Majorization for Changes in Angles Between Subspaces, Ritz Values, and Graph Laplacian Spectra

Abstract

Many inequality relations between real vector quantities can be succinctly expressed as "weak (sub)majorization" relations. We explain these ideas and apply them in several areas: angles between subspaces, Ritz values, and graph Laplacian spectra, which we show are all surprisingly related... An application of our Ritz values weak majorization result for Laplacian graph spectra comparison is suggested, based on the possibility to interpret eigenvalues of the edge Laplacian of a given graph as Ritz values of the edge Laplacian of the complete graph. We prove that $ \sum_k |\lambda1_k - \lambda2_k| \leq n l,$ where $\lambda1_k$ and $\lambda2_k$ are all ordered elements of the Laplacian spectra of two graphs with the same $n$ vertices and with $l$ equal to the number of differing edges.