a

c

d

c

charakteristisches Polynom

Eigenwerte und Eigenvektoren im R²

Werner Knobloch, 2012

b

a

charakteristisches Polynom

Eigenvektoren

Eigenvektoren

Eigenvektoren (normiert)

Fix(punkt)geraden

Fixgeraden

Fixgeraden

Die Summe der Eigenwerte ist gleich der Summe der Diagonalelemente
(=Spur), Ihr Produkt ist gleich der Determinante. (s. Lösungsformel)

A = (5.3722343105589685, -1.1048326079748356)
B = Intersect(xAxis, yAxis)
u = Vector(B, A)
C = (-2.5259658458596173, 4.0311918186306634)
v = Vector(B, C)
a: x(A)
b: y(A)
c: x(C)
d: y(C)
D: (a, 0)
E: (0, d)
w = Vector(B, D)
z = Vector(D, A)
e = Vector(B, E)
f = Vector(E, C)
M: {{a, b}, {c, d}}
g(x) = x^(2) - ((a + d) * x) - ((c * b) - (a * d))
N_1, N_2 = Root(g)
λλ_1: x(N_1)
λλ_2: x(N_2)
λ_1: (d + a + sqrt((d + a)^(2) - (4 * ((a * d) - (c * b))))) / 2
λ_2: (d + a - sqrt((d + a)^(2) - (4 * ((a * d) - (c * b))))) / 2
e_{10} = Vector((1, (λλ_1 - a) / b))
e_{20} = Vector((1, (λλ_2 - a) / b))
r = 1.1 (slider from 0 to 5, step 0.1)
h: x^(2) + y^(2) = r^(2)
F = Point(h)
i = Vector(B, F)
G: (M * i)
j = Vector(B, G)
Ortslinie1 = Locus(G, F)
k = Vector(E, C)
l = Vector(B, D)
Liste2 = Sequence({{(r * (sin(((360 * ° / 40) * i)), cos(((360 * ° / 40) * i))))}, Vector[((M * r) * (sin(((360 * ° / 40) * i)), cos(((360 * ° / 40) * i))))]}, i, 0, 39)
Testrosette_1 = true
H = (1.8340545772057832, 2.1056103902332124)
I = (1.5978868042897407, 0.41618517475243216)
J = (-0.03712085435978453, 1.7786248646562868)
Vieleck1, j_1, h_1, i_1 = Polygon(H, I, J)
H' = ApplyMatrix(M, H)
I' = ApplyMatrix(M, I)
J' = ApplyMatrix(M, J)
Vieleck1', j', h', i' = Polygon(H', I', J')
charakteristischesPolynom = true
e_1 = UnitVector(e_{10})
e_2 = UnitVector(e_{20})
Dreiecke = false
Testvektoren = true
Eigenvektoren = true
Fixgeraden = true
m = Line(B, e_1)
n = Line(B, e_2)
Hilfstexte = true
Liste3 = Sequence(Segment[((r * sin(((360 * ° / 40) * i))), (r * cos(((360 * ° / 40) * i)))), (M * ((r * sin(((360 * ° / 40) * i))), (r * cos(((360 * ° / 40) * i)))))], i, 0, 39)
Liste4 = Sequence({((r * sin(((360 * ° / 40) * i))), (r * cos(((360 * ° / 40) * i)))), (M * ((r * sin(((360 * ° / 40) * i))), (r * cos(((360 * ° / 40) * i)))))}, i, 0, 39)
h2: x^(2) + y^(2) = r^(2)
Ortslinie2 = Locus(G, F)
Testrosette_2 = false
Text3: "Problemstellung: \\
\left(  \begin{array}{} a & b \\ c & d \end{array}  \right)    \left(   \begin{array}{} x \\ y\end{array}  \right) =\textcolor{ff00ff} \lambda \cdot  \left(   \begin{array}{} x \\ y\end{array}\right) \\ 
\;\\
\left(  \begin{array}{} " + (LaTeX[a]) + " & " + (LaTeX[c]) + " \\ " + (LaTeX[b]) + " & " + (LaTeX[d]) + " \\ \end{array}  \right)  \left(   \begin{array}{} x \\ y\end{array}  \right) = \textcolor{ff00ff} \lambda \cdot  \left(   \begin{array}{} x \\ y\end{array}  \right)\\ 
\;\\
\;\;a \cdot x + b \cdot y = \textcolor{ff00ff} \lambda \cdot x\\
\;\;c \cdot x + d \cdot y = \textcolor{ff00ff} \lambda \cdot y"
Text5: "Eigenwerte:\\
x= \frac{b}{\lambda-a}\;\cdot y \;    ∧  \; x= \frac{\lambda-d}{c}\cdot  y\\
 ⇒ (\lambda-a)(\lambda-d) -c\cdot b = 0 \\⇒ \lambda^2 -(a+d)\cdot \lambda+(a \cdot d-c \cdot b) = 0\\
⇒ Charakteristisches \;Polynom\; mit\;  \lambda=x :\\
 \;  \;  \;  \;  \;  y =  x^2 -(a+d)\cdot x+(a\cdot d-c\cdot b)\\
 \;  \;  \;  \;  \;  y =  x^2 -(" + (LaTeX[a + d]) + ")\cdot x-(" + (LaTeX[(c * b) - (a * d)]) + ")\\
\lambda_{1/2} = \frac{a + d ± \sqrt{(a+d)^2 - 4(a \cdot d - c\cdot  b)} }{2}\\
\lambda_{1} =" + (LaTeX[λ_1]) + "   \;  \;    \;  \;  \;  \;    \lambda_{2} =" + (LaTeX[λ_2]) + ""
Text6: "M = \left(  \begin{array}{} a & b \\ c & d \\ \end{array} \right)

  =   \left(  \begin{array}{} " + (LaTeX[a]) + " & " + (LaTeX[c]) + " \\ " + (LaTeX[b]) + " & " + (LaTeX[d]) + " \\ \end{array}  \right)\\
M - \lambda E= \left(  \begin{array}{} a -\lambda& b \\ c & d-\lambda \\ \end{array} \right)

  =   \left(  \begin{array}{} " + (LaTeX[a]) + " -\lambda & " + (LaTeX[b]) + "  \\ " + (LaTeX[c]) + "  &" + (LaTeX[d]) + "-\lambda \\ \end{array}  \right)\\
\;\\
Spur(M) = (a + d ) =" + (LaTeX[a + d]) + " \\
\;\\
Det(M) = (a \cdot d - b \cdot c ) =" + (LaTeX[(a * d) - (b * c)]) + "\\
\;\\
Det(M-\lambda E)=  (\lambda-a)(\lambda-d) -c\cdot b = 0 \\⇒ \lambda^2 -(a+d)\cdot \lambda+(a \cdot d-c \cdot b) = 0\\ ⇒ sonst\;nur\; triviale\;  Lösungen"
Text2: "Eigenvektoren:\\
\;\;a \cdot x + b \cdot y = \lambda_1 \cdot x \\
\;\;c \cdot x + d \cdot y = \lambda_1\cdot y \\
\text{⇒ homogenes Gleichungssystem    (x=1 frei wählbar) }\\
\;\;(a -\lambda_1)\cdot x + b \cdot y = 0 \\
\;\;c \cdot x + (d -\lambda_1) \cdot y =0 \\
⇒ y =  \frac{\lambda_1 - a}{b} =  " + (LaTeX[(λ_1 - a) / b]) + "  \\
 ⇒ e_{10} =\left(   \begin{array}{}1 \\ " + (LaTeX[(λ_1 - a) / b]) + "  \end{array}  \right)  \; \; \; entsprechend: \; e_{20} =\left(   \begin{array}{}1 \\  " + (LaTeX[(λ_2 - a) / b]) + "  \end{array}  \right) "