Chapter 12 - Eigenvalues and Eigenvectors

Section 12.1 - What Are Eigenvalues and Eigenvectors, and Why Do We Need Them?

Example 12.1.1

(A={{5,0,0},{0,-9,0},{0,0,7}})//MatrixForm

(B={{π,0,0},{0,4/11,0},{0,0,-3}})//MatrixForm

A.B//MatrixForm

B.A//MatrixForm

Inverse[A]//MatrixForm

Inverse[B]//MatrixForm

MatrixPower[A,5]//MatrixForm

{{(5^5),0,0},{0,(((−9))^5),0},{0,0,(7^5)}}//MatrixForm

(Diag={{a,0,0},{0,b,0},{0,0,c}})//MatrixForm

MatrixPower[Diag,10]//MatrixForm

(G={{10,0,0},{0,100,0},{0,0,1000}})//MatrixForm

(FourthRootG={{(10^(.25)),0,0},{0,(100^(.25)),0},{0,0,(1000^(.25))}})//MatrixForm

MatrixPower[FourthRootG,4]//MatrixForm

Example 12.1.3

DiagD={{a,0,0},{0,b,0},{0,0,c}};

Q={{d,e,f},{g,h,i},{j,k,l}};

A={{1,2,3},{4,5,6},{7,8,9}};

Q.DiagD//MatrixForm

A.Q//MatrixForm

(QColumn1=Q[[All,1]])//MatrixForm

A.QColumn1//MatrixForm

(QColumn2=Q[[All,2]])//MatrixForm

A.QColumn2//MatrixForm

(QColumn3=Q[[All,3]])//MatrixForm

A.QColumn3//MatrixForm

Example 12.1.4

(A={{12,-2},{-7,-1}})//MatrixForm

(AMinusλI=A-λ IdentityMatrix[2])//MatrixForm

EigenPoly=Det[AMinusλI]

Solve[EigenPoly==0,λ]

λ1=13;λ2=-2;

(X={{x},{y}})//MatrixForm

(AMinusλI/.{λ->λ1})//MatrixForm

AMinusλI.X/.{λ->λ1}//MatrixForm

Solve[{Evaluate[(AMinusλI.X/.{λ->λ1})==0]},{x,y}]

AMinusλI.X/.{λ->λ2}//MatrixForm

Solve[{Evaluate[(AMinusλI.X/.{λ->λ2})==0]},{x,y}]

(Diagλ={{λ1,0},{0,λ2}})//MatrixForm

(Q={{-2,1},{1,7}})//MatrixForm

Q.Diagλ.Inverse[Q]//MatrixForm

A//MatrixForm

A.A.A//MatrixForm

(Q.MatrixPower[Diagλ,3].Inverse[Q])//MatrixForm

N[Sum[MatrixPower[A,n]/n!,{n,0,25}]]//MatrixForm

(eDiagλ={{(E^(λ1)),0},{0,(E^(λ2))}})//MatrixForm

N[Q.eDiagλ.Inverse[Q]]//MatrixForm

N[MatrixExp[A]]//MatrixForm