I want to write the system of linear equations
\begin{align} y_1 &= \mathbf{x}_1'(\mathbf{a}/3 + \mathbf{b}/3 + \mathbf{c}/3), \\ y_2 &= \mathbf{x}_2'(\mathbf{a}/4 + \mathbf{b}/4 + \mathbf{c}/4 + \mathbf{d}/4), \\ y_3 &=\mathbf{x}_3'(\mathbf{c}/2 + \mathbf{d}/2), \end{align}
as the matrix
$\begin{bmatrix} y_1\\ y_2\\y_3 \end{bmatrix} = \begin{bmatrix} \mathbf{x}_1'(\mathbf{a}/3 + \mathbf{b}/3 + \mathbf{c}/3) \\ \mathbf{x}_2'(\mathbf{a}/4 + \mathbf{b}/4 + \mathbf{c}/4 + \mathbf{d}/4) \\ \mathbf{x}_3'(\mathbf{c}/2 + \mathbf{d}/2) \end{bmatrix},$ where $\mathbf{x}_i$, $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$, and $\mathbf{d}$ are $p\times 1$ vectors. Denote $n=3$.
Right now, I can do this as $\mathbf{y}=\text{diag}(\mathbf{X}\mathbf{L}\mathbf{W}),$ where $\mathbf{y}_{n\times1}=\begin{bmatrix} y_1\\ y_2\\ y_3 \end{bmatrix}$, $\mathbf{X}_{n\times p}=\begin{bmatrix} \mathbf{x}'_1\\ \mathbf{x}_2'\\ \mathbf{x}_3' \end{bmatrix}$, $\mathbf{L}_{p\times q}=\begin{bmatrix} \mathbf{a} & \mathbf{b} & \mathbf{c} & \mathbf{d} \end{bmatrix}$, and $\mathbf{W}_{4\times n}=\begin{bmatrix} 1/3 & 1/4 & 0\\ 1/3 & 1/4 & 0\\ 1/3 & 1/4 & 1/2\\ 0 & 1/4 & 1/2 \end{bmatrix}$.
However, the computation of $\mathbf{XLW}$ is incredibly intensive as $n\to\infty $. Right now, I set $n=3$ for ease of demonstration, but my $n$ is likely 500+). It seems computationally inefficient to compute an $n\times n$ matrix $\mathbf{XLW}$ just to use its diagonal.
My question: Is there a more computationally efficient way to obtain the desired matrix equation?