I have a very general question, hopefully not too general. Assume that we have real numbers $a_{ij}, b_{ij}$ $(1 \leq i, \: j \leq n)$ such that $-1 \leq a_{ij}, b_{ij} \leq 1$ for all $i,j,$ for instance. I would like to get an inequality of the following type: $$ \sum_{i=1}^n \left| \sum_{j=1}^n a_{ij} b_{ji}\right| \leq \sum_{i=1}^n \left| \sum_{j=1}^n a_{ij} \right|.$$
Anybody is aware of a known inequality which has the same structure? What can be a good approach to say anything about $a_{ij}$ and $b_{ij}$?