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Is there a term for the operation $A B A^T$? In colloquial terms, I might call this a "sandwich" of a matrix between another matrix and the transpose of that other matrix.

How about for the special case where $B$ (as well as $A B A^T$) is symmetric? This shows up often in Kalman filtering.

I would like to look up some properties of this kind of operation but I'm not sure what to search for.

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  • $\begingroup$ If $B$ is the matrix of a bilinear form then this is just a change of basis. $\endgroup$ Sep 29, 2021 at 3:24
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    $\begingroup$ The term is "similarity transformation". A and B are called similar if $B=XAX^T$. When we have a quadratic form represented by A in the standard basis, then B represents the same form in a new basis. $\endgroup$ Sep 29, 2021 at 4:07
  • $\begingroup$ @AlexandreEremenko I thought that a similarity required $X$ to be invertible? Otherwise this isn't an equivalence relation. $\endgroup$
    – Yemon Choi
    Sep 29, 2021 at 20:21
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    $\begingroup$ @Yemon Choi: you are right. If $X$ is not invertible, I do not know a name for this. $\endgroup$ Sep 30, 2021 at 4:26

2 Answers 2

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If $B=XAX^T$ classically one says that A and B are congruent.

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If you rewrite to $\boldsymbol{X}\boldsymbol{A}\boldsymbol{X}^T$ there is an answer here

Names for the product will depend on the properties of $\boldsymbol{X}$ and $\boldsymbol{A}$

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