k
σ
=
=
=
⎛ ⎛⎜ ⎜⎜ ⎜⎜ ⎜
⎡ ⎛⎜ ⎜⎢ ⎜⎜ ⎜ ⎢ ⎝⎣⎜ ⎜⎜⎜ ⎝⎝
⎛ ⎞⎜ ⎟ ⎡ ⎤− −⎛ ⎞−⎜ ⎟= ⊗ − −⎢ ⎥⎜ ⎟⎜ ⎟ − −⎛ ⎞ ⎢ ⎥⎝ ⎠⎣ ⎦−⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
∑
∑∑
+ 1
⎛ ⎞⎛ ⎞⎜ ⎜ ⎟⎜ ⎜ ⎟⎜ ⎜ ⎟⎜ ⎜ ⎟⎜ ⎜ ⎟⎜ ⎜ ⎟⎜ ⎜ ⎟
⎜ ⎟⎜ ⎝ ⎠⎝ ⎠Second item:
( )
( )
( )
21 1 12 2 12
112
22 21
2 21
1
1
111
1
Ns s sM
s Ms s sN
s s n Nn ns nsN s s sMs s sss s s s
s ss ss s
u M nu M n uu N M N MM N u
u M nu M nk Nv uk N N M N Mk
σ
σ σ
=
=
==
⎡ −⎛ ⎞−−⎛ ⎞⎡ ⎤ ⎢⎛ ⎞ ⎜⎛ ⎞
⎜ ⎟⎟
⎢− ⎝ ⎠⎢ ⎥⎜ ⎟ ⎣⎜ ⎟−⎜ ⎟⎝ ⎠⎢ ⎥⎜ ⎟⊗ ⊕ + ⊗ −⎛ ⎞−⎜ ⎟⎢ ⎥⎜ ⎟⎛ ⎞ − ⎜ ⎟⎜ ⎟−⎢ ⎥⎜ ⎟⎜ ⎟⎜ ⎟ ⎝ ⎠⎜ ⎟⎢ ⎥⎝ ⎠⎝ ⎠⎣ ⎦⎝ ⎠
∑
∑∑
1I I + J
( )
( )2 2
21
1
1 1
1
s
Ns s sn s Ms s s
s nsNss s ss
ss s
u M nu ns u M n uN k N M N Mvk N
k=
=
⎛ ⎛⎜ ⎜⎜ ⎜⎜ ⎜
⎛⎜ ⎜⎜⎜ ⎜⎜⎝⎜ ⎜⎜⎜ ⎝⎝
⎛ ⎞⎜ ⎟
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞−⎛ ⎞−⎜ ⎟⎜ ⎟= ⊗ −⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞⎜ ⎟⎝ ⎠⎝ ⎠−⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎜ ⎟
⎜ ⎟⎝ ⎠
∑∑
1 1 + 1
Third item:
C07bx06 4/17/2007 39
( )
( )( ) (
( )( ) (
212 1 1
1 1112
221
11
1 11 1
1 11 1 11 1 1
Ns s s
s sN
s s Nn ns jN s s s ss sss ss s s
s ss ss s
u M n MM nu uN M N N M M
n u e u M n MM nn n kv u uk N N M N N M Mk
σ
σ
=
=
==
⎡ −⎛ ⎞−− −⎛ ⎞ ⎢⎛ ⎞ ⎜ ⎟− −⎢⎜ ⎟ ⎝ ⎠⎜ ⎟ ⎣
⎜ ⎟⎜ ⎟⊗ ⊕ + ⊗ ⎡ −⎛ ⎞−⎜ ⎟⎜ ⎟⎛ ⎞ − −⎜ ⎟⎜ ⎟−⎜ ⎟⎜ ⎟ − −⎜ ⎟ ⎝ ⎠⎜ ⎟⎝ ⎠⎝ ⎠⎝ ⎠
∑
∑∑J J
( ) ( )( ) ( )22
22
1
1
1 1 1 1 111 1
1
Ns s s s s ss s s s
n s sNss s s ss
ss s
u M n M nn u M nu un k N M N N M M Nvk N
k
σ
=
=
⎛ ⎛⎜ ⎜⎜ ⎜⎜ ⎜⎜ ⎜
ns
⎢⎜ ⎜ ⎢⎣⎜ ⎜⎜⎜ ⎝⎝
⎛⎜
⎛ ⎞⎜⎜ ⎟⎜ ⎡ ⎤− −⎛ ⎞−⎜ ⎟⎜= ⊗ + − −⎢ ⎥⎜ ⎟⎜ ⎟ − −⎛ ⎞⎜ ⎢ ⎥⎝ ⎠⎣ ⎦−⎜ ⎟⎜ ⎟⎜ ⎜ ⎟⎝ ⎠⎝ ⎠⎜
⎜⎝
∑∑
1 1
Fourth item:
( )
( )
( )
21 1 1
1 12112
21
21
1
1
1 11 11
Ns s sM
ns s
Ns s n Nn nsN s s sMs s ss
s ss s nss ss s
s s
u M nu M n uN M N M
n uu M nu M nn n k Nv uk N N M N Mk
=
=
==
⎛ ⎛ ⎞⎡ ⎤−⎛ ⎞−⎜ −⎜ ⎟⎛ ⎞ ⎢ ⎥⎛ ⎞ ⎜⎜ ⎢ ⎥⎜ ⎟⎜ ⎟ ⎝ ⎠⎜ ⎟ ⎣ ⎦⎜ ⎜ ⎟⎜ ⎟⎜ ⎟⊗ ⊕ + ⊗ ⎛ ⎞−⎛ ⎞−⎜ ⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞ −⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟−⎜ ⎟⎜ ⎟⎜ ⎟ ⎝ ⎠⎜ ⎟ ⎝ ⎠⎜ ⎟⎝ ⎠⎝ ⎠⎝ ⎠ ⎜ ⎟
⎝ ⎠⎝
∑
∑∑
1
1J J1
⎟
( )
( )2 2
21
1
1 111
Ns s sn s s Ms s s
s nsNss s ss
ss s
u M nn u u M n uN k N M N Mvk N
k=
=
⎞⎟⎟⎟⎟
⎜ ⎟⎜ ⎟⎜ ⎟
⎠⎛ ⎞⎜ ⎟
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞−⎛ ⎞−⎜ ⎟⎜ ⎟= ⊗ + −⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞⎜ ⎟⎝ ⎠⎝ ⎠−⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎜ ⎟
⎜ ⎟⎝ ⎠
∑∑
1 1
1-2-3+4 = 4-2-3+1 4-2
C07bx06 4/17/2007 40
( )
( )
( )
2 2 2
21
1 1
1 1 111 1
Ns s sn s s Ms s s n s
s nsN Nss s s ss s
s ss ss s
u M nn u u M n u nsuN k N M N M N kv vk N k N
k k=
= =
⎛ ⎞⎜ ⎟
⎛ ⎞ ⎛⎜ ⎟⎜ ⎟ ⎜⎜ ⎟⎛ ⎞−⎛ ⎞−⎜ ⎟ ⎜⎜ ⎟⊗ + − − ⊗⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟ ⎜⎛ ⎞ ⎛⎜ ⎟⎝ ⎠⎝ ⎠− −⎜ ⎟ ⎜⎜ ⎟ ⎜⎜ ⎟⎜ ⎟ ⎜⎝ ⎠ ⎝⎝ ⎠ ⎝⎜ ⎟
⎜ ⎟⎝ ⎠
∑∑ ∑
1 11 +
0
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝
=
1-3 =
( ) ( )( ) ( )
( )
222
21
1
2
1
1 1 1 1 11 1
1
1 1 11
Ns s s s s ss s s
j s sNss s s ss
ss s
s sn N
s ss
s s
u M n M nu n M ne u uk N M N N M M Nvk N
k
n un k vk N
k
σ
=
=
=
⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎡ ⎤− −⎛ ⎞−⎜ ⎟⎜ ⎟⎜ ⎟⊗ − −⎢ ⎥⎜ ⎟⎜ ⎟ − −⎛ ⎞⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦−⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎜ ⎟⎜ ⎟
⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
⎛⎜⎜− ⊗ +⎜ ⎛ ⎞
−⎜ ⎜ ⎟⎜ ⎝ ⎠⎝
∑∑
∑
1 + 1
1
ns
( )( ) ( )
( ) ( )( )
22
21
22
1
1
1 1 11 1
1 1 1 1 11 1
1
Ns s s s s ss s
s sss s s
Ns s s ss s s
j n s sNss s ss
ss s
u M n M nM nu uN M N N M M N
u M nu n M ne u un k N M N N Mvk N
k
σ
σ
=
=
=
⎛ ⎞⎜ ⎟
⎞⎜ ⎟⎟⎜ ⎟⎡ ⎤− −⎛ ⎞−⎟⎜ ⎟− −⎢ ⎥⎜ ⎟⎟ − −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎟⎜ ⎟⎟⎠⎜ ⎟
⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟
−⎛ ⎞−⎛ ⎞ ⎜ ⎟= − ⊗ − −⎜ ⎟⎜ ⎟ ⎜ ⎟ − −⎛ ⎞⎝ ⎠ ⎝ ⎠−⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
∑
∑∑
1
1 1 + ( )
ns
2
2
1s sns
s
M nM N
⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎡ ⎤−⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟
⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
1
( )
( ) ( )( ) ( )
11 1 1 1
,
222
21
1
1 1 1 1 11 1
1
I I I I I I I I I II iII
Ns s s s s ss s s s
j n s s nsNss s ss
ss s
u M n M nu n M ne u un k N M N N M M Nvk N
k
σ
−− − − −
=
=
⎡ ⎤′ ′−⎢ ⎥⎣ ⎦⎛ ⎛ ⎞⎜ ⎜ ⎟
⎛ ⎞⎜ ⎜ ⎟⎜ ⎟⎜ ⎜ ⎟⎡ ⎤− −⎛ ⎞−⎛ ⎞ ⎜ ⎟⎜ ⎜ ⎟= − ⊗ − −⎢ ⎥⎜ ⎟⎜ ⎟ ⎜ ⎟ − −⎛ ⎞⎜⎝ ⎠ ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦−⎜ ⎟⎜ ⎟⎜ ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎜ ⎟
⎜ ⎟⎝ ⎠⎝
∑∑
V V X X V X X V V g
1 1 + 1
⎞⎟⎟⎟⎟⎟⎟
⎜ ⎟⎜ ⎟
⎠
s
For
C07bx06 4/17/2007 41
( )
( )1 2
1 2
11 2
1 2
1
1 211 2
11 1 ... 0
1
1 1 1...
II iII iII II
N
n Ms nss
j M n M n N n N
N n Mss
N
j M n M n ns
Ne
M MN
eM M N
−=
− − −
−=
− −=
′′ ′=
′⎛ ⎞⎛ ⎞⎡ ⎤⎛ ⎞⊗⎜ ⎟⎜ ⎟⎜ ⎟⎢ ⎥⎛ ⎞⎛ ⎞ ⎝ ⎠⎣ ⎦⎜ ⎟⎜ ⎟′ ′ ′ ′= ⊗⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎡ ⎤⎛ ⎞⎝ ⎠⎝ ⎠⎜ ⎟⎜ ⎟⊗ ⎜ ⎟⎢ ⎥⎜ ⎟⎜ ⎟⎝ ⎠⎣ ⎦⎝ ⎠⎝ ⎠
⎡ ⎤⎛ ⎞′ ′ ′= ⊗ ⊗⎢ ⎥⎜ ⎟
⎝ ⎠⎣ ⎦
⊕
⊕
⊕
X g g X
1 11 1
1 1
1 1 1
1 ...
1
Ms ns
s s
s
s s
s
M nN M
M nN M
−
′⎛ ⎞⎡ ⎤⎛ ⎞⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦⎝ ⎠
′⎛ ⎞⎛ ⎞−= ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎛ ⎞⎜ ⎟
⎡ ⎤−⎜ ⎟= ⎢ ⎥⎜ ⎟⎣ ⎦⎜ ⎟⎜ ⎟⎝ ⎠
1
So for ( ) 11 1
I I I I I II iI
−− −′ ′V X X V X X Ig
C07bx06 4/17/2007 42
( )
( )
( )
( )
11 1
22 2
21
1 1
2
1 1 11 1
I I I I I II iII
Ns ss s Ms
n nsN Nss s ss s
s ss ss s
N n n un u nuN k N k lv wk N l N N
k l
Nn
−− −
=
= =
′ ′
⎡ ⎤⎛ ⎞⎡ ⎤⎡ ⎤⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟⎢ ⎥⎢ ⎥⎜ ⎟ ⎜ ⎟
−⎢ ⎥⎜ ⎟⎢ ⎥⎢ ⎥⎜ ⎟ ⎜ ⎟= ⊗ ⊕ − −⎢ ⎥⎜ ⎟⎢ ⎥⎢ ⎥⎜ ⎟ ⎜ ⎟⎛ ⎞ ⎡ ⎤⎢ ⎥⎜ ⎟⎢ ⎥− −⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟⎢ ⎥⎢ ⎥⎢ ⎥⎝ ⎠ ⎣ ⎦⎝ ⎠ ⎝ ⎠⎣ ⎦⎣ ⎦⎝ ⎠⎣ ⎦
⊕
∑ ∑
V X X V X X g
1 1 + 1
( )
( )
( )
( )
1
22 2 2
2 2
1 1
2
1
1 1
11 1
1
1
Ns s
ss
s ss s s Ms MsN N
s s ss ss s
s ss s
Ns s
n ss
ss s
M nN M
N n n un n u nu nuk N l N lv wk N l N N
k l
n uN vk N
k
=
= =
=
=
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎛ ⎞⎢ ⎥⎜ ⎟
⎡ ⎤−⎢ ⎥⎜ ⎟⎢ ⎥⎢ ⎥⎜ ⎟⎛ ⎞⎛ ⎞ ⎣ ⎦⎢ ⎥⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎢ ⎥− ⎝ ⎠⎜ ⎟⎜ ⎟− +⎢ ⎥⎜ ⎟⎜ ⎟⎛ ⎞ ⎡ ⎤⎢ ⎥− −⎜ ⎟⎜ ⎟⎜ ⎟ ⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠ ⎣ ⎦⎝ ⎠⎝ ⎠⎣ ⎦
= ⊗ ⊕−
∑ ∑1 +
1 1 + ( )
( )
( )
22
2
1 1
22
1
1 11
1 1
1
s sMsnsN N
s s s ss
s s
s s
s
s s sN
s ss
s s
N n n unuk N k l wl N N
l
M nN M
Nn n unk vk N
k
=
=
⎡ ⎤⎛ ⎞⎡ ⎤⎡ ⎤⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟⎢ ⎥⎢ ⎥⎜ ⎟ ⎜ ⎟
−⎢ ⎥⎜ ⎟⎢ ⎥⎢ ⎥⎜ ⎟ ⎜ ⎟− −⎢ ⎥⎜ ⎟⎢ ⎥⎢ ⎥⎜ ⎟ ⎜ ⎟⎛ ⎞ ⎡ ⎤⎢ ⎥⎜ ⎟⎢ ⎥−⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟⎢ ⎥⎢ ⎥⎢ ⎥⎝ ⎠ ⎣ ⎦⎝ ⎠ ⎝ ⎠⎣ ⎦⎣ ⎦⎝ ⎠⎣ ⎦
⎡ ⎤−⎢ ⎥ ⎛⎣ ⎦
⎜⎜⎜ ⎛ ⎞
−⎜ ⎜ ⎟⎜ ⎝ ⎠⎝
∑ ∑
∑
1
1 + ( )
( )
22 2
2 2
1
11
1 1
s sMs MsN
s s ss
s s
s sn ns
s s
N n n unu nuN l N l wl N N
l
M nn M n
=
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎞⎛ ⎞⎜ ⎟⎟⎜ ⎟
−⎜ ⎟⎟⎜ ⎟− +⎜ ⎟⎟⎜ ⎟⎡ ⎤⎜ ⎟−⎟⎜ ⎟⎢ ⎥⎟⎜ ⎟⎜ ⎟⎣ ⎦⎠⎝ ⎠⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎡ ⎤−⎜ ⎟= ⊗ ⎢ ⎥⎜ ⎟⎣ ⎦⎜ ⎟⎜ ⎟⎝ ⎠
∑
1 1
C07bx06 4/17/2007 43
( ) ( )
( ) ( )( ) ( )
1 1* 1 1 1 1 1 1
,
222
21
1
ˆ
1 1 1 1 11 1
1
I I I I I I I I I II iII I I I I I II iII
Ns s s s s ss s s s
j n s s nNss s ss
ss s
u M n M nu n M ne u un k N M N N M M Nvk N
k
σ
− −− − − − − −
=
=
⎡ ⎤′ ′ ′ ′= − +⎢ ⎥⎣ ⎦
⎛ ⎞⎜ ⎟ ⎡ ⎤− −⎛ ⎞−⎛ ⎞ ⎜ ⎟= − ⊗ − −⎢ ⎥⎜ ⎟⎜ ⎟ ⎜ ⎟ − −⎛ ⎞⎝ ⎠ ⎢ ⎥⎝ ⎠⎣ ⎦−⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
∑∑
a V V X X V X X V V g V X X V X X g
1 1 + 1s
1 1
s
s sn ns
s s
M nn M n
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟⎡ ⎤−⎜ ⎟+ ⊗ ⎢ ⎥⎜ ⎟⎣ ⎦⎜ ⎟⎜ ⎟⎝ ⎠
1 1
C07bx06 4/17/2007 44
( ) ( )( ) ( )
*
11
222
21
1
ˆ
1
1 1
1
1 1 1 11 1
1
I
n
s sj n ns
s s
nNN
Ns s s s s ss s s s
j n s sNss s s ss
ss s
g
MM ne
n M n
M
u M n M nu n M ne u un k N M N N M Mvk N
k
σ
=
=
= +
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎡ ⎤−⎜ ⎟⎜ ⎟= ⊗ + ⊗ ⎢ ⎥⎜ ⎟⎜ ⎟ ⎣ ⎦⎜ ⎟⎜ ⎟ ⎜ ⎟′⎜ ⎟ ⎝ ⎠⎝ ⎠
⎛ ⎞⎜ ⎟ ⎡ − −⎛ ⎞−⎛ ⎞ ⎜ ⎟+ − ⊗ − −⎜ ⎟⎜ ⎟ ⎜ ⎟ − −⎛ ⎞⎝ ⎠ ⎝ ⎠−⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
∑∑
L a
1
1 1
1
1 1 +
( ) ( )( )
11
22
1
1
1 1
1
1 1 1 11 1
1
n
s sj n ns
s s
nNN
s s ss s s sj n sN
s s sss
s s
MM ne
n M n
M
u M nu n M ne un k N M N N Mvk N
k=
⎛⎜⎜⎜⎜ ⎢⎜ ⎢⎣⎜⎜⎜⎝
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎡ ⎤−⎜ ⎟⎜ ⎟= ⊗ + ⊗ ⎢ ⎥⎜ ⎟⎜ ⎟ ⎣ ⎦⎜ ⎟⎜ ⎟ ⎜ ⎟′⎜ ⎟ ⎝ ⎠⎝ ⎠
⎛ ⎞⎜ ⎟
−⎛ ⎞−⎛ ⎞ ⎜ ⎟+ − ⊗ − ⎜⎜ ⎟ ⎜ ⎟ − −⎛ ⎞⎝ ⎠ ⎝ ⎠−⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠∑
1
1 1
1
1 1 + ( )2
21
Ns s s
ss s
M nu
Mσ
=
⎛⎜⎜⎜ ⎡ −⎜ −⎢ ⎟⎜ ⎢⎣⎜⎜⎜⎝
∑
C07bx06 4/17/2007 45
( )
*
11
* *
22
1
ˆ
1
1 1
1
1 1 1 11
1
j I
n
s sj I n ns I
s s
nNN
s s s sj n sN
s sss
s s
P
MM ne
n M n
M
u n M ne un k N M Nvk N
k=
′= =
′⎛ ⎞⎛ ⎞ ′⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟ ⎡ ⎤−⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟= ⊗ + ⊗ ⎢ ⎥⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟ ⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟′ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎜ ⎟ ⎝ ⎠⎝ ⎠⎝ ⎠
⎛ ⎞⎜ ⎟
−⎛ ⎞ ⎜ ⎟+ − ⊗ −⎜ ⎟ ⎜ ⎟ −⎛ ⎞⎝ ⎠ −⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠∑
L Y
1
Y 1 1 Y
1
1 1 +( )
( ) (2
211
Ns s s s s s
ss s s
u M n M nu
N M Mσ
=
⎛ ⎛⎜ ⎜⎜ ⎜⎜ ⎜ ⎡ − −⎛ ⎞⎜ ⎜ −⎢ ⎜ ⎟−⎜ ⎜ ⎢ ⎝ ⎠⎣⎜ ⎜⎜ ⎜
⎜⎜ ⎝⎝
∑
The first item:
11
*
1
1
1
1
n
nj
j I jiij
nNN
Me Y
M
M
=
′⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⊗ =⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟′⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
∑
1
Y
1
Second
*
1 1
1 1 N ns s s s
n ns Is is s s s
M n M n Yn M n n M= =
′⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟
⎡ ⎤ ⎡ ⎤− −⎜ ⎟⎜ ⎟⊗ =⎢ ⎥ ⎢ ⎥⎜ ⎟⎜ ⎟⎣ ⎦ ⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
1 s
si∑ ∑1 1 Y
The third
C07bx06 4/17/2007 46
( ) ( )( ) ( )
( )
222 *
21
1
2
1
1 1 1 1 11 1
1
1
1
Ns s s s s ss s s s
j n s s nsNss s ss
ss s
s sj N
s ss
s s
u M n M nu n M ne u un k N M N N M M Nvk N
k
u nek vk N
k
σ
=
=
=
Is
′⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎡ ⎤− −⎛ ⎞−⎛ ⎞ ⎜ ⎟⎜ ⎟⎜ ⎟− ⊗ − −⎢ ⎥⎜ ⎟⎜ ⎟ ⎜ ⎟ − −⎛ ⎞⎜ ⎟⎝ ⎠ ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦−⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎜ ⎟⎜ ⎟
⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
= ⊗−
∑∑
∑
1 1 + 1 Y
1 +( )
( ) ( )
( )
22 *
21
22
1
1 1 11 1
1 1 1 11
1
Ns s s s s ss s
s sss s s
s s s sn sN
s sss
s s
u M n M nM nu uN M N N M M N
u n M nun k N M Nvk N
k
σ
=
=
⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎡ ⎤− −⎛ ⎞−⎜ ⎟⎜ ⎟⎜ ⎟− −⎢ ⎥⎜ ⎟⎜ ⎟ − −⎛ ⎞⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎜ ⎟⎜ ⎟
⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
⎛ ⎞⎜ ⎟
−⎜ ⎟− ⊗ −⎜ ⎟ − −⎛ ⎞−⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
∑
∑
1 Y
1 1 +( )
ns I
( ) ( )2*
21
11
Ns s s s s s
s nss s s
u M n M nu
N M M Nσ
=
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎡ ⎤− −⎛ ⎞⎜ ⎟−⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
∑ 1 YI
( ) ( )( ) ( )22
2 *2
1
1
1 1 1 1 ./1 1
1
Ns s s s s ss s s s
j s sNss s s ss
ss s
u M n M nu n M ne u uk N M N N M M Nvk N
k
σ
=
=
′⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎡ ⎤− −⎛ ⎞−⎜ ⎟⎜ ⎟⎜ ⎟⊗ − −⎢ ⎥⎜ ⎟⎜ ⎟ − −⎛ ⎞⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦−⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎜ ⎟⎜ ⎟
⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
∑∑
1 + 1 Yns I
C07bx06 4/17/2007 47