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Vector Majorization Via Hessenberg Matrices
Author(s) -
Brualdi Richard A.,
Hwang SukGeun
Publication year - 1996
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/53.1.28
Subject(s) - majorization , combinatorics , permutation matrix , mathematics , permutation (music) , regular polygon , matrix (chemical analysis) , physics , chemistry , geometry , chromatography , circulant matrix , acoustics
In this paper it is proved that, for real n ‐vectors x and y, x is majorized by y if and only if x = PHQ y 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 ( n 2 − n + 2)/2 permutations of y.