  Spectral clustering is a popular algorithm that clusters points using the
eigenvalues and eigenvectors of Laplacian matrices derived from the data. For
years, spectral clustering has been working mysteriously. This paper explains
spectral clustering by dividing it into two categories based on whether the
graph Laplacian is fully connected or not. For a fully connected graph, this
paper demonstrates the dimension reduction part by offering an objective
function: the covariance between the original data points' similarities and the
mapped data points' similarities. For a multi-connected graph, this paper
proves that with a proper $k$, the first $k$ eigenvectors are the indicators of
the connected components. This paper also proves there is an equivalence
between spectral embedding and PCA.
