Let x = (x_1, x_2, ..., x_n) and y = (y_1, y_2, ..., y_n) be nonincreasing sequences of real numbers. Then x majorizes y if, for each k = 1, 2, ..., n, sum_(i = 1)^k x_i>= sum_(i = 1)^k y_i, with equality if k = n. 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.

Birkhoff's theorem | doubly stochastic matrix | Horn's theorem