Suppose I have a square matrix $M$. If $P$ is another square, invertible matrix, then $PMP^{-1}$ is said to be similar to $M$.
Now consider a transformation of the form $AMA^T$, where $A$ can be rectangular. Is there a name for this kind of transformation?
If $A$ is square and orthogonal, then we are back to the previous case because $A^T = A^{-1}$. My question is about the general case, where $A$ is not orthogonal, it need not be invertible, and can even be rectangular.
Hint:
If you do a Singular Value Decomposition of $A$ $$ A = U\Sigma V^T \quad A^T = V\Sigma ^T U^T $$ then you get $$ AMA^T = U\Sigma V^T MV\Sigma ^T U^T $$ and if A is square ...
Or you can do a QR decomposition ...