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Majorization via generalized Hessenberg matrices

โœ Scribed by Suk-Geun Hwang

Book ID
104156587
Publisher
Elsevier Science
Year
1999
Tongue
English
Weight
157 KB
Volume
292
Category
Article
ISSN
0024-3795

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โœฆ Synopsis

In this paper we give a method of constructing generalized Hessenberg matrices from those of smaller orders, with a simple combinatorial justiยฎcation. Making use of this construction, we also prove that, for real n-vectors x and y whose components are arranged in nonincreasing order, x is majorized by y if and only if there exists a doubly stochastic matrix e of order n with x ey whose support is permutation similar to a direct sum of generalized Hessenberg matrices.

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It is well known that for real n-vectors y and x, y majorizes x if and only if Ay = x for some doubly stochastic matrix A of order n. If the components of each of y and x are in nonincreasing order, then it is known that the mat~x A can be chosen to be positive semidefinite symmetric. We characteriz