Is there an intuitive interpretation of $ABA^T$?

Question

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.

Answer 1

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 $BAv = Aw \Leftrightarrow A^{-1} BAv = w$.