Let $A$ a matrix having a set of eigenvectors $\{v_1,\ldots,v_n\}$ linearly independent with $\{\lambda_1,\ldots,\lambda_n\}$ eigenvalues associated. Let $\lambda$ eigenvalue of the perturbed matrix $A+ \delta A$. Prove that $\lambda$ satisfies the following inequality.
$$|\lambda - \lambda_i| \leq K(P) \|\delta A\| \text{ for } i=1,\ldots,n$$
where $K(P):= \|P^{-1}\|\cdot \|P\| $ the condition number of a matrix $P$, and $\|\delta A\| $ is the norm of the perturbation of $A$.