Question: ```
\documentclass{article}
\usepackage{amsmath}
\usepackage{array}
\usepackage{mathtools}
\begin{document}
\renewcommand{\arraystretch}{1.2}
\newcommand{\minus}{\scalebox{0.4}[1.0]{$-$}}
\[
\begin{bmatrix*}[r]
  0&  \minus\frac{1}{2} &\frac{1}{2} \\
  \minus\frac{1}{2}&  0&\minus\frac{1}{2}\\
  \frac{1}{2}&  \minus\frac{1}{2}&0
\end{bmatrix*}
\]
\end{document}

```

[![enter image description here](https://i.stack.imgur.com/1yvlc.png)](https://i.stack.imgur.com/1yvlc.png)

How to reduce the number “0” size to fit fraction "1/2"? (reduce height of "0" ?)

---

update sample

As below photo shows the proportion close to 1/3

[![enter image description here](https://i.stack.imgur.com/Ofr0I.jpg)](https://i.stack.imgur.com/Ofr0I.jpg)

Answer: 
I would second Mico's recommendation to not do this. But the solution would be to make the fractions display style rather than shrink the zeros (your example was likely set by naïve software which doesn't know how to properly resize fractions. Here's another alternative setting adapted from your MWE:

```
\documentclass{article}
\usepackage{amsmath}
\usepackage{array}
\usepackage{mathtools}
\begin{document}
\newcommand{\half}{{\displaystyle\frac{1}{2}\vphantom{\frac{1}{2}^1_1}}}
\[
\begin{bmatrix*}[r]
  0&  -\half &\half \\
  -\half&  0&-\half\\
  \half&  -\half&0
\end{bmatrix*}
\]
\end{document}

```

[![enter image description here](https://i.stack.imgur.com/XkbLi.png)](https://i.stack.imgur.com/XkbLi.png)

I put the fraction in `\displaystyle` and used a `\vphantom` to add extra spacing—perhaps not enough—above and below the fractions.
I would say that your sample actually suggests the opposite: "I want to preserve regular size even for numerators and denominators". Use `\dfrac` instead of `\frac`.

Also, why do you have special treatment for minus? Why the `-` isn't enough?

```
\documentclass{article}

\usepackage{amsmath}
\usepackage{array}
\usepackage{mathtools}

\begin{document}
\renewcommand{\arraystretch}{2}
\[
  \begin{bmatrix*}[r]
    0             &  -\dfrac{1}{2} & \dfrac{1}{2}  \\
    -\dfrac{1}{2} &  0             & -\dfrac{1}{2} \\
    \dfrac{1}{2}  &  -\dfrac{1}{2} & 0
  \end{bmatrix*}
\]
\end{document}

```

[![enter image description here](https://i.stack.imgur.com/QLU1E.png)](https://i.stack.imgur.com/QLU1E.png)