Problem 1308 (difficulty: 8/10)

Prove that

\(\displaystyle \max_{1\le j\le p} \sqrt{\sum_{i=1}^q a_{ij}^2} \le \left\Vert\begin{pmatrix} a_{11}&\ldots&a_{1p} \\ \vdots&&\vdots\\ a_{q1}&\ldots&a_{qp} \\ \end{pmatrix}\right\Vert \le\sqrt{\sum_{i=1}^q\sum_{j=1}^pa_{ij}^2}. \)

Give an example when equality does not hold.

Give me another random problem!