Suppose I have vectors $\mathbf{x}_n$ and $\mathbf{y}_n$ for $n \in \{1, \ldots, N\}$ who are all elements of $\mathbb{R}^P$ with $P > N$.
I would like to find a symmetric matrix $\mathbf{A}$ such that $\mathbf{A}\mathbf{x}_n = \mathbf{y}_n \forall n$.
Is there an algorithm that constructs such a matrix, or determines that no such matrix exists for a given set of vectors? Is it possible to find one of rank at most $N$? If not, how do I get the matrix with lowest possible rank satisfying the relations (in the solvable cases)?