Is there a standard name for doing $B^T A B$ when $B$ is not necessarily square? Given a (generic, rectangular) matrix $B$ and a (square) matrix $A$, is there a name for doing:
$$
B^T A B\ ?
$$
My memory wanted to call this "conjugating $A$ with $B$," but 
according to mathworld this is used to refer to 
$$
B^{-1} A B\ .
$$
(Sometimes $B^T = B^{-1}$, but obviously not usually in the general case).
 A: If $B$ is also square and nonsingular, and $A$ is symmetric then $A$ and $B^TAB$ are said to be congruent. 
In my area, when $A$ and $B$ have integer entries, we say that the quadratic form (with Gram matrix $A$) represents the form with Gram matrix $B^TAB.$ In particular, there has been a good deal of success within the past few years when $B$ is to be rectangular. The names on the original research are Ellenberg and Venkatesh  https://arxiv.org/abs/math/0604232 
This was simplified and improved by Schulze-Pillot. 
https://arxiv.org/abs/0804.2158
A: When $B$ is invertible, this relation is called a matrix congruence.
A: In case of non-invertible matrix $B$, a natural way to call this operation would be a weak congruence. It is not a canonical terminology, but it seems very appropriate.
However make sure that your definition is very clear (especially if $B$ can be rectangular), and use a terminology that emphasizes the fact that this is not an equivalence relation. 
E.g. $$"\text{$B^TAB$ is weakly congruent to $A$}"$$ rather than $$"\text{$A$ and $B^TAB$ are weakly congruent}".$$
