$\ell_2$ vs $\ell_{\infty}$ induced norm of a square matrix

Question

I'm wondering if the result of this post $\| A \|_{L^2} \le \| A \|_{\infty}$ for symmetric matrices $A$ applies to square matrices that are not symmetric. Of course, for an asymmetrical square matrice $A$ with $\Vert A \Vert_{\infty} \geq \Vert A \Vert_1$, then the same result still applies, but if $\Vert A \Vert_{\infty} \leq \Vert A \Vert_1$ then it is indeterminate.

After testing with a few random matrices of various sizes, it seems to me that the induced $\ell_{\infty}$ norm is indeed an upper bound of the induced Euclidean norm, but I'm not sure how to prove or disprove this... Any ideas?

Answer 1

This isn't true. E.g. $$ \left\|\pmatrix{0&1\\ 0&1}\right\|_2=\sqrt{2}> 1=\left\|\pmatrix{0&1\\ 0&1}\right\|_\infty. $$