Let $A \in \mathbb{R}^{d \times d}$ be a stable matrix, i.e. $\rho(A)<1$, where $\rho(\cdot)$ is the spectral radius. For $t=0,1,\ldots,T,\ldots,$ we define $$ \Gamma_t \triangleq \sum_{s=0}^{t} A^s (A^s)^\top. $$
I am wondering if it's possible to uniquely recover $A$ (up to some permutation and sign flips of columns) if we have access to all the matrices $\Gamma_0, \Gamma_1,\ldots,\Gamma_T,\ldots$.
For simplicity I chose $A$ to be a symmetric matrix, i.e. $A=A^\top$. In this case, we can see that its possible to only recover $A^2$ uniquely and recovering $A$ seems hard as there can be many possible solutions.
So my question is if people studied Gramians in the context of recovering the matrices in the literature or Gramians alone do not suffice as we saw in the case of symmetric matrices above.