Majorization

Summary

Majorization compares two vectors of real numbers with the same sum by how spread out their entries are. A vector $x$ majorizes $y$ when, for each $k$, the sum of the $k$ largest entries of $x$ is at least that of $y$. Equivalently, $y$ is obtained from $x$ by a doubly stochastic matrix, that is, by averaging. It organises many inequalities, including the Schur–Horn theorem on diagonals and eigenvalues of Hermitian matrices.