Lösungen/Solutions/F1S2Ue/Ex13

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

Remove["Global`"]

m[x_] := MatrixForm[x]

Aufgabe/ Problème 1

A1 = {{1, 2, 3}, {2, 3, 4}, {3, 4, 12}};  m[A1]

Det[A1]

a

ew1 = Eigenvalues[A1] //  Simplify

ew1 // N   // Chop

b

ev1 = Eigenvectors[A1]

ev1 //  N  // Chop

m[Transpose[ev1]] //  N  // Chop

Aufgabe/ Problème 2

A2 = {{1, 3}, {0, 5}};  m[A2]

a

ew2 = Eigenvalues[A2]

b

ev2 = Eigenvectors[A2]

m[Transpose[ev2]]

Aufgabe/ Problème 3

a

x1 = {4, -2};  x2 = {3, 2};
X = Transpose[{x1, x2}] ;  X // m

λ1 = 1; λ2 = -2;
d = {{λ1, 0}, {0, λ2}};   d // m

A3 = X . d . Inverse[X];  m[A3]

b

imag[p_] :=  A3 . Transpose[{p}]  // m;

imag[{4, -2}]

imag[{3, 2}]

imag[{4, -1}]

imag[{6, 2}]

imag[{1, 4}]

Aufgabe/ Problème 4

a =. ; b =. ; c =. ; d =. ;

A4 = {{a, b}, {c, d}};  m[A4]

v = {{2}, {6}}

A4 . v  // m

a = 8; b = 2; c = 7; d = 1;  m[A4]

A4 . {{2}, {6}}  // m

Inverse[A4] // m

Inverse[A4] . v  // m

Aufgabe/ Problème 5

a

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

ew2 = Eigenvalues[B]

b

Remove[b]

b[1] = B;  b[n_] := b[n - 1] . b[1]

b[2] // m

b[10] // m

ew = Eigenvalues[b[10]]

c

ev1 = Eigenvectors[B];  Transpose[ev1] // MatrixForm

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

d

ewi = Eigenvalues[Inverse[B]]

e

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

d

ewt = Eigenvalues[Transpose[B]]

e

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

Aufgabe 6

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

a

Eigenvalues[c]

b

ev6 = Eigenvectors[c];  Transpose[ev6]  // MatrixForm

Inverse[c]  // MatrixForm

Transpose[c]  // MatrixForm

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

x= beliebig/ quelconque

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

x= E