(Old-Question) Given a $n\times n$ symmetric matrix $X$, I would like to factor it using a vector $c$ of size $n \times 1$ such that: $\sum_{i,j} [X_{ij} \cdot c_i\cdot c_j]$ is minimum. How can I find the optimal vector $c^*$?
Constraint: The entries in the vector $c$ should sum to $1$.
Also, feel free to make notes about any optimality conditions/ assumptions that might go around this problem.
(New-Edited Question)
w.r.t the above question, this is an updated problem: If for any 3 distinct indices i,j,k if the motive is to preserve the ordering between $X_{ij}$, $X_{jk}$ , $X_{ki}$ after the approximation with the vector c, defined in the old question above, what would be a suitable loss function containing X and c?
ex: If $X_{12}$ > $X_{23}$ < $X_{31}$ for a chosen i=1,j=2,k=3 then I would like to have $c_1.c_2$ > $c_2.c_3$ < $c_3.c_1$ after the approximation. What is a suitable loss function?