Spectral decomposition is pretty cool.
Application 1 - spectral clustering - an alternative to k-means for nonlinear clusters. Get a Distance matrix of your data, spectral decomp, run k-means on your k top eigen vectors and that's your clusters.
Application 2 - graph clustering - (run spectral clustering on adj matrix!)
There's some tricks to getting it to work in practice like normalizing but it's a simple and powerful method. Also the matrices can get big so it helps a lot to use sparse matrix libraries for the computations.
[1] https://towardsdatascience.com/spectral-clustering-aba2640c0....