how to add caption above and alongside on Matrix

The following example based on the kbordermatrix package might serve as a point to start from:

documentclass{article}
usepackage{amsmath}
usepackage{kbordermatrix}

begin{document}

[
  kbordermatrix{
          & (2,2) & (2,3) & (2,4) 
    (2,2) & a_1   & a_2   & 0     & cdots & 0 & a_3 & a_4 & 0 & cdots & 0 & cdots & 0
    (2,3) & a_5   & 0     & a_6   & cdots & 0 & a_7 & 0 & a_8 & cdots & 0 & cdots & 0  
          & 
          &
          &
          &vdots         &       & & & & ddots & & & & & & vdots 
          & 
          &cdots         &       & & & & cdots & & & & & & cdots 
  }
]

end{document}