BauerFike theorem
Let $\stackrel{~}{\lambda}$ be a complex number^{} and $\stackrel{~}{u}$ be a vector with ${\parallel \stackrel{~}{u}\parallel}_{p}=1$, and let $r=A\stackrel{~}{u}\stackrel{~}{\lambda}\stackrel{~}{u}$ (usually, $\stackrel{~}{\lambda}$ and $\stackrel{~}{u}$ 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:
$$\lambda \stackrel{~}{\lambda}\le {\kappa}_{p}(X){\parallel r\parallel}_{p}$$ 
see also:

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Wikipedia, http://en.wikipedia.org/wiki/BauerFike_TheoremBauerFike Theorem
Title  BauerFike theorem 

Canonical name  BauerFikeTheorem 
Date of creation  20130322 14:48:31 
Last modified on  20130322 14:48:31 
Owner  Andrea Ambrosio (7332) 
Last modified by  Andrea Ambrosio (7332) 
Numerical id  12 
Author  Andrea Ambrosio (7332) 
Entry type  Theorem 
Classification  msc 15A42 