Lösungen/Solutions/F1S2Ue/Ex11

L=Lektion/Leçon-- Sprechstunde/heures de consultation
W=Work/Arbeit/Travail
A  =Anhang/  Annexe / Appendice

Remove["Global`"]

m[x_] := MatrixForm[x]

Aufgabe/ Problème 1

A = {{1, 1, 1}, {0, 2, 2}, {0, 0, 3}};  m[A]

a

ew = Eigenvalues[A]

b

ev = Eigenvectors[A]

ev1 = Transpose[{ev[[1]]}]

A . ev1 == ew[[1]] ev1

ev2 = Transpose[{ev[[2]]}]

A . ev2 == ew[[2]] ev2

ev3 = Transpose[{ev[[3]]}]

A . ev3 == ew[[3]] ev3

c

v = Transpose[{{-1, 4, 3}}]

Av = A . v; MatrixForm[Av]

MatrixForm[μ1 ev1 + μ2 ev2 + μ3 ev3] == 
      MatrixForm[v]

s1 = 
      Solve[v == μ1 ev1 + μ2 ev2 + μ3 ev3, {μ1, μ2, μ3}] // Flatten

d

p[z_] := Point[{z[[1]], z[[2]], z[[3]]}];

lin[z_] := Line[{{0, 0, 0}, z}]

Show[
        Graphics3D[
{PointSize[
              0.03], 
p[{-1, 4, 3}], p[Flatten[Av]], 
            p[ev[[1]]], p[ev[[2]]], p[ev[[3]]], 
            p[{0, 0, 0}], 
lin[ev[[1]]], 
            lin[ev[[2]]], lin[ev[[3]]], lin[{-1, 4, 3}], 
            lin[Flatten[
                Av]], 
Line[{{-2, 0, 0}, {5, 0, 0}}], 
            Line[{{0, -2, 0}, {0, 5, 0}}], 
            Line[{{0, 0, -2}, {0, 0, 5}}]
}], 
        AspectRatio -> 01];

e

A1 = Inverse[A];  MatrixForm[A1]

ew1 = Eigenvalues[A1]

ew1 = Eigenvectors[A1]

Aufgabe/ Problème 2

X = 
      Inverse[A] . (Inverse[
              Transpose[A]] . (A . A - A + Transpose[A]));  
    m[X]

Aufgabe/ Problème 3

B = {{1, 1, 1}, {0, 1, 1}, {0, 0, 1}}; m[B]

Bi = Inverse[B];  m[Bi]