QMGreensite_merged

21.1. REVIEWOFVECTORSANDMATRICES 307

or,foreachcomponentofthetransformedcolumnvector,

vi′=

∑D

j=1

mijvj (21.9)

Likewise,amatrixtransformsrowvectorsaccordingtotherule

[w 1 ′,w′ 2 ] = [w 1 ,w 2 ]

[

m 11 m 12

m 21 m 22

]

= [(w 1 m 11 +w 2 m 21 ),(w 1 m 12 +w 2 m 22 )] (21.10)

or,foreachcomponentofthetransformedrowvector

wi′=

∑D

j=1

wjmji (21.11)

TwomatricesAandBcanbemultipliedtoformathirdmatrixC=ABaccording
totherule
[
c 11 c 12
c 21 c 22

]

=

[

a 11 a 12

a 21 a 22

][

b 11 b 12

b 21 b 22

]

=

[

a 11 b 11 +a 12 b 21 a 11 b 12 +a 12 b 22

a 21 b 11 +a 22 b 21 a 21 b 12 +a 22 b 22

]

(21.12)

or,intermsofmatrixcomponents,

cij=

∑D

k=1

aikbkj (21.13)

TheHermitian ConjugateM† ofa matrixM, withcomponents mij is the
transposecomplexconjugateofM,i.e.

M†=

[

m∗ 11 m∗ 21

m∗ 12 m∗ 22

]

(21.14)

or,ingeneral,ifwedenoteby[A]ijthei,jcomponentofthematrixA,

[M†]ij=[M]∗ji (21.15)

TheDeterminantofamatrixMisthesumofproductsofcomponents

det(M)=

∑D

i 1 =1

∑D

i 2

...

∑D

iD=1

!i 1 i 2 ...iDm 1 i 1 m 2 i 2 m 3 i 3 ....mDiD (21.16)