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[PDF] MATHS IMPORTAANT !!

3.MatrixAlgebra

Unitmatrices

zero,i.e.,(I)ij=

Products

(AB)ij=lå k=1A ikBkj

IngeneralAB6=BA.

Transposematrices

Inversematrices

(A1)ij=transposeofcofactorofAij jAj

Determinants

Aj=å

i,j,k,... ijk...A1iA2jA3k...

22matrices

IfA=ab

cd then, j

Aj=adbcAT=ac

bd A 1=1 jAj db ca

Productrules

(AB...N)T=NT...BTAT j

Orthogonalmatrices

matrixQ, Q

1=QT,jQj=1,QTisalsoorthogonal.

Solvingsetsoflinearsimultaneousequations

x=ATb.

Hermitianmatrices

Eigenvaluesandeigenvectors

Theneigenvalues

)=jAIj.IfAisHermitianthentheeigenvalues matrixA.

TrA=å

i i,alsojAj=Õ ii.

IfSisasymmetricmatrix,

U TSU= andS=UUT. correspondingeigenvalue.

Commutators

[A,B]ABBA [A,B]=[B,A] [A,B]y=[By,Ay] [A+B,C]=[A,C]+[B,C] [AB,C]=A[B,C]+[A,C]B [A,[B,C]]+[B,[C,A]]+[C,[A,B]]=0

Hermitianalgebra

b y=(b 1,b

2,...)

MatrixformOperatorformBra-ketform

HermiticitybAc=(Ab)cZ

O=Z (O )h jOji

Eigenvalues,

realAui=(i)uiO i=(i) iOjii=ijii

Orthogonalityuiuj=0Z

i j=0hijji=0(i6=j)

Completenessb=å

iu i(uib) i i Z i ij iihiji

Rayleigh-Ritz

Lowesteigenvalue

0bAbbb0Z

O Z h jOj i h j i 6

Paulispinmatrices

x=01 10 ,y=0i i0 ,z=10 01 xy=iz,yz=ix,zx=iy,xx=yy=zz=I

4.VectorCalculus

Notation

polarcoordinates =(r,,');incaseswithradialsymmetry=(r). areindependentfunctionsofx,y,z.

InCartesiancoordinatesr(`del')i

x+j y+k z2

6 6 6 6 6 6 6 4 x y

z3

7 7 7 7 7 7 7 5 grad =r,divA=rA,curlA=rA

Identities

grad(

1+2)grad1+grad2div(A1+A2)divA1+divA2