Does there exist a sequence of matrices $A_i$ such that $$\sum^\infty_{i=0}A_iA^i=A^T$$
i tried inputing $A= 0, I$ but these don't give any substantial information except $$\sum^\infty_{i=0}A_i=I$$ if we take the classical example of a nilpotent matrix $\left[\begin{matrix} 0 & 1 \\ 0 & 0 \\ \end{matrix}\right]$ $$A_1 \left[\begin{matrix} 0 & 1 \\ 0 & 0 \\ \end{matrix}\right] = \left[\begin{matrix} 0 & 0 \\ 1 & 0 \\ \end{matrix}\right]$$ I don't know how to progress from here, any help?
That is impossible in the last case of $A$ you consider. Just multiply both sides by $A$ on the right. The LHS becomes 0, and the RHS becomes non-zero.