Is it possible to write this operation as a (series of) matrix operation(s)

Question

Consider the symmetric matrix $T\in \mathbb{R}^{n\times n}$. For the sake of simplicty I will use the example for $n=2$. I was wondering if we have the full matrix as $$ T = \begin{pmatrix} a & c\\ c& b\end{pmatrix},$$ it is possible to construct this from a vector $t\in\mathbb{R}^{\frac{1}{2}n(n+1)}$, such that $$ t = \begin{pmatrix} a\\b\\c\end{pmatrix}.\quad (n=2)$$ I was thinking that this might be possible by writing it as some kind of strange permutation in the form of $$AtB=T,$$
where $A\in \mathbb{R}^{n\times \frac{1}{2}n(n+1)}$ and $B\in \mathbb{R}^{1\times n}$. However, working this out component wise gives contradicting requirements for $a_{ij}$ ($a_{13}=0$ and $1$), so this seems impossible.

$\textbf{Question:}$ Is there a way to construct $T$ when $t$ is given, which uses only matrix-vector operations?

Answer 1

The rank of a product of matrices is at most as large as the rank of each of the factors. Therefore, the rank of $AtB$ can be at most $1.$

$$ T = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix} t \begin{pmatrix} 1 & 0 \end{pmatrix} + \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix} t \begin{pmatrix} 0 & 1 \end{pmatrix} $$

Answer 2

This is not possible for arbitrary $a$, $b$, and $c$. Notice that \begin{align} \begin{bmatrix} d_{11}&d_{12}&d_{13}\\d_{21}&d_{22}&d_{23}\end{bmatrix}\begin{bmatrix} a\\b\\c\end{bmatrix}\begin{bmatrix} e_1&e_2\end{bmatrix}=&\; \begin{bmatrix} e_1(ad_{11}+bd_{12}+cd_{13})&e_{2}(ad_{11}+bd_{12}+cd_{13})\\e_{1}(ad_{21}+bd_{22}+cd_{23})&e_{2}(ad_{21}+bd_{22}+cd_{23})\end{bmatrix}\\ \equiv&\; \begin{bmatrix} r_{11}&r_{12}\\r_{21}&r_{22}\end{bmatrix} \end{align}

which implies that the resulting matrix always has the feature that $$\frac{r_{11}}{r_{21}}=\frac{r_{12}}{r_{22}}.$$ Hence, for the operation to work it would be necessary that $$\frac{a}{c}=\frac{c}{b}.$$