Aufgabe 9 : Berechne die Eigenwerte der (3×3)-Matrix
A=
0 1 0 1 0 1 0 1 0
Zusatz: Berechne die zugeh¨origen Eigenprojektoren Beweis.
f(λ) = det(A−λE)
= det
0 1 0 1 0 1 0 1 0
−
λ 0 0 0 λ 0 0 0 λ
!
= det
−λ 1 0
1 −λ 0
0 1 −λ
= −(λ)∗(−λ)∗(−λ) + 1∗1∗0 + 0∗1∗1−0∗(−λ)−(−λ)∗1∗1
= −λ3+λ+λ
= −λ3+ 2λ
= 0
= λ3−2λ
= λ(λ2−2)
λ1= 0, λ1=√
2, λ3=−√ 2
1
Zusatz:
P1 = (A−λ2E)(A−λ3E) (λ1−λ2)(λ1−λ3)
=
0 1 0 1 0 1 0 1 0
−
√
2 0 0
0 √
2 0
0 0 √
2
!
0 1 0 1 0 1 0 1 0
−
−√
2 0 0
0 −√
2 0
0 0 −√
2
!
(0−√
2)(0−(−p 2)
=
1
2 0 −12 0 12 0
−12 0 12
P2 = (A−λ1E)(A−λ3E) (λ2−λ1)(λ2−λ3)
=
0 1 0 1 0 1 0 1 0
√2 1 0
1 √
2 1
0 1 √
2
√2(√ 2 +√
2)
=
1 4
√1 8
1 4
√1 8
1 2
√1 1 8
4
√1 8
1 4
P3 = (A−λ1E)(A−λ2E) (λ3−λ1)(λ3−λ2)
=
0 1 0 1 0 1 0 1 0
−√
2 1 0
1 −√
2 1
0 1 −√
2
−√ 2(−√
2−√ 2)
=
−14 √1
8 −14
√1 8
1 2
√1 1 8
4
√1 8
1 4
=
−14 √1
8 −14
−√1
8 −12 −√1
8
−14 √18 −14
2
