# Bauer-Fike theorem

Let $\tilde{\lambda}$ be a complex number and $\tilde{u}$ be a vector with $\|\tilde{u}\|_{p}=1$, 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 $A=XDX^{-1}$, with $D$ a diagonal matrix. Then the matrix $A$ has an eigenvalue $\lambda$ which satisfies the inequality:

 $|\lambda-\tilde{\lambda}|\leq\kappa_{p}(X)\|r\|_{p}$