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 .