I am wondering how exactly these two matrices (source (p.16)) were constructed from the set of affine inequalities:
$$ \begin{align} i &\geq 0\\ j &\geq 0\\ −i + N − 1 &\geq 0\\ −j + N − 1 &\geq 0 \end{align} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ D= \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ −1 & 0 & 1 & −1 \\ 0 & −1 & 1 & −1 \end{bmatrix} \begin{bmatrix} i\\ j\\ N\\ 1 \end{bmatrix} \geq 0 $$
I can't tell how the cells in the first matrix got there.
In trying to understand affine inequalities and polytopes, I have these definitions:
A polytope is defined as:
$$S = \{ x \in \mathbb{R}^n \mid Ax \leq b \}$$ $$S = \{ x \in \mathbb{R}^n \mid Ax + b \geq 0 \}$$
where $A \in \mathbb{R}^{m\times n}$ and $b \in \mathbb{R}^m$.
The elements of $A$ and $b$ can be taken as integers if we want a $\mathbb{Z}$-polytope.
Let $a \in \mathbb{R}^n$ and $b \in R$ be given.
- The set $\{x \in \mathbb{R}^n \mid a^T x = b\}$ is called a hyperplane.
- The set $\{x \in \mathbb{R}^n \mid a^T x \leq b\}$ is called a half-space.
A polyhedron $P \subset \mathbb{R}^n$ is bounded if there exists a constant $K$ such that $|x_i| < K\ \forall x \in P, \forall i \in [1, n]$.
A polytope is a bounded polyhedron.
The image of a polytope $P$ under an affine map $π:x \mapsto Ax + b$ is also a polytope.
You can define a polytope by either (1) affine inequalities or (2) a convex hull.
So in there I see $Ax + b \geq 0$, where $A \in \mathbb{R}^{m\times n}$ and $b \in \mathbb{R}^m$. That seems similar to the matrix above (correct me if I'm wrong), where $A$ is the first matrix and $b$ is the second (not sure where $x$ is). Trying to get an understanding of how the affine inequality equation is converted into matrix form. Not sure what field goes where.