$\left\| A \right\| \le \varepsilon \Rightarrow \left\| {\mathop A\limits^{\_\_} } \right\| \le \varepsilon$ Suppose


*

*$A \in {C^{n \times n}}$

*$\left\| A \right\| \le \varepsilon$ such that $\left\| . \right\|$ is matrix norm subordinate to the euclidean vector norm.
Is this true that $\left\| {\mathop A\limits^{\_\_} } \right\| \le \varepsilon $ and $\left\| {{A^*}} \right\| \le \varepsilon $?

Note that 
$$
\|A\| = \sqrt{\lambda_{max}(A^*A)} = \max_{\|x\| = 1} \|Ax\|
$$
In order to show that $\|A^*\| = \|A\|$, it suffices to note that $AA^*$ and $A^*A$ have the same non-zero eigenvalues.
In order to show that $\|\overline A\| = \|A\|$, it suffices to note that $\|\overline A x\| = \|A \overline x\|$.
