Is there a standard name for doing $B^T A B$ when $B$ is not necessarily square?

Question

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).

Answer 1

When $B$ is invertible, this relation is called a matrix congruence.

Answer 2

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

Answer 3

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 $$"\text{$A$ and $B^TAB$ are weakly congruent}".$$