Editing Laplacian matrix (section)

=== Magnetic Laplacian ===
There are other situations in which entries of the adjacency matrix are complex-valued, and the Laplacian does become a [[Hermitian matrix]]. The Magnetic Laplacian for a directed graph with real weights <math>w_{ij}</math>  is constructed  as the [[Hadamard product (matrices)|Hadamard product]] of the [[Symmetric_matrix#Real_symmetric_matrices|real symmetric matrix]] of the symmetrized Laplacian and the Hermitian phase matrix with the [[complex number|complex]] entries
:<math>\gamma_q(i, j) = e^{i2 \pi q(w_{ij}-w_{ji})}</math>
which encode the edge direction into the phase in the complex plane.
In the context of quantum physics, the magnetic Laplacian can be interpreted as the operator that describes the phenomenology of a free charged particle on a graph, which is subject to the action of a magnetic field and the parameter <math>q</math>  is called electric charge.<ref>{{cite conference|title=Graph Signal Processing for Directed Graphs based on the Hermitian Laplacian | conference=ECML PKDD 2019: Machine Learning and Knowledge Discovery in Databases |pages=447–463 |year=2020|doi= 10.1007/978-3-030-46150-8_27|url=https://ecmlpkdd2019.org/downloads/paper/499.pdf |author1=Satoshi Furutani |author2=Toshiki Shibahara|author3= Mitsuaki Akiyama|author4= Kunio Hato|author5=Masaki Aida }}</ref>
In the following example <math>q=1/4</math>:
{|class="wikitable"
! [[Adjacency matrix]]
! Symmetrized Laplacian
! Phase matrix
! Magnetic Laplacian
|-
| <math display="inline">\left(\begin{array}{rrrr}
   0 &  1 &  0 &  0\\
   1 &  0 &  1 &  0\\
   0 &  0 &  0 &  0\\
   0 &  0 &  1 &  0\\
\end{array}\right)</math>
| <math display="inline">\left(\begin{array}{rrrr}
   2 &  -2 &  0 &  0\\
   -2 &  3 &  -1 &  0\\
   0 &  -1 &  2 &  -1\\
   0 &  0 &  -1 &  1\\
\end{array}\right)</math>
| <math display="inline">\left(\begin{array}{rrrr}
   1 &  1 &  1 &  1\\
   1 &  1 &  i &  1\\
   1 &  -i &  1 & -i\\
   1 &  1 &  i &  1\\
\end{array}\right)</math>
| <math display="inline">\left(\begin{array}{rrrr}
   2 & -2  &  0 & 0\\
   -2 & 3  & -i  &  0\\
  0 &   i &   2 & i\\
  0 &   0 &   -i & 1\\
\end{array}\right)</math>
|}