I tried to solve the following equation in matrix ${\bf A} \in \Bbb R^{N \times N}$. There are a total of $M$ known vectors.
$$\textbf{x}^{(i)}=(x_1^{(i)},x_2^{(i)},...,x_N^{(i)})^T \in \Bbb R^N,i=1,2,...,M$$
Let a constant vector $$\textbf{d}=(d^{(1)},d^{(2)},...,d^{(M)})^T \in \Bbb R^M$$
$$\sum_{i=1}^M(\textbf{x}^{(i)}(\textbf{x}^{(i)})^T)((\textbf{x}^{(i)})^T\textbf{A}\textbf{x}^{(i)})+M\textbf{A}+\sum_{i=1}^M(\textbf{x}^{(i)}(\textbf{x}^{(i)})^T)\textbf{d}^{(i)}=\textbf{0}$$