= Inverse infinity-norm bound from diagonal dominance
{title2=$\|A^{-1}\|_{\ell^\infty}\le\delta^{-1}$}
If $\delta=\min_i(|a_{ii}|-\sum_{j\ne i}|a_{ij}|)>0$, then $A$ is invertible and its <operator norm> on $\ell^\infty$ satisfies the displayed bound. For $Ax=b$, choose $i$ maximizing $|x_i|$. The <reverse triangle inequality> gives $|b_i|\ge\delta\|x\|_{\ell^\infty}$. Taking $b=0$ proves injectivity and hence invertibility in finite dimensions; taking arbitrary $b$ proves the bound. The estimate is useful for converting <strict diagonal dominance> into quantitative <matrix inverse> stability.
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