Getting alignments with so inhomogeneous objects requires a bit of manual adjusting.
I raised the first row vector “by eye”; it would be possible to compute the amount of raising exactly, but I did it just to show that it's really awful and unhelpful for the reader. Why the top and not the bottom? There's no reason for choosing either, so the right place is at the center.
More importantly, I added some shorthands that help both in typing and in reading code.
Some “phantoms” make the objects align nicely.
\documentclass{article}
\usepackage{amsmath,bm}
\newenvironment{rowvector}
{\bm{[}\begin{matrix}}
{\end{matrix}\bm{]}}
\newcommand{\pder}[2]{\frac{\partial #1}{\partial #2}}
\newcommand{\ol}{\overline}
\newcommand{\ul}{\underline}
\begin{document}
\[
\frac{dz}{dt} =
\raisebox{3ex}{$\begin{rowvector} 1 & -1 \end{rowvector}$}
\begin{bmatrix}
\pder{\ul{y}}{\ul{x}} & \pder{\ol{y}\vphantom{\ul{y}}}{\ul{x}} \\[0.5em]
\pder{\ul{y}}{\ol{x}} & \pder{\ol{y}\vphantom{\ul{y}}}{\ol{x}}
\end{bmatrix}
\begin{bmatrix}
1 \vphantom{\pder{\ul{y}}{\ul{x}}} \\[0.5em]
-1 \vphantom{\pder{\ul{y}}{\ul{x}}}
\end{bmatrix}
\]
\[
\frac{dz}{dt} =
\begin{rowvector} 1 & -1 \end{rowvector}
\begin{bmatrix}
\pder{\ul{y}}{\ul{x}} & \pder{\ol{y}\vphantom{\ul{y}}}{\ul{x}} \\[0.5em]
\pder{\ul{y}}{\ol{x}} & \pder{\ol{y}\vphantom{\ul{y}}}{\ol{x}}
\end{bmatrix}
\begin{bmatrix}
1 \vphantom{\pder{\ul{y}}{\ul{x}}} \\[0.5em]
-1 \vphantom{\pder{\ul{y}}{\ul{x}}}
\end{bmatrix}
\]
\end{document}

\partial
symbols are not aligned horizontally which I don't love, I guess they're being pushed up by the\underline{}
?\partial
s is a nightmare. I didn't notice. Any fixes for that?