Not entirely sure whether this satisfies all your constraints (since I'm not putting the sub-captions in the LoF):

\documentclass{article}
\begin{document}
\begin{figure}
\begin{minipage}{.5\linewidth}
\centering
\[A=\left[\begin{array}{cc}
a & b \\
c & d
\end{array}\right]\]
Short caption for matrix~$A$
\end{minipage}%
\begin{minipage}{.5\linewidth}
\centering
\[A'=\left[\begin{array}{cc}
e & f \\
g & h
\end{array}\right]\]
Short caption for matrix~$A'$
\end{minipage}
\caption{A caption to the entire figure}
\end{figure}
\end{document}
The idea is to place two minipages of .5\linewidth inside a figure. Yes, the figure environment can contain anything inside of it, even mathematics, even a tabular, even an image. It merely acts as a place holder that floats around.
Within each minipage the contents can be supplied in whichever way you like. As I've done in the above example, a display equation is used, together with regular centred text below. You can modify the "short captions" to be \small or \footnotesize, if needed.
If the requirement is to have the "short captions" form part of the LoF, some more work is required.