principal component analysis

Principal component analysis (PCA) is not a new decomposition but an application of the ones you already know. Given a cloud of data points, it finds the directions along which the data varies the most: the first principal component is the single direction of greatest spread, the second is the direction of next-greatest spread perpendicular to the first, and so on. Mathematically these directions are the top eigenvectors of the data's covariance matrix, equivalently the top singular vectors from an SVD.

The practical payoff is compression and visualization. If most of the variation in a 50-dimensional dataset lives in just two or three directions, you can keep those few principal components and throw the rest away, plotting high-dimensional data on a flat page or shrinking it before feeding it to another algorithm.

Be honest about what PCA does and does not do. It simply rotates your coordinate axes to line up with the directions of highest variance; it does not understand the meaning of your data. High variance is not the same as importance, the components can be hard to interpret, and PCA only captures linear structure, so curved or clustered patterns can be missed entirely.