Does the expression $\sum_{i=1}^n \sum_{j=1}^n a_{ij}b_{ij} $ have a name? Say, $A=(a_{ij})$ and $B=(b_{ij})$ are two $n$ by $n$ matrices. Does the expression $$\sum_{i=1}^n \sum_{j=1}^n a_{ij}b_{ij} $$ have a name? Is there a neat way to write it in terms of $A$ and $B$? 
I encountered the expression in the context of optimization. Is there any other context it comes up?

Can't comment yet... 
Supposing your matrices have real entries, this is the Frobenius inner product https://en.wikipedia.org/wiki/Frobenius_inner_product.
