Border matrix with multiple indices on top and bottom

A TikZ based solution! By means of the comment percusse suggested, it is possible to make the foreach loop simpler: it iterates on rows/columns of the names that matrix of math nodes gives to each entry of the matrix and reuses that number also in the border index.

documentclass{article}
usepackage{amsmath,amssymb}
usepackage{tikz}
usetikzlibrary{matrix,calc}
begin{document}

begin{align*}
P&=
begin{tikzpicture}[baseline=-thedimexprfontdimen22textfont2relax ]
matrix(m)[matrix of math nodes,left delimiter=(,right delimiter=),inner sep=4pt,ampersand replacement=&]
{
x_1 &  y_1 & s_1 &  z_1 
x_2 &  y_2 & s_2 &  z_2 
x_3 &  y_3 & s_3 &  z_3 
x_4 &  y_4 & s_4 &  z_4 
};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
foreach s in {1,2,...,4}{
% bottom index
node[red,shift=(m-4-s.south),yshift=-0.4cm](0,0) {$a_{s}$};
}
foreach n in {1,2,...,4}{
% top index
node[blue,shift=(m-1-n.north),yshift=0.4cm](0,0) {$b_{n}$} ;
}
end{tikzpicture}
end{align*}

end{document}

Following the strategy, it is also possible to define indices on the matrix's left or right side:

documentclass{article}
usepackage{amsmath,amssymb}
usepackage{tikz}
usetikzlibrary{matrix,calc}
begin{document}

begin{align*}
P&=
begin{tikzpicture}[baseline=-thedimexprfontdimen22textfont2relax ]
matrix(m)[matrix of math nodes,left delimiter=(,right delimiter=),inner sep=4pt,ampersand replacement=&]
{
x_1 &  y_1 & s_1 &  z_1 
x_2 &  y_2 & s_2 &  z_2 
x_3 &  y_3 & s_3 &  z_3 
x_4 &  y_4 & s_4 &  z_4 
};
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
foreach s in {1,2,...,4}{
% bottom index
node[blue,shift=(m-4-s.south),yshift=-0.4cm,text height=1ex,](0,0) {$a_{s}$};
}
foreach n in {1,2,...,4}{
% right index
node[green!70!blue,shift=(m-n-4.west),xshift=1.5cm,text height=1ex,](0,0) {$b_{n}$} ;
}
end{tikzpicture}
end{align*}

end{document}

The result:

Maybe this work

documentclass{article}
usepackage{amsmath}
usepackage{xcolor}
begin{document}
begin{align*}
begin{array}{c}
textcolor{blue}{ begin{array} {c@{hspace{13pt}}c@{hspace{13pt}}c@{hspace{13pt}}c@{hspace{13pt}}c@{hspace{13pt}}c@{hspace{-10pt}}}
y_1 & y_2 & y_3 & y_4 & y_5 & y_6    
end{array}}[5pt]
begin{matrix}
x_1  x_2  x_3 x_4  x_5
end{matrix}quad
begin{bmatrix}
0.9 &  0  &  0  &  0  &  0  &  0 
1  & 0.4 & 0.5 &  0  &  0  &  0 
0  &  0  &  1  &  0  &  0  &  0 
0  &  0  & 0.2 &  1  &  0  &  0 
0  &  0  &  0  & 0.4 & 0.5 & 0.2
end{bmatrix}
end{array}
end{align*}
end{document}

begin{align*} P=begin{array}{c} textcolor{blue}{ begin{array}{c@{hspace{13pt}}c@{hspace{13pt}}c@{hspace{13pt}}c@{hspace{13pt}}c@{hspace{13pt}}c}
    y_1 & y_2 & y_3 & y_4 & y_5 & y_6     end{array}}[5pt] begin{array}{c} abcde end{array} begin{bmatrix}
    0.9 &  0  &  0  &  0  &  0  &  0 
    1  & 0.4 & 0.5 &  0  &  0  &  0 
    0  &  0  &  1  &  0  &  0  &  0 
    0  &  0  & 0.2 &  1  &  0  &  0 
    0  &  0  &  0  & 0.4 & 0.5 & 0.2
    end{bmatrix}   end{array} end{align*}

The package nicematrix provides tools to do such things easily.

documentclass{article}
usepackage{nicematrix}

begin{document}
NiceMatrixOptions{code-for-first-row = color{red},
                   code-for-first-col = color{blue},
                   code-for-last-row = color{green},
                   code-for-last-col = color{magenta}}
$begin{pNiceArray}{cc|cc}[first-row,last-row=5,first-col,last-col,nullify-dots]
       & C_1    & Cdots &        & C_4    &        
L_1    & a_{11} & a_{12} & a_{13} & a_{14} & L_1    
Vdots & a_{21} & a_{22} & a_{23} & a_{24} & Vdots 
hline
       & a_{31} & a_{32} & a_{33} & a_{34} &        
L_4    & a_{41} & a_{42} & a_{43} & a_{44} & L_4    
       & C_1    & Cdots &        & C_4    &     
end{pNiceArray}$
end{document}