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A Birkhoff Theorem for Doubly Stochastic Matrices with Vector Entries
Author(s) -
Clapp M. H.,
Shiflett R. C.
Publication year - 1980
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1002/sapm1980623273
Subject(s) - parallelogram , mathematics , extreme point , regular polygon , combinatorics , matrix (chemical analysis) , unit square , extension (predicate logic) , convex set , square (algebra) , square matrix , set (abstract data type) , characterization (materials science) , discrete mathematics , symmetric matrix , convex optimization , geometry , eigenvalues and eigenvectors , computer science , nanotechnology , materials science , physics , quantum mechanics , artificial intelligence , robot , composite material , programming language
The concept of a doubly stochastic matrix whose entries come from a convex subset of the unit square is defined. It is proved that the only convex subsets of the unit square which contain (0,0) and (1, 1) and allow an extension of Birkhoff's characterization of the extreme points of the set of doubly stochastic matrices are parallelograms. A sufficient condition is given for a matrix to be extreme when the convex subset is not a parallelogram.