How did this matrix get created from the inequality

Question

I am new to matrices and to systems of inequalities. When I look at a matrix it's difficult to tell where the $1's$, $0's$ and $-1's$ come from. I know they somehow come from the inequality itself, but for some reason I'm not seeing how.

My questions are:

1. How these inequalities become (get plugged into) these matrices (source): $$ 1 \leq i \leq n : \begin{bmatrix} 1 & 0\\ -1 & 0 \end{bmatrix} \begin{pmatrix} i\\ j \end{pmatrix} + \begin{pmatrix} -1\\ n \end{pmatrix} \geq 0 $$ $$ 1 \leq j \leq n : \begin{bmatrix} 0 & 1\\ 0 & -1 \end{bmatrix} \begin{pmatrix} i\\ j \end{pmatrix} + \begin{pmatrix} -1\\ n \end{pmatrix} \geq 0 $$
2. Specifically, how $1 \leq i \leq n$ becomes \begin{bmatrix}1 & 0\\-1 & 0\end{bmatrix}
3. And how $1 \leq i \leq n$ becomes \begin{pmatrix}-1\\n\end{pmatrix}

---

Now I'll describe the problem I'm having in more detail if that's helpful.

(I assume this equation is from $\textbf{A}\vec{x} + \vec{b} \geq 0$).

Take for example these two parts:

$$ (a):1 \leq i \leq n\ \ \ \ (b):\begin{bmatrix} 1 & 0\\ -1 & 0 \end{bmatrix} $$

I know (a) can be rewritten into a set of two inequalities (though I'm not even sure I'm doing that right):

\begin{align} 0 &\leq -1 + i\\ 0 &\leq -i + n \end{align}

So then in my attempt to figure out how the inequality goes into the matrix, I do this:

\begin{align} 0 &\leq -1 + i \mapsto -1, 1\\ 0 &\leq -i + n \mapsto -1, 1 \end{align}

since $\mathbf{-1} + i$ is like $(\mathbf{-1} + (\mathbf{1} \times i))$ and $\mathbf{-}i + n$ is like $((\mathbf{-1} \times i) + (\mathbf{1} \times n))$. But that would lead to a matrix like this:

$$ \begin{bmatrix} -1 & 1\\ -1 & 1 \end{bmatrix} $$

when it should be

$$ \begin{bmatrix} 1 & 0\\ -1 & 0 \end{bmatrix} $$

So I'm confused how that matrix (b) gets created from the inequality (a). The same goes for the second matrix in the first diagram.

Also, I am not sure where this comes from either:

$$ \begin{pmatrix} -1\\ n \end{pmatrix} $$

The full set of inequalities/matrices (if it's helpful) is below. I'm basically just trying to understand how they got those matrix equations from the inequalities in the diagram:

They got the inequalities out of the for loop if that helps.

Another example is the following diagram. Even though they have the colors showing how they mapped the values, I still don't see how they did it (went from inequality $\to$ matrix).

I would simply like to know how to do it for one of these inequality/matrices pairs, so I can apply it to all of them. Thank you so much for the help.

Answer 1

Indeed it is basically breaking $1\leq i\leq n$ into two parts $1\leq i$ and $i\leq n$.

You then make both of them as nonnegativity conditions $i-1\geq 0$ and $-i+n\geq0$ so they are now

$$ \begin{bmatrix}1\\-1\end{bmatrix}i + \begin{bmatrix}-1\\n\end{bmatrix} \geq 0 $$ Since nothing depends on $j$ for these inequalities we can add the dummy constraint $0j\geq 0$ and everything stays the same

$$ \begin{bmatrix}1&0\\-1&0\end{bmatrix}\begin{bmatrix}i\\j\end{bmatrix} + \begin{bmatrix}-1\\n\end{bmatrix} \geq 0 $$

You can do the same for $j$.