How to draw a matrix with some framed elements, and some arrows between two frames?

An all TiKZ solution is easy with a matrix of math nodes:

\documentclass[tikz,border=2mm]{standalone}
\usetikzlibrary{matrix} 
\begin{document}

\begin{tikzpicture}

\matrix (A) [matrix of math nodes, right delimiter = ), left delimiter =(] {
0      &        &        & & \\
a_1    & \ddots &        & & \\
\vdots &        & \ddots & & \\
a_2    &        &        & \ddots &  \\
\vdots & b_1    & \ddots & & \\
\vdots & \vdots &        & &  \\
\vdots & b_2    &        & & \\
\vdots & \vdots & c_1    & &  \\
\vdots & \vdots & \vdots & & \\
\vdots & \vdots & c_2    & &  \\
\vdots & \vdots & \vdots & & 0 \\
};

\draw[red] (A-2-1.north west) rectangle (A-4-1.south east);
\draw[red] (A-5-2.north west) rectangle (A-7-2.south east);
\draw[red] (A-8-3.north west) rectangle (A-10-3.south east);

\draw[->,blue] (A-4-1.south) --++(-90:1mm) -| (A-5-2.north);
\draw[->,blue] (A-7-2.south) --++(-90:1mm) -| (A-8-3.north);
\end{tikzpicture}

\end{document}

Adding some styles and using matrix from PGF can make this:

Code:

\documentclass[tikz]{standalone}
\usepackage{tikz}
\usetikzlibrary{matrix}

\begin{document}
    \begin{tikzpicture}[
          block/.style={text height=0.5em
                      , minimum width=1em,
                      , outer sep=0pt}]
    \matrix [matrix of math nodes
           , left delimiter={(},right delimiter={)}
           , nodes in empty cells
           , nodes=block
           , row sep=1em
           , column sep=1em] at (0,0)
    {
        0 & & & & & & & & & & \\
        |(a1)| a_1 & \ddots &  \\
        \vdots & & \ddots & & & & & & & & \\
        |(an)| a_n & & & & & & & & & & \\
        \vdots & |(b1)| b_1 & & & & & & & & & \\
        \vdots & \vdots & & & & & & & & & \\
        \vdots & |(bn)| b_n & & & & & & & & & \\
        \vdots & \vdots & |(c1)| c_1 & & & & & & & & \\
        \vdots & \vdots & \vdots & & & & & & & & \\
        \vdots & \vdots & |(cn)| c_n & & & & & & & & \\
        \vdots & \vdots & \vdots & & & & & & & & 0 \\
    };

    \draw[dashed,red] (a1.north west) rectangle (an.south east);
    \draw[dashed,red] (b1.north west) rectangle (bn.south east);
    \draw[dashed,red] (c1.north west) rectangle (cn.south east);

    \draw[-latex,blue] (an.south) --++(-90:1mm) -| (b1.north);
    \draw[-latex,blue] (bn.south) --++(-90:1mm) -| (c1.north);
    \end{tikzpicture}
\end{document}