= Diagonal dominance of paired square global-relation collocation
{title2=$\delta_n=e^{-2\omega_n}$}
For <sine collocation of square modified Helmholtz global relations>, each opposite-side mode has a scaled matrix
$$
\begin{pmatrix}1&\delta_n\\\delta_n&1\end{pmatrix},\qquad\delta_n=e^{-2\sqrt{k^2+(n\pi/2)^2}}<e^{-\pi}.
$$
The assembled system has <strict diagonal dominance>, and its <spectral condition number of a positive-definite matrix> is below $(1+e^{-\pi})/(1-e^{-\pi})<1.091$. The conclusion concerns the explicit paired sine rows, not arbitrary uncombined complex rows.
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