Editing Laplacian matrix (section)

===Symmetric Laplacian for a directed graph===
The Laplacian matrix of a directed graph is by definition generally non-symmetric, while, e.g., traditional [[spectral clustering]] is primarily developed for undirected graphs with symmetric adjacency and Laplacian matrices. A trivial approach to applying techniques requiring the symmetry is to turn the original directed graph into an undirected graph and build the Laplacian matrix for the latter. 

In the matrix notation, the adjacency matrix of the undirected graph could, e.g., be defined as a [[OR gate|Boolean sum]] of the adjacency matrix <math>A</math> of the original directed graph and its [[matrix transpose]] <math>A^T</math>, where the zero and one entries of <math>A</math> are treated as logical, rather than numerical, values, as in the following example:
{|class="wikitable"
! [[Adjacency matrix]]
! Symmetrized adjacency
! Symmetric Laplacian matrix
|-
| <math display="inline">\left(\begin{array}{ccc}
  0 &  1 &  1\\
  0 &  0 &  1\\
  1 &  0 &  0\\
\end{array}\right)</math>
| <math display="inline">\left(\begin{array}{ccc}
  0 &  1 &  1\\
  1 &  0 &  1\\
  1 &  1 &  0\\
\end{array}\right)</math>
| <math display="inline">\left(\begin{array}{ccc}
   2 &  -1 &  -1\\
  -1 &   2 &  -1\\
  -1 &  -1 &   2\\
\end{array}\right)</math>
|}