Reshaping a matrix and rearranging the elements lexicographically into a vector

Question

Let us say that I have a $3 \times 3$ matrix $\bf X$, that has to be reshaped into a vector and rearranged as follows:

$$ {\bf v} ({\bf X}) = \begin{bmatrix} x_{22} & (x_{23}+x_{32}) & (x_{21}+x_{12}) & x_{33} & (x_{13}+x_{31}) & x_{11} \end{bmatrix} $$

where the indices indicate the position of the elements in the matrix $\bf X$. How to do this rearrangement automatically for any size of matrix $\bf X$?

The ${\bf v} ({\bf X})$ represents the coefficients of monomials of a vector $y = \begin{bmatrix} a_1 & a_2 & b\end{bmatrix}^T$ of degree 2 i.e. the elements of ${\bf v} ({\bf X})$ are coefficients of $z = \begin{bmatrix} a_1^2 & a_1a_2 & a_1b & a_2^2 & a_2b & b^2 \end{bmatrix}^T$. So, the matrix ${\bf X}$ contains coefficients of z in a fashion that, the diagonal elements are monomials without cross product and the off-diagonal elements are monomials with two variable.

$$ {\bf X} = \operatorname{coeff} \left( \begin{bmatrix} b^2 & \frac{a_1b}{2} & \frac{a_2b}{2} // \frac{a_1b}{2} & a_1^2 & \frac{a_1a_2}{2} // \frac{a_2b}{2} & \frac{a_1a_2}{2} & a_2^2\end{bmatrix} \right).$$

For $y = \begin{bmatrix} a_1 & a_2 & a_3 & b\end{bmatrix}^T$ and degree 2, $z = \begin{bmatrix} a_1^2 & a_1a_2 & a_1a_3 & a_1b & a_2^2 & a_2a_3 & a_2b & a_3^2 & a_3b & b^2 \end{bmatrix}^T$ and the matrix ${\bf X}$ will be of size $4 \times 4$.

Answer 1

$${\bf X} \mapsto \left( {\bf P} \operatorname{vech} ({\bf X}) \right)^\top$$

where $\operatorname{vech} : {\Bbb S}_n (\Bbb R) \to {\Bbb R}^m$, where ${\Bbb S}_n (\Bbb R)$ is the set of $n \times n$ real symmetric matrixes and $m := \binom{n+1}{2}$, is the half-vectorization operator and $\bf P$ is a $m \times m$ permutation matrix.