Set of Bauer Fike

The Bauer-Fike theorem (after Friedrich Ludwig Bauer and Charles Theodore Fike , 1960) is a sentence from the numerical mathematics . It provides an estimate of the change in the eigenvalues ​​of a matrix due to disturbances.

Let be a submultiplicative matrix norm , a diagonalizable matrix with the eigenvalues and a perturbation of . Then every eigenvalue in the spectrum has at most the following distance from the spectrum of : ${\displaystyle \|\cdot \|}$ ${\displaystyle A\in \mathbb {C} ^{n\times n}}$ ${\displaystyle \lambda _{i}}$ ${\displaystyle \delta A\in \mathbb {C} ^{n\times n}}$ ${\displaystyle A}$ ${\displaystyle \lambda }$ ${\displaystyle A+\delta A}$ ${\displaystyle A}$

${\displaystyle \min _{i}|\lambda -\lambda _{i}|\,\leq \,\|S^{-1}\delta AS\|\,\leq \,\kappa (S)\,\|\delta A\|}$

with the condition number and a matrix that has the eigenvectors of as columns, d. H. . ${\displaystyle \kappa (S)=\|S\|\|S^{-1}\|}$ ${\displaystyle S=(e_{1},\dotsc ,e_{n})}$ ${\displaystyle A}$ ${\displaystyle S^{-1}AS=\mathrm {diag} (\lambda _{1},\dots ,\lambda _{n})}$

literature

• Robert Plato: Numerical Mathematics compact. Basic knowledge for study and practice. 3. Edition. Vieweg, Braunschweig et al. 2006, ISBN 978-3-8348-0277-4 , section 12.2.1, p. 312f.