Fike

Fike

Der Satz von Bauer-Fike (nach Friedrich Ludwig Bauer und C.T. Fike, 1960) ist ein Satz aus der numerischen Mathematik. Er liefert eine Abschätzung über die Veränderung der Eigenwerte einer Matrix auf Grund von Störungen.

Sei  \|\cdot\| eine submultiplikative Matrixnorm,  A \in \mathbb{C}^{n \times n} eine diagonalisierbare Matrix mit den Eigenwerten λi und  \delta A \in \mathbb{C}^{n \times n} eine Störung von A. Dann hat jeder Eigenwert λ im Spektrum von A + δA höchstens den folgenden Abstand zum Spektrum von A:

 \min |\lambda - \lambda_i | \,\le\, \|S^{-1}\,\delta\!A \,S\| \,\le\, \kappa (S)\,\|\delta\!A\|

mit der Konditionszahl \kappa (S)=\|S\|\|S^{-1}\| und S = (e1,...,en) eine Matrix, die die Eigenvektoren von A als Spalten hat, d.h. S^{-1}AS=\mathrm{diag}(\lambda_1,\dots,\lambda_n).

Der Beweis benutzt den Satz von Gerschgorin.


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