# What is the difference between affinity matrix eigenvectors and graph Laplacian eigenvectors in the context of spectral clustering?

In spectral clustering, it's standard practice to solve the eigenvector problem

$$L v = \lambda v$$

where $L$ is the graph Laplacian, $v$ is the eigenvector related to eigenvalue $\lambda$.

My question: why bother taking the graph Laplacian? Couldn't I just solve the eigenvector problem for the graph (affinity matrix) itself, like the guy did in this video?

• How would you use PCA on a graph?!? – Anony-Mousse Nov 28 '15 at 12:55
• I meant PCA on the affinity matrix (a square matrix with pairwise affinities) representing a graph, just like that video shows. – felipeduque Nov 28 '15 at 13:51
• You mean singular value decomposition then? PCA is computing the covariance matrix, then decomposing it. – Anony-Mousse Nov 28 '15 at 16:33
• You're completely right. My bad. Just edited. – felipeduque Nov 28 '15 at 16:47