Lösungen/Solutions/E1S2AlUe/Ex13

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

Remove["Global`"]

m[x_] := MatrixForm[x]

vecLen[u_] := 
        Sqrt[Flatten[u] . Flatten[u]];
vecLen[{{u1}, {u2}, {u3}}]

Aufgabe 1

a

mA = {{1, 3}, {3, 1}};  m[mA]

ew = Eigenvalues[mA]

b

ev = Eigenvectors[mA]

ev = Eigenvectors[mA]/Sqrt[2]

c

u = Transpose[ev]

m[u]

d = {{ew[[1]], 0}, {0, ew[[2]]}};  m[d]

Inverse[u] // m

mA1 = u . d . Inverse[u];  mA1 // m

mA == mA1

d

xVec = Transpose[{{x1, x2}}]; m[xVec]

mA . xVec

res = Transpose[xVec] . mA . 
          xVec //Expand

res = res[[1]][[1]]

res == 4

e

{y1, y2} = Inverse[u] . xVec ;  y = {y1, y2}

y // m

Transpose[xVec].u

Transpose[xVec] . u   // m

Transpose[xVec] . u  ==  Transpose[y]

Transpose[y] . d . y

Transpose[y] . d . y //  Expand

Transpose[{{r}, {s}}] . d . {{r}, {s}}

Transpose[{{r}, {s}}] . d . {{r}, {s}} /. {r -> y1,  
            s -> y2}  // Expand  // Flatten

Aufgabe 2

x1 = {1, 3, 2}; x2 = {1, 3, 5}; 
    x3 = {1, 1, 1}; 
X = Transpose[{x1, x2, x3}];  m[X]

Det[X]

λ1 = 1; λ2 = -1; λ3 = 2;

d = {{λ1, 0, 0}, {0, λ2, 0}, {0, 0, λ3}};  
    d // m

A = X . d . Inverse[X];  A // m

A . {{3}, {2}, {1}}  // m