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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).

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  • $\begingroup$ I've taken to calling it "A sandwiched with B" as in "Salami sandwiched with Rye"(?) $\endgroup$ Commented Sep 20, 2021 at 15:47

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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

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

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  • $\begingroup$ I found this, but wikipedia suggests that $B$ should be invertible... $\endgroup$ Commented Jun 13, 2018 at 0:34
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    $\begingroup$ @AlecJacobson If $B$ is not invertible, then this doesn't define an equivalence relation, and the output matrix has little to do with the original one. $\endgroup$ Commented Jun 13, 2018 at 0:36
  • $\begingroup$ What's the "original one"? A ? Meanwhile, I'm not trying to define an equivalence relation. $\endgroup$ Commented Jun 13, 2018 at 0:40
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    $\begingroup$ Example of Context: This type of operation involving a non-square $B$ shows up, for example, in linearly constrained quadratic optimization. If I'm minimizing $x^T A x$ subject to a linear constraint $x = B y$ then I can minimize $y^T B^T A B y$ instead. In this way the matrix $\tilde{A} = B^T A B$ is indeed related to $A$ in the sense that it represents the quadratic form $x^T A x$ restricted to the subspace spanned by columns of $B$. $\endgroup$ Commented Jun 13, 2018 at 0:50
  • $\begingroup$ @AlecJacobson Yes the original one is $A$. I think that you could then give more information about what your purposes are, because it does not seem very natural in itself to allow non-invertible $B$. $\endgroup$ Commented Jun 13, 2018 at 0:50
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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 $$"\text{$A$ and $B^TAB$ are weakly congruent}".$$

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  • $\begingroup$ Is this a standard term? Does it still make sense if $B$ is rectangular, thus $A$ and $B^TAB$ are not the same size? $\endgroup$ Commented Jun 13, 2018 at 13:15
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    $\begingroup$ @AlecJacobson What is standard is to add the adjective weak when you extend a notion by relaxing some constraints. It absolutely makes sense even if the relation is not limited to matrices of the same size. $\endgroup$ Commented Jun 13, 2018 at 14:09
  • $\begingroup$ I don't understand the downvotes though. $\endgroup$ Commented Jun 13, 2018 at 18:49

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