Lösungen

Aufgabe 1

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

Aufgabe 2

aA = {{0, 1, 0}, {0, 0, 1}, {1, 0, 0}}; aA  // MatrixForm

dz[φ_] = {{Cos[φ], -Sin[φ], 
            0}, {Sin[φ], Cos[φ], 0}, {0, 0, 
            1}};
dz[φ]  // MatrixForm

aA . (dz[φ] - x) . Transpose[aA] == 
      aA . x . aA

Solve[
      aA . (dz[φ] - x) . Transpose[aA] == 
        aA . x . aA, {x}]

Inverse[aA]

(dz[φ] - x) . Transpose[aA] == x . aA

dz[φ] . Transpose[aA] - x . Transpose[aA] == 
      x . aA

dz[φ] . Transpose[aA] == 
      x . aA + x . Transpose[aA]

dz[φ] . Transpose[aA] == 
      x . (aA + Transpose[aA])

Inverse[aA + Transpose[aA]]

% // N

dz[φ] . Transpose[aA] . Inverse[aA + Transpose[aA]] == 
      x

dz[φ] . Transpose[aA] . Inverse[aA + Transpose[aA]] // 
      MatrixForm

% // N // MatrixForm

Aufgabe 3

A = {{4, u}, {1, 3}}; MatrixForm[A]

v = Transpose[{{2, -1}}]; v // MatrixForm

Transpose[A] . v

Flatten[Transpose[A] . v]

Solve[Flatten[Transpose[A] . v] == {7, 9}, {u}]

u = 6

Inverse[A] // MatrixForm

% // N // MatrixForm

w = Transpose[{{2, 6}}]; w // MatrixForm

Inverse[A] . w // MatrixForm

% // N // MatrixForm

Aufgabe 4

B = {{1, 0, 9}, {0, 2, 0}, {0, 8, 3}}; 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]]

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

Eigenvalues[Inverse[B]]

N[%]

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

N[%] // MatrixForm

Aufgabe 5

cc = {{1, 0, 3}, {0, 2, 0}, {0, 8, 9}} ; cc // MatrixForm

ev3 = Eigenvectors[cc]; Transpose[ev3] // MatrixForm

Eigenvalues[cc]

rot[φ_] := {{1, 0, 0}, {0, 
            Cos[φ], -Sin[φ]}, {0, Sin[φ], 
            Cos[φ]}};

rot[φ] // MatrixForm

rot[Pi/2] . cc  // MatrixForm

Eigenvectors[rot[Pi/2] . cc] // Simplify //  Transpose  //  
      MatrixForm

Eigenvectors[N[rot[Pi/2] . cc]] // Simplify //  
        Transpose  //  MatrixForm

Eigenvalues[rot[Pi/2] . cc] // Simplify

Eigenvalues[rot[Pi/2] . cc] // Simplify // N