Trjtdtk

I am working on my dissertation and I have no idea how I should do this in LaTeX.

And the shading definitely is challenging.

I have only been able to do this so far, I am still an amateur at it.

usepackagecolor
newcommandxtimes
newcommandycellcolorgreen

begindocument
beginequation*
 left[
 beginarrayccc
y & y & y \
 & y & y R \
 & & y \
endarray
 right]left[
 beginarrayc
 \
 a_j \
 \
 endarray
 right]=
endequation*
enddocument

With TikZ matrix:

documentclassarticle
usepackageamsmath
usepackagetikz
usetikzlibrarymatrix, positioning, patterns, backgrounds
tikzset
 every matrix/.append style=
 matrix of math nodes,
 nodes in empty cells,
 inner sep=0pt,
 outer sep=0pt,
 column sep=-pgflinewidth,
 row sep=-pgflinewidth,
 ,
 mylargenode/.style=
 text centered,
 text width=30pt,
 text height=20pt,
 text depth=10pt,
 ,
 mylongnode/.style=
 text centered,
 text width=20pt,
 text height=40pt,
 text depth=30pt,
 ,
 mysmallnode/.style=
 text centered,
 text width=20pt,
 text height=20pt,
 text depth=10pt,
 ,
 mylonglargenode/.style=
 text centered,
 text width=30pt,
 text height=40pt,
 text depth=30pt,
 ,


begindocument
With delimiters like in your image:
[
begintikzpicture
matrix[
 nodes=mylargenode,
 column 6/.style=nodes=mysmallnode,
 row 6/.style=nodes=mylonglargenode
 ] (A) %
 &&&&&;
beginscope[on background layer]
foreach ind in 1,2,3,5
 
 draw[green!70!black,fill=green!40!lightgray] (A-ind-ind.north west) -- (A-ind-ind.north east) -- (A-ind-ind.south east) -- cycle;
 node[anchor=north east] at (A-ind-ind.north east) R;
 
endscope
draw[dotted, green!70!black] (A-4-4.north west) -- (A-4-4.north east) -- (A-4-4.south east) -- cycle;
node[font=Huge,shift=(-1,-2)] at (A) $0$; 
node[font=Huge,shift=(1,3)] at (A) $0$;
matrix[
 right =of A.north east,
 anchor=north west,
 nodes=mysmallnode
 ] (B) %
 underlinea_1 \
 underlinea_2 \
 underlinea_3 \
 \
 underlinea_M\
 underlineb\
 ;
foreach ind in 1,2,...,5
 draw (B-ind-1.south west) -- (B-ind-1.south east);

draw[dotted] (B-4-1.north) -- (B-4-1.south);
matrix[
 right =4em of B.north east,
 anchor=north west,
 nodes=draw,mysmallnode
 ] (C) \
 ;
path (B-4-1) -- node[midway] $=$ (C-4-1);
draw[dotted] (C-4-1.north) -- (C-4-1.south);
% delimiters:
foreach mymatr in A,B,C

draw[very thick] ([xshift=4pt]mymatr.north west) -- (mymatr.north west) -- (mymatr.south west) -- ([xshift=4pt]mymatr.south west);
draw[very thick] ([xshift=-4pt]mymatr.north east) -- (mymatr.north east) -- (mymatr.south east) -- ([xshift=-4pt]mymatr.south east);

endtikzpicture
]

Or with ordinary delimiter:
[
begintikzpicture[ 
 every matrix/.append style=
 left delimiter=[,
 right delimiter=],
 
 ]
matrix[
 nodes=mylargenode,
 column 6/.style=nodes=mysmallnode,
 row 6/.style=nodes=mylonglargenode
 ] (A) %
 &&&&&;
beginscope[on background layer]
foreach ind in 1,2,3,5
 
 draw[green!70!black,fill=green!40!lightgray] (A-ind-ind.north west) -- (A-ind-ind.north east) -- (A-ind-ind.south east) -- cycle;
 node[anchor=north east] at (A-ind-ind.north east) R;
 
endscope
draw[dotted, green!70!black] (A-4-4.north west) -- (A-4-4.north east) -- (A-4-4.south east) -- cycle;
node[font=Huge,shift=(-1,-2)] at (A) $0$; 
node[font=Huge,shift=(1,3)] at (A) $0$;
matrix[
 right =of A.north east,
 anchor=north west,
 nodes=mysmallnode
 ] (B) %
 underlinea_1 \
 underlinea_2 \
 underlinea_3 \
 \
 underlinea_M\
 underlineb\
 ;
foreach ind in 1,2,...,5
 draw (B-ind-1.south west) -- (B-ind-1.south east);

draw[dotted] (B-4-1.north) -- (B-4-1.south);
matrix[
 right =4em of B.north east,
 anchor=north west,
 nodes=draw,mysmallnode
 ] (C) \
 ;
path (B-4-1) -- node[midway] $=$ (C-4-1);
draw[dotted] (C-4-1.north) -- (C-4-1.south);
endtikzpicture
]
enddocument

It is indeed a bit tedious. (EDIT: Fixed the first R to become math mode, mille grazie to @Sebastiano!)

documentclass[tikz,border=3.14mm]standalone
usetikzlibrarypositioning,backgrounds,patterns
begindocument
begintikzpicture
 beginscope[local bounding box=mat1]
 path node (R0) $R$ foreach X [remember=X as lastX (initially 0)] in
 1,...,4
 ifnumX=3
 node[below right=1.5em of RlastX] (RX) $phantomR$
 else
 node[below right=1.5em of RlastX] (RX) $R$
 fi
 ;
 beginscope[on background layer]
 draw[fill=green!70!black] 
 foreach X in 0,1,2,4 
 ([xshift=-1em]RX.north west) -- ([yshift=-1em]RX.south east)
 coordinate(bX)
 ;
 draw[densely dotted] ([xshift=-1em]R3.north west) -- ([yshift=-1em]R3.south east)
 coordinate(b3) |- cycle;
 endscope
 node[pattern=crosshatch,pattern color=purple,anchor=north west,
 minimum height=6em] (F) at ([yshift=-1em]R4.south east) $F$; 
 draw[densely dotted] ([yshift=-1em]R3.south east-|R4.south east)
 rectangle (F.east|-R3.north);
 foreach X [evaluate=X as Y using int(40+10*X)] in 0,1,2,4 
 -RX.north);
 endscope 
 path (mat1.south west) -- (mat1.north east)
 node[pos=0.3,font=Large] $0$ node[pos=0.7,font=Large] $0$;
 draw[thick] ([xshift=0.7em]mat1.north west) -| (mat1.south west) -- ++ (0.7em,0)
 ([xshift=-0.7em]mat1.north east) -| (mat1.south east) -- ++ (-0.7em,0);
 %
 beginscope[local bounding box=mat2]
 foreach X [remember=X as lastX (initially 0)] in 1,2,3
 -RlastX.north) --
 ([xshift=2.5em]mat1.east
 path ([xshift=1em]mat1.east|-R4.north) --
 ([xshift=2.5em]mat1.east|-b4) node[midway] (aM) $a_M$;
 node[below=0.5em of aM] (b) $b$; 
 endscope
 draw[thick] ([xshift=0.5em]mat2.north west) -| (mat2.south west) -- ++ 
 (0.5em,0)
 ([xshift=-0.5em]mat2.north east) -| (mat2.south east) -- ++ (-0.5em,0);
 foreach X in 0,...,4
 -bX) -- (mat2.east
 %
 node[anchor=west] (eq) at (mat2.east|-mat1.center) $=$;
 %
 beginscope[local bounding box=mat3]
 node[pattern=dots,anchor=north west,pattern color=yellow!60!orange,
 minimum height=6em] (f) at ([xshift=0.1em]eq.east|-F.north) $f$; 
 foreach X [remember=X as lastX (initially 0),
 evaluate=X as Y using int(30*X+10)] in 1,2,3
 fill[yellow!Y] ([xshift=0.1em]eq.east
 fill[yellow] ([xshift=0.1em]eq.east|-R4.north) rectangle
 (f.east|-b4); 
 draw[densely dotted] (f.center|-b2) -- (f.center|-b3);
 endscope
 foreach X in 0,...,4
 draw (mat3.west
 draw[thick] ([xshift=0.3em]mat3.north west) -| (mat3.south west) -- ++ 
 (0.3em,0)
 ([xshift=-0.3em]mat3.north east) -| (mat3.south east) -- ++ (-0.3em,0);
endtikzpicture
enddocument