QMGreensite_merged

21.1. REVIEWOFVECTORSANDMATRICES 309

whereHisanHermitianmatrix.
TheEigenvalueEquationforamatrixM istherelationMu=λu,or
[
m 11 m 12
m 21 m 22

][

u 1

u 2

]

= λ

[

u 1

u 2

]

(21.28)

Incomponents,theeigenvalueequationreads

∑D

j=1

mijuj=λui (21.29)

Thesolutionsofthisequationareasetofnormalizedeigenvectorsu(n)andcorre-
spondingeigenvaluesλ

{u(n),λn} =

{[

u( 1 n)

u( 2 n)

]

λ(n)

}

(21.30)

wherethesuperscriptn= 1 , 2 ,..,Dlabelsthedifferent, linearlyindependentsolu-
tions. Iftherearek linearlyindependentsolutionswiththesameeigenvalue,then
thateigenvalueissaidtobek-folddegenerate.^1 Theeigenvalueequationcanalsobe
written

(M−λI)u= 0 (21.31)

• ToFind theEigenvalues: TakethedeterminantofM−λI andsetitto
  zero,i.e.

det(M−λI)=det

[

m 11 −λ m 12

m 21 m 22 −λ

]

= 0 (21.32)

IngeneralthisisaD-thorderpolynomialinλ,withDsolutionsλ 1 ,λ 2 ,...,λD,which
areingeneralcomplex. Ifkeigenvaluesarethesame,thereisak-folddegeneracy.

• ToFindtheEigenvectors: Tofindtheeigenvectoru(n)correspondingtoa
  giveneigenvalueλn,solvethesimultaneoussetofequations

m 11 u( 1 n)+m 12 u( 2 n) = λnu( 1 n)

m 21 u( 2 n)+m 22 u( 2 n) = λnu( 2 n) (21.33)

(^1) Ofcourse,ifD=2,thenaneigenvaluecanbenomorethan2-folddegenerate.