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By Eduard L. Stiefel

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Durch Ausquadrieren ergibt sich n Φ(ξ1 , . . , ξn ) = E 2 ξk yi−k − yi k=1 n = E|yi |2 − n ξk E(yi−k yi ) − k=1 n ξk E(yi yi−k ) + k=1 ξj ξk E(yi−j yi−k ) , j,k=1 wobei lediglich vorausgesetzt werden muß, daß die Varianz E|yi |2 aller Meßwerte endlich ist. Mit den folgenden Vektoren x, b ∈ x = ξk n k=1 , n und der Matrix T ∈ b = E(yi yi−k ) n k=1 , n×n , T = E(yi−j yi−k ) n j,k=1 , kann das Fehlerfunktional auch folgendermaßen geschrieben werden: Φ(x) = E|yi |2 − x∗ b − b∗ x + x∗ T x = E|yi |2 − 2 Re x∗ b + x∗ T x .

36 I Zentrale Grundbegriffe 9. Betrachten Sie f¨ ur Matrizen A ∈ Radius r(A) = sup 0=x∈ Ã n×n den Spektralradius (A) und den numerischen x∗ Ax . x∗ x n Ist einer dieser beiden Radien eine Norm in tr¨ aglichkeit mit der Euklidnorm. Ã n×n ¨ ? Uberpr¨ ufen Sie gegebenenfalls die Ver- 10. Sei · eine Vektornorm und ||| · ||| die induzierte Matrixnorm. (a) Zeigen Sie, daß f¨ ur jede nichtsingul¨are Matrix S ∈ n×n durch x S = Sx eine orige induzierte Matrixnorm ist. Vektornorm definiert wird und |||A|||S = |||SAS −1 ||| die zugeh¨ (b) Sei A ∈ n×n und V −1 AV = J die Jordan-Normalform von A.

Es ist L−1 i = I + li ei und L = I + l1 e1 + . . + ln−1 en−1 . Beweis. Aufgrund der Nulleintr¨age in den Vektoren lj und ei ist e∗i lj = 0 f¨ ur 1 ≤ i ≤ j ≤ n . 5) Daraus folgt zun¨achst die erste Behauptung, denn (I − li e∗i )(I + li e∗i ) = I − li e∗i + li e∗i − li e∗i li e∗i = I − li e∗i li e∗i = I . Die spezielle Form von L ergibt sich induktiv: Dazu nehmen wir an, daß −1 L−1 = I + l1 e∗1 + . . + lk e∗k 1 · · · Lk 48 II Lineare Gleichungssysteme f¨ ur ein k mit 1 ≤ k < n gilt (f¨ ur k = 1 ist dies nach dem ersten Teil des = I + lk+1 e∗k+1 folgt dann Lemmas erf¨ ullt).

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An Introduction to Numerical Mathematics by Eduard L. Stiefel

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