Majorization
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 
.
REFERENCES:
Bhatia, R. Matrix Analysis. New York: Springer-Verlag, 1997.
Horn, R. A. and Johnson, C. R. Matrix Analysis, Repr. with Corrections. Cambridge, England: Cambridge University Press, 1987.
Marshall, A. W. and Olkin, I. Inequalities: The Theory of Majorizations and Its Applications. New York: Academic Press, 1979.
Nielsen, M. A. "Conditions for a Class of Entanglement Transformations." Phys. Rev. Lett. 83, 436-439, 1999.
