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I want to draw the following matrix in LaTeX:

I found that I should use the TikZ package, and I have spent a lot of time looking at the package. However, I have no experience of using the package before, and I do not know how to draw it. How can I do it?

Just arrays:

documentclassarticle
usepackageamsmath,array

begindocument

[
newcommandblocke
 beginmatrix
 hphantom0 & hphantom0 & hphantom0 & hphantom0 \
 \ \
 endmatrix

newcommandblockr
 beginmatrix
 hphantom0 & hphantom0 & hphantom0 &
 hatNsmash[b]vphantomBig \ &&& vdots \ &&& 2 \ &&& 1
 endmatrix

newcommandblockl
 beginmatrix
 1 & hphantom0 & hphantom0 & hphantom0 \
 2 \ vdots \ hatNsmash[b]vphantomBig
 endmatrix

newcommanddddots
 beginmatrix
 hphantom0 & hphantom0 & ddots \
 \
 \
 ddots
 endmatrix

beginpmatrix
hspace*0.3em
beginarray
hline
& blockl & & & blocke \
hline
blockr & & blockl & & \
hline
& blockr & & blockl & \
hline
& & & dddots & \
hline
& & & blockr & \
hline
endarray
hspace*0.3em
endpmatrix
]

enddocument

I locally defined a few commands to ease input.

Welcome to TeX.SE! Actually, you do not need TikZ for this.

documentclassarticle
usepackageamsmath
begindocument
[left(beginarray*5ccccc
hline
& & & & & 1 & & & & & & & & & & & & & & & & & & & \
& & & & & 2 & & & & & & & & & & & & & & & & & & & \
& & & & & vdots & & & & & & & & & & & & & & & & & & & \
& & & & & vdots & & & & & & & & & & & & & & & & & & & \
& & & & & widehatN & & & & & & & & & & & & & & & & & & & \
hline
& & & & widehatN & & & & & & 1 & & & & & & & & & & & & & & \
& & & & vdots & & & & & & 2 & & & & & & & & & & & & & & \
& & & & vdots & & & & & & vdots & & & & & & & & & & & & & & \
& & & & 2 & & & & & & vdots & & & & & & & & & & & & & & \
& & & & 1 & & & & & & widehatN & & & & & & & & & & & & & & \
hline
& & & & & & & & & widehatN & & & & & & 1 & & & & & & & & & \
& & & & & & & & & vdots & & & & & & 2 & & & & & & & & & \
& & & & & & & & & vdots & & & & & & vdots & & & & & & & & & \
& & & & & & & & & 2 & & & & & & vdots & & & & & & & & & \
& & & & & & & & & 1 & & & & & & widehatN & & & & & & & & & \
hline
& & & & & & & & & & & & & & & & & bullet & & & & & & & \
& & & & & & & & & & & & & & & & & & bullet & & & & & & \
& & & & & & & & & & & & & & & bullet & & & & bullet & & & & & \
& & & & & & & & & & & & & & & & bullet & & & & & & & & \
& & & & & & & & & & & & & & & & & bullet & & & & & & & \
hline
& & & & & & & & & & & & & & & & & & & widehatN & & & & & \
& & & & & & & & & & & & & & & & & & & vdots & & & & & \
& & & & & & & & & & & & & & & & & & & vdots & & & & & \
& & & & & & & & & & & & & & & & & & & 2 & & & & & \
& & & & & & & & & & & & & & & & & & & 1 & & & & & \
hline
endarrayright)]
enddocument

However, you could use TikZ to improve the appearance.

documentclassarticle
usepackageamsmath
usepackagetikz
usetikzlibrarytikzmark
tikzset% https://tex.stackexchange.com/a/156990/121799
 dots/.style args=#1per #2%
 line cap=round,
 dash pattern=on 0 off #2/#1
 

begindocument
[left(beginarray*5ccccc
hline
& & & & & 1 & & & & & & & & & & & & & & & & & & & \
& & & & & tikzmarknode212 & & & & & & & & & & & & & & & & & & & \
& & & & & & & & & & & & & & & & & & & & & & & & \
& & & & & & & & & & & & & & & & & & & & & & & & \
& & & & & tikzmarknodeN1widehatN & & & & & & & & & & & & & & & & & & & \
hline
& & & & tikzmarknodeN2widehatN & & & & & & 1 & & & & & & & & & & & & & & \
& & & & & & & & & & tikzmarknode232 & & & & & & & & & & & & & & \
& & & & & & & & & & & & & & & & & & & & & & & & \
& & & & tikzmarknode222 & & & & & & & & & & & & & & & & & & & & \
& & & & 1 & & & & & & tikzmarknodeN3widehatN & & & & & & & & & & & & & & \
hline
& & & & & & & & & tikzmarknodeN4widehatN & & & & & & 1 & & & & & & & & & \
& & & & & & & & & & & & & & & tikzmarknode252 & & & & & & & & & \
& & & & & & & & & & & & & & & & & & & & & & & & \
& & & & & & & & & tikzmarknode242 & & & & & & & & & & & & & & & \
& & & & & & & & & 1 & & & & & & tikzmarknodeN5widehatN & & & & & & & & & \
hline
& & & & & & & & & & & & & & & & & bullet & & & & & & & \
& & & & & & & & & & & & & & & & & & bullet & & & & & & \
& & & & & & & & & & & & & & & bullet & & & & bullet & & & & & \
& & & & & & & & & & & & & & & & bullet & & & & & & & & \
& & & & & & & & & & & & & & & & & bullet & & & & & & & \
hline
& & & & & & & & & & & & & & & & & & & tikzmarknodeN6widehatN & & & & & \
& & & & & & & & & & & & & & & & & & & & & & & & \
& & & & & & & & & & & & & & & & & & & & & & & & \
& & & & & & & & & & & & & & & & & & & tikzmarknode262 & & & & & \
& & & & & & & & & & & & & & & & & & & 1 & & & & & \
hline
endarrayright)]
begintikzpicture[overlay,remember picture]
foreach X in 1,...,6
draw[very thick, dots=8 per 1cm,shorten >=3pt,shorten <=3pt] (2X) -- (NX);
endtikzpicture
enddocument

ADDENDUM: In an attempt to make Ryan Reich a bit happy, here is a version in which you only have to specify the nonempty entries of this sparse array. These entries are determined by the list 6/1/1,6/2/2,6/5/widehatN,... , where each entry has the structure <X>/<Y>/<entry> with X and Y specifying the position. Notice that one could shorten this further if you really have only one type of columns, but the point here is to allow to build arbitrary sparse matrices of that type. This code uses this answer.

documentclassarticle
usepackagetikz
usetikzlibrarymatrix,calc
usepackageetoolbox
tikzset% https://tex.stackexchange.com/a/156990/121799
 dots/.style args=#1per #2%
 line cap=round,
 dash pattern=on 0 off #2/#1
 

begindocument
edefmylist6/1/1,6/2/2,6/5/widehatN,%
5/6/widehatN,5/9/2,5/10/1,%
11/6/1,11/7/2,11/10/widehatN,%
10/11/widehatN,10/12/2,10/15/1,%
16/11/1,16/12/2,16/15/widehatN,%
20/21/widehatN,20/22/2,20/25/1%

begintikzpicture
 letmymatrixcontentempty
 foreach Y in 1,...,25%
 foreach X in 1,...,25
 xdeftmp
 foreach XX/YY/ZZ in mylist
 ifnumXX=X
 ifnumYY=Y
 xdeftmpZZ
 fi
 fi
 ifnumX=25
 expandaftergapptoexpandaftermymatrixcontentexpandaftertmpspace\% 
 else
 expandaftergapptoexpandaftermymatrixcontentexpandaftertmpspace&space%
 fi 
 
 
 matrix [matrix of math nodes,ampersand replacement=&,left delimiter=(,
 right delimiter=),draw,row sep=3pt] (mat) 
 mymatrixcontent
 ;
 foreach X [evaluate=X as Y using int(X+1)]in 5,10,...,20
 aux1) -- (mat.south -
 foreach X/Y [evaluate=Y as Z using int(Y+3)] in
 6/2,5/6,11/7,10/12,16/12,20/22 
 draw[very thick, dots=8 per 1cm,shorten >=3pt,shorten <=3pt]
 (mat-Y-X) -- (mat-Z-X);
 draw[line width=1.2mm, dots=4 per 1cm]
 (mat-16-18.center) -- (mat-18-20.center)
 (mat-18-16.center) -- (mat-20-18.center);
endtikzpicture
enddocument