I have $2$ sets of $3D-\text{Pointclouds}$ $P_1$ and $P_2$ (shown in Red and Blue respectively) that lie on a common plane $(\text{Fig.} 1).$ The orientation of the plane (with respect to origin) is known and is defined by a transformation matrix (Rotation and Translation) say $\mathbf{T}_0 = [\mathbf{R}_0| \mathbf{t}_0]$. Using the local coordinates of the points on the plane and $\mathbf{T}_0$, I can compute the global coordinates of each point in the pointcloud with respect to origin.
By connecting the elements from $P_2$ with its corresponding element in $P_1$ I obtain a vector field $\mathbf{V}$ which when projected in the $xy-$Plane appears as illustrated in the $\text{Fig.2}.$
[EDIT]
Since I don't know the scalar function that describes the vectorfield but only the coordinates of each vector in the $xy$-plane, how can one split $\mathbf{V}$ into $3$ matrices: a diagonal matrix, a symmetric matrix (with zero trace) and an anti-symmetric matrix as stated in this answer, which says
I can describe my vector field as $$ \mathbf{V} = \begin{pmatrix} \mathbf{v}_{11} & \mathbf{v}_{12} & \dots & \mathbf{v}_{1N}\\ \mathbf{v}_{21} & \mathbf{v}_{22} & \dots & \mathbf{v}_{2N}\\ \vdots & \vdots & \ddots & \vdots \\ \mathbf{v}_{M1} & \mathbf{v}_{12} & \dots & \mathbf{v}_{MN}\\ \end{pmatrix} $$
where $\mathbf{v}_{ij}$ is the vector from $i^\mathrm{th}$ row $(i \in 1,\dots,M)$ and $j^\mathrm{th}$ column $(j \in 1,\dots,N)$ whose $x$ and $y$ components are $u$ and $v$ respectively.
How do I proceed to decompose $\mathbf{V}$?
