An important application of principal component analysis is "low-rank approximation of matrices"; very roughly, this can be used for "dimension reduction": Substituting a system of hundreds of thousands of equations by a system with only hundreds.

This is neatly illustrated by applying the technique to compress images, even if all the usual compression techniques actually employ vastly different methods. An interactive demo is here: https://timbaumann.info/svd-image-compression-demo/