nonzero vector $x\in\Re^n$ and choose $i$ such that $\abs{x_i}=\| x\|_\infty.$ Then by strict diagonal dominance, $$\begin{aligned}
\abs{(Ax)_i} = \ABS{\sum_{j=1}^n A_{ij}x_j}
\geq \abs{A_{ii}} \|x\|_\infty - 
\sum_{\substack{j=1\\j\ne i}}^n \abs{A_{ij}} \|x\|_\infty
>0,\end{aligned}$$ so that $Ax\ne 0.$

Rational approximation. {#rational-approximation. .unnumbered}
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The degree of a rational function $p(x)/q(x),$ where $p$ and $q$ are polynomials on $\Re^n,$ is the maximum of the degrees of $p$ and $q.$ Consider a function $f\colon X\to\moo,$ where $X\subseteq \Re^n.$ For $d\geq0,$ define $$\begin{aligned}
 R(f,d) \,= \,\inf_{\rule{0pt}{7pt}p,q} \,\sup_{x\in X} 
   \left\lvert f(x) - \frac{p(x)}{q(x)} \right\rvert,\end{aligned}$$ where the infimum is over multivariate polynomials $p$ and $q$ of degree up to $d$ such that $q$ does not vanish on $X.$ In words, $R(f,d)$ is the least error in an approximation of $f$ by a multivariate rational function of degree up to $d.$ A closely related quantity is $$\begin{aligned}
 R^+(f,d) \,= \,\inf_{\rule{0pt}{7pt}p,q} \,\sup_{x\in X} 
   \left\lvert f(x) - \frac{p(x)}{q(x)} \right\rvert,\end{aligned}$$ where the infimum is over multivariate polynomials $p$ and $q$ of degree up to $d$ such that $q$ is positive on $X.$ These two quantities are related in a straightforward way: $$\begin{aligned}
 R^+(f,2d) \leq R(f,d) \leq R^+(f,d). \end{aligned}$$ The second inequality here is trivial. The first follows from the fact that every rational approximant 