Drawing this matrix using Tikz

Question

I have a matrix which has certain properties. To illustrate these properties, I would like to generate the following image using Tikz:

Additionally, I would like the following things on the image:

1. The upper triangle should be shaded light blue
2. The lower triangle should be shaded light red
3. The red and blue lines should be dotted exactly where there are dots in the zeros along the horizontal and vertical directions.

I am new to using Tikz and any help with this would be greatly appreciated!

Answer 1

For fun: that is easy with pstricks:

\documentclass[11pt,x11names]{standalone}
\usepackage{pst-node}
\usepackage{auto-pst-pdf}
\usepackage{amsmath}
\usepackage{sansmath}

\begin{document}

 \mathversion{sans}\setlength\fboxsep{0pt}
\begin{postscript}
    $\begin{pmatrix}
        0 & 0 & 0 & 0 & \dots & 0 \\ 0 & \pnode[0,1.5ex]{A} & & &\pnode[0,1.5ex]{Dh} \\0\\0\\\vdots &\pnode{Dv} \\0 & & & & & \pnode{B}
    \end{pmatrix}$
    \psset{arrowinset=0, linewidth=1.5pt, linearc=0.2pt, nodesep = 6pt,, arrows =c- }
    \psline{<->}(A)(B)
    \psset{linecolor=RoyalBlue2, fillcolor=RoyalBlue1!15!, arrows =c-}
    \ncangle[angleB=90, linestyle = none, fillstyle=solid,]{A}{B}
    \ncline[nodesepB=9.5pt]{A}{Dh}
    \ncangle[angleA=90, nodesepB=7.5pt]{B}{Dh}
    \rput(Dh){\color{RoyalBlue2}\dots}
    \psset{angleA=-90, angleB=180, linecolor=Red1!60!, fillcolor=Red1!10!, arrows =-c}
    \ncangle[linestyle=none, fillstyle=solid]{A}{B}
    \ncangle[nodesepA=3pt,]{Dv}{B}
    \ncline[nodesepA=12pt]{Dv}{A}
    \rput[B](Dv){\color{Red1!60!}\vdots}
\end{postscript}

\end{document} 

Answer 2

With TikZ is easy to:

\documentclass[border=3mm,
               tikz,
               prewiev]{standalone}
\usetikzlibrary{matrix}
\begin{document}
    \begin{tikzpicture}
\matrix[matrix of math nodes,
        left delimiter=(,
        right delimiter=),
        nodes in empty cells] (m)
{
0       & 0 & 0 & 0 & \dots & 0 \\
0   & \vphantom{0}  
            &   &   &       &   \\
0       &   &   &   &       &   \\
0       &   &   &   &       &   \\
\vdots  &   &   &   &       &   \\
0       &   &   &   &       &   \\
};
\fill[blue!30] (m-2-2.north west) -| (m-6-6.south east);
\fill[ red!30] (m-2-2.north west) |- (m-6-6.south east);
\draw[line width=3mm,white]  (m-2-2.north west) -- (m-6-6.south east);
\draw[thick,stealth-stealth] (m-2-2.north west) -- (m-6-6.south east);

    \end{tikzpicture}
\end{document}

Edit: Gonzalo Medina remind me on the forgotten dots, Sorry for this. Now I add them in the first row and column and also appearance of diagonal line with use white background line.

Edit (2): The matrix will be centered in text, if you enclose with \begin{center} ... \end{center} or put in some other appropriate environment, for example \begin{figure}\centering\begin{tikzpicture} ... \end{tikzpicture}\end{figure}, etc.

For rounded shadow fill you need to add option rounded corners. Se new MWE below.

\documentclass{article}
    \usepackage{tikz}
\usetikzlibrary{matrix}

\usepackage[active,tightpage]{preview}% just for show only picture
\PreviewEnvironment{center}%
\setlength\PreviewBorder{1em}


\begin{document}
    \begin{center}
    \begin{tikzpicture}
\matrix[matrix of math nodes,
        left delimiter=(,
        right delimiter=),
        nodes in empty cells] (m)
{
0       & 0 & 0 & 0 & \dots & 0 \\
0   & \vphantom{0}
            &   &   &       &   \\
0       &   &   &   &       &   \\
0       &   &   &   &       &   \\
\vdots  &   &   &   &       &   \\
0       &   &   &   &       &   \\
};
\fill[rounded corners,blue!30] (m-2-2.north west) -| (m-6-6.south east);
\fill[rounded corners, red!30] (m-2-2.north west) |- (m-6-6.south east);
\draw[line width=3mm,white]  (m-2-2.north west) -- (m-6-6.south east);
\draw[thick,stealth-stealth] (m-2-2.north west) -- (m-6-6.south east);
    \end{tikzpicture}
    \end{center}
\end{document}