Transformation of vector into a symmetric matrix !!!

Let us consider a transformation $T$, which converts a vector into a symmetric matrix, e.g.
Xh=[x1,x2,x3]T

I want to have this type of decomposition:

Here, $Q\in \mathbb{R}^{m\times \frac{q(q+1)}{2}}$, $\bar{Q}\in \mathbb{R}^{p \times q}$, $T(X_h)\in \mathbb{R}^{q \times q}$ and $X_h \in \mathbb{R}^{q(q+1)/2}$.

For example:

Consider a known constant matrix $Q \in \mathrm{R}^{10 \times 3}$. A vector  $X_h = [x_{11}, x_{12}, x_{22}]^T \in \mathrm{R}^3$.

Decomposition must be like,

where, $T(QX_h)\in \mathrm{R}^{4 \times 4}$, $\bar{Q}\in \mathrm{R}^{4 \times 2}$ and $T(X_h)\in \mathrm{R}^{2\times 2}$

Given $Q$, I want to find $\bar{Q}$ such that the equation holds for all $X_h$.

Is such decomposition possible?
 
 

Edited February 2, 20223 yr by Shivan
Latex editng