I have a variables $\gamma\in \mathbb R^n$ and $W\in \mathbb R^{d\times n}$, and a fixed projection matrix $\Pi\in \mathbb R^{d\times d}$. I want to find $\gamma$ such that $W^T\Pi W\gamma=\beta \mathbf{1}$ for some $\beta\in \mathbb R$. The problem, is I would like $\gamma_i$ to be denoted as a function of $w_i^T\Pi w_j$, where $w_k$ denotes the $k$th column of $W$.
Attempt: So I could use the pseudoinverse and pick any $\gamma$ such that $(W^T\Pi W)^+W^T\Pi W\gamma=\beta (W^T\Pi W)^+\mathbf{1}$, but that's not very helpful because I don't know much about $W$, only the inner products of their columns under $\Pi$.
For this reason, I wanted to find a concise and explicit expression for $\gamma_i$. Here's an example. Let $\delta_{ij}=w_i^T\Pi w_j$. Then for $n=2$, we have $$\gamma_1\delta_{11}+\gamma_2\delta_{12}=\beta$$ and $$\gamma_1\delta_{21}+\gamma_2\delta_{22}=\beta.$$ We have $\gamma_1=\frac{\beta-\gamma_2\delta_{12}}{\delta_{11}},$ and \begin{align} \frac{\beta-\gamma_2\delta_{12}}{\delta_{11}}\delta_{21}+\gamma_2\delta_{22}&=\beta\\ \gamma_2(\delta_{22}-\delta_{21}\delta_{12}/\delta_{11})&=\beta(1-1/\delta_{11})\\ \gamma_2&=\frac{\beta(1-1/\delta_{11})}{\delta_{22}-\delta_{21}\delta_{12}/\delta_{11}}. \end{align} Substituting $\gamma_2$, we get an expression for $\gamma_1$ as well. However, this becomes increasingly complex as $n$ grows.
I'm wondering if there's a nice expression for each $\gamma_i$?