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I see expressions like this all the time in technical literature. The only $A$ and $B$ can be any size matrices as long as the expression is legal. I believe that the transposition is usually conjugate for complex vector spaces.

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Ah, congruent matrices... – J. M. Oct 3 '11 at 18:19

If $B$ describes a bilinear form $\langle v, Bw \rangle$, then $A^T B A$ describes the same bilinear form in a different set of coordinates $\langle Av, BAw \rangle = \langle v, A^T B A w \rangle$ (for $A$ invertible). This is completely analogous to the way in which, if $B$ describes a linear transformation $Bv = w$, then $A^{-1} BA$ describes the same linear transformation in a different set of coordinates $BAv = Aw \Leftrightarrow A^{-1} BAv = w$.

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And in complex cases the transposition is conjugated because then the form to be expressed in different coordinates is often Hermitean rather than bilinear. – Henning Makholm Oct 3 '11 at 18:42
If $A$ is not a square matrix, then the interpretation of Qiaochu Yuan does not hold, but can be generalized, namely, $\langle v, B w \rangle$, but $w = A^T \omega$ and $A^T$ maps from $\mathbb{R}^n \mapsto \mathbb{R}^m$. – Sasha Oct 3 '11 at 19:20

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