Generalized doubly stochastic matrices and linear preservers

A real or complex n×n matrix is generalized doubly stochastic if all of its row sums and column sums equal one. Denote by the linear space spanned by such matrices. We study the reducibility of under the group Γ of linear operators of the form A↦PAQ, where P and Q are n×n permutation matrices. Using...

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Bibliographic Details
Published inLinear & multilinear algebra Vol. 53; no. 1; pp. 1 - 11
Main Authors Chiang †, Hanley, Li, Chi-Kwong
Format Journal Article
LanguageEnglish
Published Taylor & Francis Group 01.01.2005
Subjects
AMS Subject Classifications: 15A04
Linear maps
Permutation matrices
Symmetric doubly stochastic matrices
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ISSN0308-1087
1563-5139
DOI10.1080/03081080410001681599

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Summary:A real or complex n×n matrix is generalized doubly stochastic if all of its row sums and column sums equal one. Denote by the linear space spanned by such matrices. We study the reducibility of under the group Γ of linear operators of the form A↦PAQ, where P and Q are n×n permutation matrices. Using this result, we show that every linear operator mapping the set of generalized doubly stochastic matrices into itself is a linear combination of the operators in Γ followed by a translation of a fixed matrix in . We compare our results with those from related studies by Sinkhorn and Benson. We also consider similar problems for the generalized symmetric doubly stochastic matrices.
ISSN:0308-1087
1563-5139
DOI:10.1080/03081080410001681599