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