Lösungen

Aufgabe 1

A = Transpose[{{8, 7}, {2, 1}}]; MatrixForm[A]

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

A . v

Inverse[A] // MatrixForm

% // N

Inverse[A] . v // MatrixForm

% // N

Aufgabe 2

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

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

Eigenvalues[B]

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]]

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

Aufgabe 3

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

Aufgabe 4

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

Eigenvalues[d]

ev4 = Eigenvectors[d];  Transpose[ev4]  // MatrixForm

Inverse[d]  // MatrixForm

Transpose[d]  // MatrixForm

x . (d - Inverse[d]) == Transpose[d]-Inverse[
          d]

x . (d + Inverse[d]) == Transpose[d] + Inverse[d]

Aufgabe 5

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

Aufgabe 6

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

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

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

dy[φ_] := Inverse[tzy] . dz[φ] . tzy;  
    dy[φ] // MatrixForm

dx[φ_] := Inverse[tzx] . dz[φ] . tzx;  
    dx[φ] // MatrixForm

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

res61 = dz[Pi/4] . v  ; res61 // MatrixForm

res61   // N // MatrixForm

dx[Pi/4] . dz[Pi/4] . v // MatrixForm

dx[Pi/4] . dz[Pi/4] . v // N // MatrixForm

dy[Pi/6] . dx[Pi/4] . dz[Pi/4] . v // MatrixForm

dy[Pi/6] . dx[Pi/4] . dz[Pi/4] . v // N // MatrixForm