Let $\mathbf{A}$ be a real $n \times m$ matrix. Let $\mathbf{B}$ be a real $m \times n$ matrix. How to solve the following matrix equation? $$\mathbf{A}\mathbf{B}=\mathbf{B^{t}}\mathbf{A^{t}}$$
Obviously, $\mathbf{B}=\mathbf{A^{t}}$ solves the above equations. How to find all other possible solutions if they exist?