Matrizen und Eigensysteme

Remove["Global`*"]

Uebung 1

Konstruktion der Matrix

eW1=1;eW2=2;;
d={{eW1,0},{0,eW2}};
a={2,-1}; b={3,2};
m={a,b};
mT=Transpose[m];
m1=Inverse[mT];
r=mT.d.m1

r//MatrixForm

mT//MatrixForm

MatrixPower[r,1]

MatrixPower[r,2]

mT.d.d.m1

d={{eW1,0},{0,eW2}}

e1={1,0}; e2={0,1};

Daten

MatrixRank[r]

NullSpace[r]

CharacteristicPolynomial[r,x]

CharacteristicPolynomial[d,x]

Tr[r]

Tr[d]

Det[r]

Det[d]

RowReduce[r]

Kern, Im

NullSpace[r]

RowReduce[r]

RowReduce[r].{x1,x2}

Inverse Matrix

rI=Inverse[r]

rI//MatrixForm

rI=mT.d.m1

m1.Inverse[d].mT

m1.Inverse[d].mT//MatrixForm

EW, EV

eW=Eigenvalues[r]

eV=Eigenvectors[r]

Eigenvectors[r]//MatrixForm

Eigensystem[r]

Abbildung:

r.{x1,x2}//MatrixForm

r.e1

r.e2

rI.e1

rI.e2

{a x1,x2}//MatrixForm

r.eV[[1]]

r.eV[[2]]

rI.eV[[1]]

rI.eV[[2]]

r.Transpose[{k1 eV[[1]],k2 eV[[2]]}]

rI.Transpose[{k1 eV[[1]],k2 eV[[2]]}]

Uebung 2

Konstruktion der Matrix

eW1=1;eW2=2; eW3=3; 
d={{eW1,0,0},{0,eW2,0},{0,0,eW2}};
a={2,-1,0}; b={3,2,0}; c={0,2,4};
m={a,b,c};
mT=Transpose[m];
m1=Inverse[mT];
r=mT.d.m1

r//MatrixForm

mT//MatrixForm

MatrixPower[r,1]

MatrixPower[r,2]

mT.d.d.m1

e1={1,0,0}; e2={0,1,0}; e3={0,0,1};

Daten

MatrixRank[r]

NullSpace[r]

CharacteristicPolynomial[r,x]

CharacteristicPolynomial[d,x]

Tr[r]

Tr[d]

Det[r]

Det[d]

RowReduce[r]

Kern, Im

NullSpace[r]

RowReduce[r]

RowReduce[r].{x1,x2,x3}

Inverse Matrix

rI=Inverse[r]

rI//MatrixForm

rI=mT.d.m1

m1.Inverse[d].mT

m1.Inverse[d].mT//MatrixForm

EW, EV

eW=Eigenvalues[r]

eV=Eigenvectors[r]

Eigenvectors[r]//MatrixForm

Eigensystem[r]

Abbildung:

r.{x1,x2,x3}//MatrixForm

{a x1,b x2, c x3}//MatrixForm

r.{1,0,0}

r.{0,1,0}

r.{0,0,1}

rI.{1,0,0}

rI.{0,1,0}

rI.{0,0,1}

r.eV[[1]]

r.eV[[2]]

r.eV[[3]]

rI.eV[[1]]

rI.eV[[2]]

rI.eV[[3]]

r.Transpose[{k1 eV[[1]],k2 eV[[2]],k3 eV[[3]]}]

rI.Transpose[{k1 eV[[1]],k2 eV[[2]],k3 eV[[3]]}]

Uebung 3

Konstruktion der Matrix

eW1=1;eW2=2; eW3=0; 
d={{eW1,0,0},{0,eW2,0},{0,0,eW3}};
a={2,-1,0}; b={3,2,0}; c={0,2,4};
m={a,b,c};
mT=Transpose[m];
m1=Inverse[mT];
r=mT.d.m1

r//MatrixForm

mT//MatrixForm

MatrixPower[r,1]

MatrixPower[r,2]

mT.d.d.m1

Daten

MatrixRank[r]

NullSpace[r]

CharacteristicPolynomial[r,x]

CharacteristicPolynomial[d,x]

Tr[r]

Tr[d]

Det[r]

Det[d]

RowReduce[r]

Kern, Im

NullSpace[r]

RowReduce[r]

RowReduce[r].{x1,x2,x3}

Inverse Matrix

rI=Inverse[r]

rI//MatrixForm

rI=mT.d.m1

m1.Inverse[d].mT

m1.Inverse[d].mT//MatrixForm

EW, EV

eW=Eigenvalues[r]

eV=Eigenvectors[r]

Eigenvectors[r]//MatrixForm

Eigensystem[r]

Abbildung:

r.{x1,x2,x3}//MatrixForm

{a x1,b x2, c x3}//MatrixForm

r.{1,0,0}

r.{0,1,0}

r.{0,0,1}

rI.{1,0,0}

rI.{0,1,0}

rI.{0,0,1}

r.eV[[1]]

r.eV[[2]]

r.eV[[3]]

rI.eV[[1]]

rI.eV[[2]]

rI.eV[[3]]

r.Transpose[{k1 eV[[1]],k2 eV[[2]],k3 eV[[3]]}]

rI.Transpose[{k1 eV[[1]],k2 eV[[2]],k3 eV[[3]]}]