Finding the invariant of a linear transformation

Question

Consider a linear transformation, characterized by a square matrix $A_{n\times{n}}$. We want to find the general second order "length", assumed to be in a quadratic shape $$L=X^TGX,$$ such that this "length" is equal for the original space vectors ($X$), and the mapped vectors ($Y=AX$). Specifically, it would be nice if an explicit relation could be suggested for $G$ in terms of $A$ or its derivatives (such as its eigenvectors, Eigen-decomposition matrices, ...). Any help would be appreciated and thanks in advance.