Correctly format \vdots in small matrix

I propose a \svdots macro for “scaling” vertical dots. I've never really liked the standard definition, to be honest.

\documentclass{article}
\usepackage{amsmath}

\DeclareRobustCommand{\svdots}{% s for `scaling'
  \vbox{%
    \baselineskip=0.33333\normalbaselineskip
    \lineskiplimit=0pt
    \hbox{.}\hbox{.}\hbox{.}%
    \kern-0.2\baselineskip
  }%
}

\begin{document}

$S=
\begin{pmatrix}
  \sigma_1 & 0        & 0\\ 
  0        & \sigma_2 & \svdots\\
  0        & \dots    &
\end{pmatrix}
\begin{pmatrix}
  \sigma_1 & 0        & 0\\ 
  0        & \sigma_2 & 0\\
  0        & \dots    &
\end{pmatrix}$

\scriptsize

$S=\begin{pmatrix}
  \sigma_1 & 0        & 0\\ 
  0        & \sigma_2 & \svdots\\
  0        & \dots    &
\end{pmatrix}$

\end{document}

If you want the vertical dots centered with respect to the formula axis, you can do

\documentclass{article}
\usepackage{amsmath}

\DeclareRobustCommand{\svdots}{% s for `scaling'
  \vcenter{%
    \offinterlineskip
    \hbox{.}
    \vskip0.25\normalbaselineskip
    \hbox{.}
    \vskip0.25\normalbaselineskip
    \hbox{.}%
  }%
}

\begin{document}

$S=
\begin{pmatrix}
  \sigma_1 & 0        & 0\\ 
  0        & \sigma_2 & \svdots\\
  0        & \dots    &
\end{pmatrix}
\begin{pmatrix}
  \sigma_1 & 0        & 0\\ 
  0        & \sigma_2 & 0\\
  0        & \dots    &
\end{pmatrix}$

\scriptsize

$S=\begin{pmatrix}
  \sigma_1 & 0        & 0\\ 
  0        & \sigma_2 & \svdots\\
  0        & \dots    &
\end{pmatrix}$

\end{document}

I would do that with stackengine , which has a \Vectorstack command for math mode:

\documentclass[12pt]{article}
\usepackage{amsmath}
\usepackage{stackengine}

\begin{document}

\scriptsize

$S=\begin{pmatrix}\sigma_1 & 0 & 0\\
                0 &\sigma_2 & \Vectorstack{\vdots}\\
                0 &\dots &
\end{pmatrix}$

\end{document} 

For a \vdots centred w.r.t. the middle of the 0 in the row, you can try this:

$S=\begin{pmatrix}\sigma_1 & 0 & 0\\[0.5ex]
                0 &\sigma_2 & \raisebox{-0.12\height}[0pt][0pt]{$\vdots$}\\[0.5ex]
                0 &\dots &
\end{pmatrix}$

Using the package nicematrix adjusting the values of \NiceMatrixOptions to -5pt (minimum value),

\documentclass[a4paper,12pt]{article}
\usepackage{amsmath,amssymb}
\usepackage{nicematrix}

\begin{document}
\NiceMatrixOptions{cell-space-top-limit=-5pt,cell-space-bottom-limit=-5pt}
$S=\begin{pNiceMatrix}
\sigma_{1} & 0 & 0\\
0 & \sigma_{2} & \vdots \\
0 & \ldots &
\end{pNiceMatrix}$

\end{document}

you can obtain similar output.