Given a matrix $A\in\mathbb{R}^{3\times3}$ and a symmetric, positive definite matrix $B\in\mathbb{R}^{3\times3}$, I am currently having a hard time trying to solve the following equation:
$$ (A\,\text{diag}(x_1, x_1, x_3) - \mathbf{y}\,\mathbf{1}^\top)^\top(A\,\text{diag}(x_1, x_1, x_3) - \mathbf{y}\,\mathbf{1}^\top) = B $$
Where $x_i>0\,\,\,\forall i \in\{1, 2, 3\}$ and $\mathbf{y}\in\mathbb{R}^3$ are the unkowns (6 in total). On the other hand, $\mathbf{1} \in \mathbb{R}^3$ is a vector of ones so that $\mathbf{y}\,\mathbf{1}^\top\in\mathbb{R}^{3\times3}$ and each of the columns of $\mathbf{y}\,\mathbf{1}^\top$ is the vector of unknowns $\mathbf{y}$.
Since $B$ is symmetric, there are 6 constraints on 6 the unkowns, which I believe to be linearly independent. However, I can't figure it out how to proceed to solve this system.
Any help is highly appreciated.