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 
.
