An interesting application of eigenvectors is for clustering data. Using the eigenvectors of a matrix derived from a distance matrix, unlabeled data can be separated into groups. Spectral clustering methods get their name from the use of the spectrum of this matrix. A distance matrix for 
elements (for example, the pairwise distance between data points) is an 
symmetric matrix. Given such an 
distance matrix 
with distance values 
, we can create the Laplacian matrix of the data points as follows:
Here, 
is the identity matrix and 
is the diagonal matrix containing the row sums of 
:
The data clusters are obtained from the eigenvectors of L. In the simplest case of data points with only two classes, the first eigenvector (that is, the one corresponding to the largest eigenvalue) is often enough to separate the data.
Here is an example of simple two-class clustering. The following code creates some 2D data points and clusters them...
