Existence of eigenvalue in normal matrix such that $|\lambda - \mu| < ε\,$ with $\|x\|_2= 1$ and $\|Ax − \mu x\|_2 < ε$

Question

Let $A ∈M_{n\times n}(\mathbb{C})$ be a normal matrix, $\mu\in\mathbb{C}$ and $\epsilon > 0$. Prove that, if $x\in\mathbb{C}^n$ is a vector such that $\|x\|_2 = 1$ and $\|Ax − \mu x\|_2 < \epsilon$, then $A$ has an eigenvalue $\lambda$ such that $|\lambda - \mu| < \epsilon$.

Answer 1

Hint. As $A$ is normal, it has an orthonormal eigenbasis $\{u_1,\ldots,u_n\}$. Let $Au_i=\lambda_iu_i$ and $x=\sum_{i=1}^nc_iu_i$ where $\sum_{i=1}^n|c_i|^2=1$. You may express $\|(A-\mu I)x\|_2$ in terms of $c_i,\lambda_i$ and $\mu$.