Let $M$ be a symmetric positive definite square matrix. Such a matrix defines an inner product.
Let $A$ be another square matrix. We define $B = A^t M A$.
Under which circumstances do we have $M = B$? In other words, under which conditions is the inner product defined by $M$ invariant under conjugatin by $A$?
A necessary condition is that $A$ is invertible. What else can be said?
$$ A^TMA=M \Leftrightarrow I=M^{-1/2}A^TMAM^{-1/2}=(M^{1/2}AM^{-1/2})^T(M^{1/2}AM^{-1/2}). $$ This means that $M^{1/2}AM^{-1/2}$ is real orthogonal, i.e. $A=M^{-1/2}QM^{1/2}$ for some orthogonal matrix $Q$.