Suppose for a matrix $ A \in \mathbb{R}^n$, we have $ \ ||A||_2 \leq ||A^TA||^{1/2}$, where $||.||$ is a norm on $\mathbb{R}^n$ associated to matrix norm on $\mathbb{R}^{n \times n}$ and $||.||_2$ is a matrix $2-$ norm.
Show that $ \kappa_2(A) \leq [\kappa_1(A) \kappa_{\infty} (A)]^{1/2}$, where $||.||_{\infty}$ matrix $\infty-$norm
and $\kappa$ denote the condition number .
Answer:
Since $||A||_2 \leq \sqrt{||A^T||||A||}$, we have
$\kappa_2(A) \leq \sqrt{\kappa_2(A) \kappa_{2}(A)}$,
Now if $ \kappa_2(A) \leq \kappa_1(A)$ and $\kappa_2(A) \leq \kappa_{\infty}(A)$, then we are done. But is it possible?