Problem: This problem comes from a past Ph.D. qualifying exam at my institution.
Let $\newcommand{\R}{\mathbb{R}} \newcommand{\l}{\lambda} A \in \R^{n \times n}$ be of full rank and diagonalizable as $A = XDX^{-1}$ (for $D$ a diagonal matrix, and $X \in \mathrm{GL}_n(\R)$). Let $\l_1,\cdots,\l_n$ denote the eigenvalues of $A$, and let $A':=A+E$ for some $E \in \R^{n\times n}$. Let $\l'$ be an eigenvalue of $A'$.
Prove: $\displaystyle \min_{1 \le i \le n} |\l' -\l_i| \le \kappa_2(X) \cdot \|E\|_2$
Relevant Definitions: Most of these are basic stuff, but for completion's sake:
$\mathrm{rank}(M) := \dim \mathrm{range}(M)$, and is of full rank if $M \in \R^{n \times n}$ and $\mathrm{rank}(M) = n$
$\l$ is an eigenvalue of $M$ with eigenvector $v$ when $Mv=\l v$
The $2$-norm of a matrix is induced from the $\ell^2$ norm: $$ \|M\|_2 := \sup_{0 \ne x \in \R^n} \frac{\|Mx\|_2}{\|x\|_2} \equiv \sup_{\substack{x \in \R^n \\ \|x\|_2 = 1}} \|Mx\|_2$$
The $2$-condition number $\kappa_2(M)$ of a matrix $M$ is defined by $$ \kappa_2(M) := \|M\|_2 \cdot \|M^{-1}\|_2 $$ provided $M$ is invertible. More generally, if $M$ is of full rank one may use $$ \kappa_2(M) := \|M\|_2 \cdot \|M^+\|_2 $$ where $M^+$ is the (Moore-Penrose) pseudoinverse $$ M^+ := (M^\ast M)^{-1} M^\ast $$
I haven't done much work with conditioning numbers and perturbations (since fundamentally that feels like what $E$ is "meant" to be). I do know of some basic results like
$$\kappa_2(M) = \frac{\max \{ \sigma \in \R \mid \sigma \text{ is a singular value of } M \}}{\min \{ \sigma \in \R \mid \sigma \text{ is a singular value of } M \}}$$
but ultimately I'm not seeing how I could retrieve any sort of result.
Nudges/hints in the right direction would be much appreciated; I've been doing fine on many of the problems so far, but this one has me stumped.