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.