Block matrices, separator with space

Question

I want to write block matrices like the one shown below

I tried underlining, fraction, and using __ as an entry by making three rows but they did not look nice.I want the horizontal lines inside the matrix to be gapped in between as shown. How can I do this?

Answer 1

This is one way:

\documentclass[a4paper,journal]{IEEEtran}
\usepackage{amsmath}

\begin{document}
\[
\begin{bmatrix}
  \frac{I}{\mathrm{O}}\frac{\mathrm{O}}{\mathrm{O}}
\end{bmatrix}
\begin{bmatrix}
  \frac{x}{u}
\end{bmatrix} =
\begin{bmatrix}
  \frac{A}{C}\frac{B}{D}
\end{bmatrix}
\begin{bmatrix}
  \frac{x}{u}
\end{bmatrix} +
\begin{bmatrix}
  \frac{\mathrm{O}}{-y}
\end{bmatrix}
\]
%
\[
\left[
  \frac{I}{\mathrm{O}}\frac{\mathrm{O}}{\mathrm{O}}
\right]
\left[
  \frac{x}{u}
\right] =
\left[
  \frac{A}{C}\frac{B}{D}
\right]
\left[
  \frac{x}{u}
\right] +
\left[
  \frac{\mathrm{O}}{-y}
\right]
\]
\end{document}

Instead of dynamically varying brackets (\left[ and \right]), you may use fixed size brackets like \Bigg[ and \Bigg] provided by amsmath.

Answer 2

Here's a more "matrix-like" solution:

\documentclass{article}
\usepackage{amsmath,booktabs,xparse}

\ExplSyntaxOn
\DeclareExpandableDocumentCommand{\seps}{m}
 {
  \noalign{\seps_do_arg:n { #1 } }
  \tl_use:N \g_seps_list_tl
 }
\tl_new:N \g_seps_list_tl
\cs_new_protected:Npn \seps_do_arg:n #1
 {
  \tl_clear:N \g_seps_list_tl
  \clist_map_inline:nn { #1 }
   { \tl_gput_right:Nn \g_seps_list_tl { \cmidrule(lr){##1--##1} } }
 }
\ExplSyntaxOff


\begin{document}
\[
\begin{bmatrix}
I & \textrm{O} \\
\seps{1,2}
\textrm{O} & \textrm{O}
\end{bmatrix}
\begin{bmatrix}
x \\
\seps{1}
u
\end{bmatrix}
=
\begin{bmatrix}
A & B \\
\seps{1,2}
C & D
\end{bmatrix}
\begin{bmatrix}
x \\
\seps{1}
u
\end{bmatrix}
+
\begin{bmatrix}
\mathrm{O} \\
\seps{1}
-y
\end{bmatrix}
\]
\end{document}

You simply give the list of separators you need. It would unfortunately be quite difficult to know how many columns your matrix has.