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Journal of the London Mathematical Society 1996 53(1):28-38; doi:10.1112/jlms/53.1.28
© 1996 by London Mathematical Society
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© The London Mathematical Society

Vector Majorization Via Hessenberg Matrices

Richard A. Brualdi and Suk-Geun Hwang

Department of Mathematics, University of Wisconsin Madison, WI 53706, USA
Department of Mathematics Education, Kyungpook University Taegu 702-701, Republic of Korea

Received 12 April 1994.

In this paper it is proved that, for real n-vectors x and y, x is majorized by y if and only if x = PHQy for some permutation matrices P, Q, and for some doubly stochastic matrix H which is a direct sum of doubly stochastic Hessenberg matrices. This result reveals that any n-vector which is majorized by a vector y can be expressed as a convex combination of at most (n2 – n + 2)/2 permutations of y.

First author's research partially supported by NSF grant DMS-9123318. Second author supported by a grant from TGRC-Kosef under the international cooperation program. This paper was completed while the second author was visiting the University of Wisconsin at Madison.


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