Lösungen/ Solutions

Remove["Global`*"]

mF[x_] := MatrixForm[x]

Aufgabe/ Problème  1

Solve[{
x + y + z + w == 
            1, 
y - z + w == 
            1, 
z - w == 
            1, 
y - w == 1
}, {x, 
          y, z, w}]   // Flatten

Aufgabe/ Problème  2

X =  Transpose[{{1, 3}, {-3, 1}}]; X // MatrixForm

d = Transpose[{{7, 0}, {0, 2}}]; d // MatrixForm

A = X . d . Inverse[X]; A // MatrixForm

% // N

iA = Inverse[A]; iA // MatrixForm

% // N

tA = Transpose[A]; tA // MatrixForm

% // N

Aufgabe/ Problème  3

B = {{1, 0, 9, 0}, {0, 2, 0, 0}, {0, 8, 3, 5}, {0, 0, 0, 4}}; 
    B // MatrixForm

Eigenvalues[B]

ev2 = Eigenvectors[B];  Transpose[ev2] // MatrixForm

b[1] := B; b[n_] := b[n - 1] . b[1]

b[2]  // MatrixForm

B . B  // MatrixForm

B . B . B  // MatrixForm

b[3]  // MatrixForm

h = b[10]; h // MatrixForm

Eigenvalues[b[10]]

Eigenvalues[B]^10

ev10 = Eigenvectors[b[10]];  Transpose[ev10] // MatrixForm

Eigenvalues[Inverse[B]]

% // N

evInv2 = Eigenvectors[Inverse[B]];  
    Transpose[evInv2] // MatrixForm

% // N

Remove["Global`*"]

mF[x_] := MatrixForm[x]

Aufgabe/ Problème  4

rot[φ_] := { {Cos[φ], -Sin[φ]}, {Sin[φ], Cos[φ]}}; 
    rot[φ] // MatrixForm

φ = Pi/6;
mP1 = {{1}, {2}};
mP2 = {{2}, {3}};
mP3 = {{3}, {2}};

rP1 = rot[φ] . mP1

% // N

rP2 = rot[φ] . mP2

% // N

rP3 = rot[φ] . mP3

% // N

Transpose[{Flatten[mP3 - mP1], Flatten[mP3 - mP2]}]

Det[Transpose[{Flatten[mP3 - mP1], Flatten[mP3 - mP2]}]]

Det[Transpose[{Flatten[rP3 - rP1], Flatten[rP3 - rP2]}]]

Aufgabe/ Problème  5

a

Matrix/ Matrice

mD = {{1, 0}, {0, -2}};  mF[mD]

mX = Transpose[{{2, -1}, {3, 2}}];  mF[mX]

A = mX . mD . Inverse[mX];  mF[A]

% // N

Punkte abbilden, Rechnung / Appliquer des points, calcul

v[p_] := Transpose[{p}]

P[1] = {1, 2};
P[2] = {2, -4};
P[3] = {-2, -1};

Table[mF[ v[P[k]]], {k, 1, 3}]

Table[mF[A .  v[P[k]]], {k, 1, 3}]

% // N

B[k_] := Flatten[ A . v[P[k]]];  B[1]

% // N

b

Punkte abbilden, Zeichnung / Appliquer des points, dessin

Show[
        Graphics[
{PointSize[
              0.03], 
Point[P[1]], Point[P[2]], 
            Point[P[3]], 
Line[{P[1], P[2]}], 
            Line[{P[1], P[3]}], 
            Line[{P[3], P[2]}], 
Point[B[1]], Point[B[2]], 
            Point[B[3]], 
Line[{B[1], B[2]}], 
            Line[{B[1], B[3]}], 
            Line[{B[3], 
                B[2]}], 
Line[{{-2, 0}, {2, 0}}], 
            Line[{{0, -2}, {0, 
                  2}}], 
Line[{-2 {2, -1}, 
                2 {2, -1}}], 
Line[{-{3, 2}, {3, 
                  2}}]
}], 
        AspectRatio -> 0.5];

<< Graphics`Arrow`

Show[
        Graphics[
{PointSize[
              0.03], 
Point[P[1]], Point[P[2]], 
            Point[P[3]], 
Line[{P[1], P[2]}], 
            Line[{P[1], P[3]}], 
            Line[{P[3], P[2]}], 
Point[B[1]], Point[B[2]], 
            Point[B[3]], 
Line[{B[1], B[2]}], 
            Line[{B[1], B[3]}], 
            Line[{B[3], 
                B[2]}], 
Line[{{-2, 0}, {2, 0}}], 
            Line[{{0, -2}, {0, 
                  2}}], 
Line[{-2.4 {2, -1}, 
                2.4 {2, -1}}], 
Line[{-1.3 {3, 
                    2}, 1.3 {3, 2}}], 
Arrow[P[1], B[1]], 
            Arrow[P[2], B[2]], Arrow[P[3], B[3]]
}], 
        AspectRatio -> 0.5];

c, d

Eigenvalues[A]

mD = {{1, 0}, {0, -2}};  mF[mD]

Eigenvectors[A]

λ Eigenvectors[A] // Transpose // mF

2 Eigenvectors[A] // Transpose // mF

mX = Transpose[{{2, -1}, {3, 2}}];  mF[mX]

v1 = Transpose[{Eigenvectors[A][[1]]}];  v1 // mF

A . v1 == -2 v1

v2 = Transpose[{Eigenvectors[A][[2]]}];  v2 // mF

A . v2 == 1 v2

Lösungen / Solutions: Eigenvalues, Eigenvectors