Supplementary Mathematical Arguments - Electromagnetic Brain Imaging - page 246

We next show the equality of the two quadratic forms, i.e.,

1
2 ˆ

z H ʣ − 1

zz

T

ʣ − 1

z =

ˆˆ ˆ .

To show this relationship, we use the matrix inversion formula in Eq. ( C.93 ), and
rewrite ʣ − 1

ˆˆ

such that 5

− 1 ʣ − 1

xx ʣ yx − 1

ʣ − 1

ˆˆ =

, (C.20)

− − 1 − 1 ʣ yx ʣ − 1

xx

where is defined in Eq. ( C.16 ). Let us define:

M c = − 1

, (C.21)

M s =− ʣ − 1

xx ʣ yx − 1

. (C.22)

Then, M c is symmetric because is symmetric. Using

xx ʣ yx

ʣ − 1 ʣ xx + ʣ yx ʣ − 1 ʣ − 1

xx ʣ yx = ʣ yx ʣ − 1

xx ʣ yx = xx ,

we also get the relationship:

M s =− ʣ − 1

xx ʣ yx − 1

=− − 1 ʣ yx ʣ − 1

xx − 1

=− − 1 ʣ yx ʣ − 1

xx

− 1 T ʣ yx − 1 T ʣ − 1

xx

ʣ − 1
xx

yx

M s , =− ʣ = =− (C.23)

T

where

yx =− ʣ yx is used. Thus, M s is skew-symmetric.
We define a matrix M as

ʣ

M = ( M c + i M s ) , (C.24)

and compute the quadratic form z H Mz . Since we have the relationship,
z H Mz H

z H M H z z H

H z

= = ( M c + i M s )

z H M c i M s z z H z H Mz

= − = ( M c + i M s ) z = , (C.25)

5 In Eq. ( C.20 ), the

− 1 , which can be shown as follows. According to Eq. ( C.93 ), ( 2 , 2 ) th element is

this element is equal to

ʣ yx ʣ yx

ʣ − 1
yy

+ ʣ − 1
yy

ʣ xx + ʣ yx ʣ − 1
xx

T
yx

ʣ − 1
yy

ʣ .

= − 1 . ʣ − 1

xx

) − 1 Using Eqs. ( C.13 )and( C.91 ), this is equal to ( ʣ + ʣ ʣ xx yx yx