Bauer-Fike theorem

Let $\tilde{\lambda }$ be a complex number and $\tilde{u}$ be a vector with ${\parallel \tilde{u}\parallel }_p=1$, and let $r=A\tilde{u}-\tilde{\lambda }\tilde{u}$ (usually, $\tilde{\lambda }$ and $\tilde{u}$ are considered to be approximation of an eigenvalue and of an eigenvector of $A$). Assume $A$ is diagonalizable and $A=XD{X}^{-1}$, with $D$ a diagonal matrix. Then the matrix $A$ has an eigenvalue $\lambda$ which satisfies the inequality:

$$|\lambda -\tilde{\lambda }|\le {\kappa }_p(X){\parallel r\parallel }_p$$

see also:

• Wikipedia, http://en.wikipedia.org/wiki/Bauer-Fike_TheoremBauer-Fike Theorem