= Diagonal scaling of Gershgorin discs
{title2=$r_i(D^{-1}AD)=\sum_{j\ne i}|a_{ij}|\,d_j/d_i$}
A positive <diagonal matrix> $D=\operatorname{diag}(d_1,\ldots,d_n)$, with $d_i>0$, is an <invertible matrix> and changes the coordinate scales through the <similarity transformation> $B=D^{-1}AD$. Its diagonal entries are still $a_{ii}$, while $b_{ij}=a_{ij}d_j/d_i$. Thus the <Gershgorin discs> have the same centres but different radii. <Matrix similarity> preserves the <characteristic polynomial> and <eigenvalues>, so the original and scaled disk unions are both valid bounds and may be intersected. Suitable scaling can compensate for badly unbalanced off-diagonal entries; it need not shrink every radius.https://cklixx.people.wm.edu/teaching/math408/408-note.pdf[Chi-Kwong Li, Math 408 lecture notes, §6.1: similarity transformations for eigenvalue localization]{ref}
For example,
$$
A=\begin{pmatrix}2&100\\0.01&3\end{pmatrix},\qquad D=\begin{pmatrix}100&0\\0&1\end{pmatrix},\qquad D^{-1}AD=\begin{pmatrix}2&1\\1&3\end{pmatrix}.
$$
The original <Gershgorin discs> are $|z-2|\leq100$ and $|z-3|\leq0.01$. Their union is simply the huge first <disk>; the small second <disk> contains no <eigenvalue>. After scaling, both radii are $1$, giving $|z-2|\leq1$ or $|z-3|\leq1$. The two <Gershgorin discs> overlap, so their individual <eigenvalue> counts are not certified. The scaled <matrix> is a real <symmetric matrix>, however, so <matrix similarity> now proves that the original <matrix> has real <eigenvalues> in $[1,4]$ and is an <invertible matrix>. Directly, both <matrices> have <characteristic polynomial> $\lambda^2-5\lambda+5$, hence
$$
\lambda_\pm=\frac{5\pm\sqrt5}{2}\approx1.381966,\ 3.618034.
$$
The scaling changes the available bounds, while these <eigenvalues> stay fixed.
\Image[/gershgorin-diagonal-scaling.png]
{title=Diagonal similarity scaling replaces a radius-100 Gershgorin disc by two radius-1 discs while preserving both eigenvalues}
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