Chapter 12 - Eigenvalues and Eigenvectors

Section 12.3 - Applications of the Diagonalizability of Square Matrices

Example 12.3.1

A={{3,-5,1},{0,2,-2},{-4,7,0}};

(AMinusλI=A-λ IdentityMatrix[3])//MatrixForm

EigenPoly=Det[AMinusλI]

λListExact=Solve[EigenPoly==0,λ]

λListApprox=N[λListExact]

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

AMinusλI/.λListApprox[[1]]//MatrixForm

Det[AMinusλI/.λListApprox[[1]]]

(Eqnsλ1=AMinusλI.X/.λListApprox[[1]])//MatrixForm

Solve[{2.54396 x-5 y+z==0,1.54396 y-2 z==0,-4 x+7 y-0.456043 z==0},{x,y,z}]

AMinusλI/.λListExact[[1]]

Simplify[Det[AMinusλI/.λListExact[[1]]]]

(Eqnsλ1Exact=Simplify[AMinusλI.X/.λListExact[[1]]])//MatrixForm

Solve[Flatten[Eqnsλ1Exact]==0,Flatten[X]]

EVλ1=N[%][[1]]

AMinusλI/.λListExact[[2]]

Simplify[Det[AMinusλI/.λListExact[[2]]]]

(Eqnsλ2Exact=Simplify[AMinusλI.X/.λListExact[[2]]])//MatrixForm

Solve[Flatten[Eqnsλ2Exact]==0,Flatten[X]]

EVλ2=N[%][[1]]

AMinusλI/.λListExact[[3]]

Simplify[Det[AMinusλI/.λListExact[[3]]]]

(Eqnsλ3Exact=Simplify[AMinusλI.X/.λListExact[[3]]])//MatrixForm

Solve[Flatten[Eqnsλ3Exact]==0,Flatten[X]]

EVλ3=N[%][[1]]

(Diag=Chop[DiagonalMatrix[Table[λ/.λListApprox[[k]],{k,1,3}]]])//MatrixForm

(Q=Join[Join[X/.EVλ1/.z->1,X/.EVλ2/.z->1,2],X/.EVλ3/.z->1,2])//MatrixForm

Chop[Q.Diag.Inverse[Q]]//MatrixForm

A//MatrixForm

MatrixPower[A,4]//MatrixForm

Chop[Q.MatrixPower[Diag,4].Inverse[Q]]//MatrixForm

(eDiag=Chop[DiagonalMatrix[Table[(E^λ)/.λListApprox[[k]],{k,1,3}]]])//MatrixForm

Chop[Q.eDiag.Inverse[Q]]//MatrixForm

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

N[MatrixExp[A]]//MatrixForm

Simplify[Eigenvalues[A,Cubics->True]]

EigVals2=N[Eigenvalues[A,Cubics->True]]

(Q2=Transpose[N[Eigenvectors[A,Cubics->True]]])//MatrixForm

(Diag2=Chop[DiagonalMatrix[EigVals2]])//MatrixForm

Chop[Q2.Diag2.Inverse[Q2]]//MatrixForm

Example 12.3.2

A={{-10,11,-6},{-15,16,-10},{-3,3,-2}};

(AMinusλI=A-λ IdentityMatrix[3])//MatrixForm

Det[AMinusλI]

Factor[%]

λList=Eigenvalues[A]

Eigenvectors[A]

Example 12.3.3

A={{2.-I,5,I,0},{3+I,-I,π I,-2},{1,2,3,4},{-2 I,0,-1,I}};

N[Eigenvalues[A]]

(AMinusλI=A-λ IdentityMatrix[4])//MatrixForm

EigenPoly=Collect[Det[AMinusλI],λ]

Factor[%]

λList=Eigenvalues[A]

EigVects=Chop[Eigenvectors[A]]

(Diag=Chop[DiagonalMatrix[λList]])//MatrixForm

Q=Transpose[EigVects];

Chop[Q.Diag.Inverse[Q]]//MatrixForm

N[A]//MatrixForm

Q.MatrixPower[Diag,5].Inverse[Q]//MatrixForm

MatrixPower[A,5]//MatrixForm

(eDiag=Chop[DiagonalMatrix[(E^(λList))]])//MatrixForm

Q.eDiag.Inverse[Q]//MatrixForm

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

N[MatrixExp[A]]//MatrixForm