Lösungen

Remove["Global`*"]

1

B = {{1, 2}, {3, 4}};   B // MatrixForm

Inverse[B] // MatrixForm

mD = {{1, 0}, {0, 2}};   mD // MatrixForm

a

A = B . mD . Inverse[B];
A // MatrixForm

b

Eigenvalues[A]

λ1 = Eigenvalues[A][[1]]

λ2 = Eigenvalues[A][[2]]

c

Eigenvectors[A]

x1 = Eigenvectors[A][[1]]

x2 = Eigenvectors[A][[2]]

Eigensystem[A]

d

v = {8, -6}

solv = 
      Solve[v == α x1 +  β x2, {α, β}] // 
        Flatten

α1 = α /. solv

β1 = β /. solv

e

A . v

f

α1 λ1 x1 + β1 λ2 x2

2

Remove["Global`*"]

B = {{1, 2, 3}, {2, 3, 4}, {3, 4, 4}};   B // MatrixForm

Inverse[B] // MatrixForm

mD = {{1, 0, 0}, {0, 2, 0}, {0, 0, 3}};  mD // MatrixForm

A = B . mD . Inverse[B];
A // MatrixForm

Eigensystem[A]

Eigenvalues[A]

λ1 = Eigenvalues[A][[1]]

λ2 = Eigenvalues[A][[2]]

λ3 = Eigenvalues[A][[3]]

Eigenvectors[A]

x1 = Eigenvectors[A][[1]]

x2 = Eigenvectors[A][[2]]

x3 = Eigenvectors[A][[3]]

v = {8, -6, 14}

solv = 
      Solve[v == α x1 +  β x2 +  γ x3, {α, β, γ}] // Flatten

α1 = α /. solv

β1 = β /. solv

γ1 = γ /. solv

A . v

α1 λ1 x1 + β1 λ2 x2 + γ1 λ3 x3

3

Selbststudium

4

Remove["Global`*"]

a Abbidlung einer Kurve

v[t_] := {Cos[t] + t, Log[t + 1] + Sin[t]^2}

ParametricPlot[v[t], {t, 0, 7}];

b

A = {{2, 1}, {1, 3}};
Eigensystem[A]

c

A . v[t]

ParametricPlot[A . v[t], {t, 0, 7}];

d   Selbststudium