Lösungen / Solutions

1

Matrixkomposition nach geometr. Vorgaben

d = {{1, 0}, {0, -1}}; 
    xM = {{2, 1}, {-1, 2}}; {MatrixForm[d], MatrixForm[xM]}

m = xM . d . Inverse[xM]; m // MatrixForm

N[%]

d = {{1, 0}, {0, -1}}; 
    xM1 = {{2, 1}, {1, 2}}; {MatrixForm[d], MatrixForm[xM1]}

m1 = xM1 . d . Inverse[xM1]; m1 // MatrixForm

N[%]

m1 . m // MatrixForm

N[%]

m . {1, 1} // MatrixForm

N[%]

2

Matrixgleichung

d = {{1, 0}, {0, -1}}; 
    xM = {{2, 1}, {-1, 2}}; {MatrixForm[d], MatrixForm[xM]}

{MatrixForm[d^2], MatrixForm[d . d]}

x = {{x1, x2}, {x3, x4}}; x // MatrixForm

x . xM == xM . (d^2 - d)

x == xM . (d^2 - d) . Inverse[xM]

MatrixForm[xM . (d^2 - d) . Inverse[xM]]

N[%]

Det[xM . (d^2 - d) . Inverse[xM]]

3

Caley-Hamilton

Remove[a, b, c, d, m]

DiagonalMatrix[{a, b, c, d}]

IdentityMatrix[3]

m[u_] := {{0, 1, 1}, {1, 0, 1}, {0, 1, u}}; 
    m[u] // MatrixForm

p[λ_] := Det[m[u] - λ IdentityMatrix[3]];  
    p[λ]

Solve[
      Det[m[u] - λ IdentityMatrix[3]] == 
        0, {λ}]

N[%]

Det[m[0] - λ IdentityMatrix[3]]  IdentityMatrix[
        3]

Solve[
      Det[m[0] - λ IdentityMatrix[3]] == 
        0, {λ}]

N[%]

Det[m[0] - λ IdentityMatrix[3]]  IdentityMatrix[
          3] /. λ -> -1

IdentityMatrix[3] + 2 m[0] - m[0] . m[0] . m[0]

4

EW,EV finden

u1 = {1, 1, 0}; u2 = {1, -1, 0}; x1 = {u1, u2, Cross[u1, u2]}; 
    h = x1 . DiagonalMatrix[{1, 2, 3}] . 
        Inverse[x1];
h // MatrixForm

N[%]

k = {{-1, 2, 3}, {2, 0, 4}, {3, 4, 0}}; k // MatrixForm

Eigenvalues[h]

Solve[
      Det[h - λ IdentityMatrix[3]] == 
        0, {λ}]

Eigenvectors[h]