The Laplacian Matrix

The Laplacial matrix is significant because the Dirichlet problem uses it. More information will be posted when more information is understood with regards to why it is needed. This is a just a step towards the complete post explaining the Dirichlet problem.

So, The Laplacian matrix is a representation of a graph. It can represent both directed and undirected graphs but directed graphs are outside the scope of this project so they will not be discussed. Mathematically, the Laplacian matrix is the difference of two matrices, the degree matrix and the adjacency matrix.

The degree matrix is a diagonal nxn matrix that holds information about the degree of each node (vertex) in the graph. The prerequisite for the implementation of this matrix is that the graph must be labelled. Taking node with label i, we write its degree in the location aii of the degree

 matrix. 

[The degree of a vertex is the number of edges adjacent to it.]

The following image (taken from Wikipedia) demonstrates a degree matrix.

Now, the adjacency matrix is a representation of all the adjacent vertices of a

 vertex. It is also an nxn matrix. Element aij is 1 if vertices i and j are adjacent, otherwise it is 0. Therefore, an adjacency matrix must always be symmetrical.  The following image (also from Wikipedia) shows a graph and its respective adjacency matrix.

So, in order to complete the Laplacian matrix, we subtract the adjacency matrix from the degree matrix. The following image (again, Wikipedia) shows the Laplacian matrix for the same graph used in the above 2 images.

This vector can be directly achieved using the following definition: