Editing Incidence structure (section)

===Incidence matrix===
{{main|Incidence matrix}}

The '''incidence matrix''' of a (finite) incidence structure is a [[(0,1) matrix]] that has its rows indexed by the points {{mvar|{p<sub>i</sub>} }} and columns indexed by the lines {{math|{''l<sub>j</sub>''} }} where the {{mvar|ij}}-th entry is a 1 if {{math|''p<sub>i</sub>'' I ''l<sub>j</sub>''}} and 0 otherwise.{{efn|The other convention of indexing the rows by lines and the columns by points is also widely used.}} An incidence matrix is not uniquely determined since it depends upon the arbitrary ordering of the points and the lines.<ref name=Beth17>{{harvnb|Beth|Jungnickel|Lenz|1986|page=17}}</ref>

The non-uniform incidence structure pictured above (example number 2) is given by:
<math display=block>\begin{align}
P &= \{ A, B, C, D, E, P \} \\[2pt]
L &= \left\{ \begin{array}{ll}
  l = \{C, P, E \}, & m = \{ P \}, \\
  n = \{ P, D \}, & o = \{ P, A \}, \\
  p = \{ A, B \}, & q = \{ P, B \} \end{array} \right\}
\end{align}</math>

An incidence matrix for this structure is:
<math display="block"> \begin{pmatrix} 
0 & 0 & 0 & 1 & 1 & 0 \\
0 & 0 & 0 & 0 & 1 & 1 \\
1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 1 & 1 & 0 & 1
\end{pmatrix} </math>
which corresponds to the incidence table:
{| class="wikitable" style="text-align:center; width=200px; height=200px;"
|-
! {{math|I}} !! {{mvar|l}} !! {{mvar|m}} !! {{mvar|n}} !! {{mvar|o}} !! {{mvar|p}} !! {{mvar|q}}
|-
! {{mvar|A}}
| 0 || 0 || 0 || 1 || 1 || 0 
|-
! {{mvar|B}}
| 0 || 0 || 0 || 0 || 1 || 1 
|-
! {{mvar|C}}
| 1 || 0 || 0 || 0 || 0 || 0
|-
! {{mvar|D}}
| 0 || 0 || 1 || 0 || 0 || 0 
|-
! {{mvar|E}}
| 1 || 0 || 0 || 0 || 0 || 0 
|-
! {{mvar|P}}
| 1 || 1 || 1 || 1 || 0 || 1 
|}

If an incidence structure {{mvar|C}} has an incidence matrix {{mvar|M}}, then the dual structure {{math|''C''<sup>∗</sup>}} has the [[transpose matrix]] {{mvar|M}}<sup>T</sup> as its incidence matrix (and is defined by that matrix).

An incidence structure is self-dual if there exists an ordering of the points and lines so that the incidence matrix constructed with that ordering is a  [[symmetric matrix]].

With the labels as given in example number 1 above and with points ordered {{math|''A'', ''B'', ''C'', ''D'', ''G'', ''F'', ''E''}} and lines ordered {{math|''l'', ''p'', ''n'', ''s'', ''r'', ''m'', ''q''}}, the Fano plane has the incidence matrix:
<math display="block"> \begin{pmatrix} 
1 & 1 & 1 & 0 & 0 & 0 & 0\\
1 & 0 & 0 & 1 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 1 & 1 \\
0 & 1 & 0 & 1 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 & 1 & 0 & 1 \\
0 & 0 & 1 & 1 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 & 1 & 1 & 0
\end{pmatrix} . </math>
Since this is a symmetric matrix, the Fano plane is a self-dual incidence structure.