How do I show that the given matrix can be decomposed?

Question

Suppose $P\subseteq\mathbb R^n$ is a polyhedron given by $m$ constraints $\langle a_i,x\rangle\leq b_i, i=1,2,...,m$ and let $w_1,w_2,...w_n$ be its vertices. Define $S=(s_{ij})$ by $s_{ij}=b_i-\langle a_i,w_j\rangle$. Now suppose that there is a matrix $Q$ such that $Q=\{(x,y)\in \mathbb R^{n+r}|Bx+Cy=d,y\geq 0\}$ and that $\pi(Q)=P$, where $\pi(x,y)=x$. Then how do I show that $S=UV$ where $U_{m\times r}, V_{r\times n}$ are non-negative matrices?