Rectangle box around elements of matrix

Question

Here, I wish to draw a rectangle around the principal diagonal elements (red colored) of the below matrix.

MWE:

\documentclass{article}

\usepackage{amsmath,xcolor}

\begin{document}

Here, I wish to draw a rectangle around the principal diagonal elements 
(red colored) of the below matrix.
\[
  A = \begin{bmatrix} 
    \textcolor{red}{1} & 2 & 3 & 4\\
    1 & \textcolor{red}{2} & 3 & 4\\
    1 & 2 & \textcolor{red}{3} & 4\\
    1 & 2 & 3 & \textcolor{red}{4}
  \end{bmatrix}
\]

\end{document}

Answer 1

Like this?

With use of nicematrix package you obtain above result after two compilation ot he following MWE:

\documentclass{article}
\usepackage{nicematrix}

\begin{document}
\[
\begin{bNiceArray}{>{\strut}llll}[margin=3mm]
\Block[draw=red]{}{1} & 2 & 3 & 4 \\
1 & \Block[draw=red]{}{2} & 3 & 4 \\
1 & 2 & \Block[draw=red]{}{3} & 4 \\
1 & 2 & 3 & \Block[draw=red]{}{4} \\
\end{bNiceArray}
\]
\end{document}

or like this:

\documentclass{article}
\usepackage{nicematrix,tikz}
\usetikzlibrary{fit}

\begin{document}
\[
\begin{bNiceArray}{>{\strut}cccc}[margin,extra-margin = 1pt]
1 & 2 & 3 & 4 \\
1 & 2 & 3 & 4 \\
1 & 2 & 3 & 4 \\
1 & 2 & 3 & 4 \\
\CodeAfter
  \begin{tikzpicture}
  \node [draw=red, rounded corners=2pt, inner ysep = 0pt,
       rotate fit=-38, fit = (1-1) (4-4) ] {} ;
  \end{tikzpicture}
\end{bNiceArray}
\]
\end{document}

Answer 2

Two solutions with pstricks: one with a simple frame around the diagonal elements and another which adds a coloured background. The pst-node module has a dedicated command for that – \ncbox:

    \documentclass[svgnames]{article}

    \usepackage{amsmath, xcolor}
    \usepackage{pst-node}
    \usepackage{auto-pst-pdf}

    \begin{document}

    \[
      A = \begin{bmatrix}
       \rnode{B}{\color{red}{1}} & 2 & 3 & 4\\
        1 & \color{red}{2} & 3 & 4\\
        1 & 2 & \textcolor{red}{3} & 4\\
        1 & 2 & 3 & \rnode{E}{\color{red}{4}}
      \end{bmatrix}
    \ncbox[linecolor =VioletRed, boxsize=5pt, linearc=0.05]{B}{E}
    \]
    \[
      A = \begin{bmatrix}
       \rnode{B}{\color{red}{1}} & 2 & 3 & 4\\
        1 & \color{red}{2} & 3 & 4\\
        1 & 2 & \textcolor{red}{3} & 4\\
        1 & 2 & 3 & \rnode{E}{\color{red}{4}}
      \end{bmatrix}
    \ncbox[linecolor =IndianRed, fillstyle=solid, fillcolor=MistyRose, opacity=0.2, boxsize=5pt, linearc=0.05, nodesep=0.8pt]{B}{E}
    \]

    \end{document} 

Edit:

If you do no want the diagonal frame to touch the brackets, you can, omong other possibilities, change the value of the \ncbox parameter linearc=0.18to have round extremities, or nest a simple matrix in bmatrix and add some spacingon each side, like this:

      \[
      A = \begin{bmatrix}
        \:\begin{matrix}
       \rnode{B}{\color{red}{1}} & 2 & 3 & 4\\
        1 & \color{red}{2} & 3 & 4\\
        1 & 2 & \textcolor{red}{3} & 4\\
        1 & 2 & 3 & \rnode{E}{\color{red}{4}}
        \end{matrix}\:
      \end{bmatrix}
    \ncbox[linecolor =IndianRed, fillstyle=solid, fillcolor=MistyRose, opacity=0.2, boxsize=5pt, linearc=0.05, nodesep=0.8pt]{B}{E}
    \]