#StackBounty: #matrices #braces Braces for block matrix

Bounty: 500

I have a block matrix S of dimension (Lm+n)times (L+n)m

matrix

But I think it is not written in a clear way for the reader. Do you have suggestions?

this is the code I wrote so far (thanks to some answers found here in tex.stackexchange):

documentclass{article}
usepackage{amsmath}
     usepackage{xcolor}
     %%overbrace matrix
     newcommandovermat[2]{%
      makebox[0pt][l]{$smash{color{white}overbrace{phantom{%
                begin{matrix}#2end{matrix}}}^{text{color{black}#1}}}$}#2}
        newcommandbovermat[2]{%
         makebox[0pt][l]{$smash{overbrace{phantom{%
                begin{matrix}#2end{matrix}}}^{text{#1}}}$}#2}
          newcommandpartialphantom{vphantom{frac{partial e_{P,M}} 
          {partial w_{1,1}}}}

begin{document}

 begin{equation}
  S=left[phantom{begin{matrix}a_0\ b_0\ ddots\b_0 end{matrix}}
right.hspace{-1.5em}
   begin{matrix}
    bovermat{$mn$}{M_n & ... & M_1  } & bovermat{$Lm$}{0_{ntimes m} & 
       ...& 0_{ntimes m}}\
        a_nI_m & ... & a_1I_m & a_0I_m & ... & 0\
        vdots & ddots & & & ddots & vdots \
        0 & & a_nI_m & ... & ... & a_0I_m
    end{matrix}
       hspace{-1.5em}
        left.phantom{begin{matrix}a_0\ a_0\ ddots\b_0 
           end{matrix}}right]hspace{-1em}
       begin{tabular}{l}
       $left.lefteqn{phantom{begin{matrix} a_0 end{matrix}}}right}n$\
       $left.lefteqn{phantom{begin{matrix} b_0\ ddots\ b_0 
        end{matrix}}} right}Lm$
     end{tabular}
     end{equation}
  end{document}


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