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306 CHAPTER21. QUANTUMMECHANICSASLINEARALGEBRA

21.1 Review of Vectors and Matrices

AD-dimensionalcolumnvectorisasetofDcomplexnumbersarrangedinacolumn








v 1

v 2

v 3

.

.

.

vD















(21.2)

andaD-dimensionalrowvectorisasetofDcomplexnumbersarrangedinarow

[w 1 ,w 2 ,w 3 ,...,wD] (21.3)

Tosavetyping,IwilluseD= 2 whendisplayingrowandcolumnvectorsexplicitly.
Foreachcolumnvector,andoneeachrowvector,thereisdefinedaninnerprod-
uct

[w 1 ,w 2 ]·

[

v 1

v 2

]

= w 1 v 1 +w 2 v 2

=

∑D

i=1

wivi (21.4)

Toeachcolumnvector,thereisacorrespondingrowvector(andvice-versa)
[
v 1
v 2

]

=⇒[v 1 ∗,v 2 ∗] (21.5)

Thenorm|v|ofavectorvisdefinedasthesquare-rootoftheinnerproductofthe
columnvectorwithitscorrespondingrowvector

|v|^2 =[v 1 ∗,v∗ 2 ]·

[

v 1

v 2

]

=

∑D

i=1

vi∗vi (21.6)

AmatrixM isaD×Dsquarearrayofcomplexnumbers

M=

[

m 11 m 12

m 21 m 22

]

(21.7)

whichtransformscolumnvectorsvintocolumnvectorsv′accordingtotherule
[
v′ 1
v′ 2

]

=

[

m 11 m 12

m 21 m 22

][

v 1

v 2

]

=

[

m 11 v 1 +m 12 v 2

m 21 v 1 +m 22 v 2

]

(21.8)