Consider a linear system with a map $A: \mathbb{R}^n \rightarrow \mathbb{R}^m$ such that $y = A x$ with $n \geq m$

The input space $x$ is constrained by a zonotope set $\mathcal{X} \subseteq \mathbb{R}^n$ in an $\mathcal{H}$-representation given

$$ \mathcal{X} = \{ \, x \in \mathbb{R}^n\, | \, \, C_{x}^{T} x \leq d_{x} \, \} $$

We need to characterize the output variable $y$ by obtaining a set $\mathcal{Y} \subseteq \mathbb{R}^m$ such that

$$ \mathcal{Y} = \{ \, y \in \mathbb{R}^m \, | \, \, y = A x ,\, x \in \mathcal{X} \,\} $$

WE know that the set $\mathcal{Y}$ is also a Zonotope and could be expressed in $\mathcal{H}$-representation as

$$ \mathcal{Y} = \{ \, y \in \mathbb{R}^m\, | \, \, C_{y}^{T} y \leq d_{y} \, \} $$

Our questions are:

- How to mathematically prove that $\mathcal{Y}$ is a Zonotope.
- How to compute $\mathcal{Y}$ i.e. obtain $\{C_{y}, d_{y}\}$ given $A$ and $\{C_{x}, d_{x}\}$.