Let and be nonincreasing sequences of real numbers. Then majorizes if, for each , 2, ..., ,
with equality if . Note that some caution is needed when consulting the literature, since the direction of the inequality is not consistent from reference to reference. An order-free characterization along the lines of Horn's theorem is also readily available.
majorizes iff there exists a doubly stochastic matrix such that . Intuitively, if majorizes , then is more "mixed" than . Horn's theorem relates the eigenvalues of a Hermitian matrix to its diagonal entries using majorization. Given two vectors , then majorizes iff there exists a Hermitian matrix with eigenvalues and diagonal entries .