Equality in matrix norm equation.

Question

I have proved that for a matrix $A \in M_{m \times n}$ $$\frac{1}{\sqrt n} \|A\|_{\infty} \leq \|A\|_2 \leq \sqrt m \|A\|_{\infty}$$

But wanting to know example of nonzero matrix $A$ where equality occurs.

Will Identity matrix work as an example?

Answer 1

Suppose you want

$$\frac1{\sqrt{n}}\|A\|_\infty = \|A\|_2 = \sqrt{m}\|A\|_\infty$$

Then we have $$\|A\|_\infty=\sqrt{mn}\|A\|_\infty$$ and since $\|A\|_\infty >0$ if $A \neq 0$, we have $\sqrt{mn}=1$, that is $m=n=1$.

When $m=n=1$, the equality holds.

Also, when $m \neq n$, then the equality doesn't hold if $A \ne 0$.