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Majorization

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)^kx_i>=sum_(i=1)^ky_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.

x majorizes y iff there exists a doubly stochastic matrix P such that y=Px. Intuitively, if x majorizes y, then y is more "mixed" than x. Horn's theorem relates the eigenvalues of a Hermitian matrix A to its diagonal entries using majorization. Given two vectors lambda,v in R^n, then lambda majorizes v iff there exists a Hermitian matrix A with eigenvalues lambda_i and diagonal entries v_i.

See also

Birkhoff's Theorem, Doubly Stochastic Matrix, Horn's Theorem

Portions of this entry contributed by Serge Belongie

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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.

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Majorization

Cite this as:

Weisstein, Eric W., with contributions by Serge Belongie. "Majorization." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Majorization.html

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