I want to create a series of matrix operations that would allow me to combine the vectors $[x1,x2,1]$, $[y1, y2, 1]$ without using a Hadamard product to produce $[y1, y2 x1, x2] = [y1, y2, 1] \circ [1, x1,x2]$. The vectors can be rearranged in any way, i.e. $[1, x1,x2]$, and use of matrix multiplications, additions, etc... can be used, but the diagonal function or the Hadamard product cannot. Is this possible?
The problem that I want to solve is trying to "simplify" the operations. $$ x = [x_1, x_2, x_3, \dots, x_n]^T \in [0,1] \\ y = 1- x \\ x_p = [1|x]\\ y_p=[y|1]\\ r=x_p \circ y_p = [y_1,y2*x1,...,y_n*x_{n-1},x_n] $$
You can already rewrite $y_p$ as: $$ y_p = \begin{bmatrix} 0 & 1 \\ -I & 1 \end{bmatrix}x_p \\ y_p = T x_p \\ r=x_p \circ T x_p $$
But I feel like you should be able to "simplify" the problem further