the measurement equation of the $i$-th sample is:
$0=r_i + 2 x_i^T P \dot x_i$, where $r_i\in\mathbb{R}_+, 0 \prec P\in\mathbb{R}^{n\times n}, x_i\in\mathbb{R}^n,\dot x_i \in\mathbb{R}^{n}$.
My goal is to estimate the elements of $P$ out of $M$ samples with the LS Method. Since $P$ is symmetric, there are $m=\frac{n(n+1)}{2}$ unknown elements. Lets assume $M>m$.
I was able to reformulate the measurement equation to
$0 = r_i + \underbrace{2 \text{vec}^T(x_i \dot x_i^T)}_{=:u_i}\text{vec}(P) $,
but this formulation results in a non-symmetric $P$ with
$\text{vec}(\hat P) = - (U^T U)^{-1} U r,$ where $U=[u_1^T,\dots, u_M^T]^T$ and $r = [r_1,\dots,r_M]^T$.
Because $n$ is large, I have problems to implement this problems. I need either a closed formulation of the measurement equation to which the LS method can be applied or an implementation (e.g. pseudo code), which takes care of the symmetry of $P$.
Any idea how to solve this problem for arbitrary dimensions? Grateful for any help.