Convex Sets of Non-Negative Matrices

R. A. Brualdi

doi:10.4153/CJM-1968-016-9

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In (8) M. V. Menon investigates the diagonal equivalence of a non-negative matrix A to one with prescribed row and column sums and shows that this equivalence holds provided there exists at least one non-negative matrix with these row and column sums and with zeros in exactly the same positions A has zeros. However, he leaves open the question of when such a matrix exists. W. B. Jurkat and H.J. Ryser in (7) study the convex set of all m × n non-negative matrices having given row and column sums.

Footnotes

The research of the author was supported, in part, by N.S.F. contract no. GP-3993.

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Crossref Citations

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Menon, M.V. and Schneider, Hans 1969. The spectrum of a nonlinear operator associated with a matrix. Linear Algebra and its Applications, Vol. 2, Issue. 3, p. 321.

Sinkhorn, Richard 1974. Diagonal equivalence to matrices with prescribed row and column sums. II. Proceedings of the American Mathematical Society, Vol. 45, Issue. 2, p. 195.

Brualdi, Richard A. 1974. The DAD theorem for arbitrary row sums. Proceedings of the American Mathematical Society, Vol. 45, Issue. 2, p. 189.

Brualdi, Richard A. and Gibson, Peter M. 1976. Convex polyhedra of doubly stochastic matrices—IV. Linear Algebra and its Applications, Vol. 15, Issue. 2, p. 153.

Brualdi, Richard A 1977. Convex polytopes of permutation invariant doubly stochastic matrices. Journal of Combinatorial Theory, Series B, Vol. 23, Issue. 1, p. 58.

Berger, Marc A and Kelley, C.T 1979. A variational equivalent to diagonal scaling. Journal of Mathematical Analysis and Applications, Vol. 72, Issue. 1, p. 291.

1979. Nonnegative Matrices in the Mathematical Sciences. p. 298.

Brualdi, Richard A. 1980. Matrices of zeros and ones with fixed row and column sum vectors. Linear Algebra and its Applications, Vol. 33, Issue. , p. 159.

Bapat, Ravindra 1982. D1AD2 theorems for multidimensional matrices. Linear Algebra and its Applications, Vol. 48, Issue. , p. 437.

Hartfiel, D.J. 1984. A control theory for stochastic eigenvectors of stochastic matrices through variations of entries. Linear Algebra and its Applications, Vol. 56, Issue. , p. 139.

Berman, Abraham 1984. Annals of Discrete Mathematics (20): Convexity and Graph Theory, Proceedings of the Conference on Convexity and Graph Theory. Vol. 87, Issue. , p. 55.

Joe, Harry 1985. An ordering of dependence for contingency tables. Linear Algebra and its Applications, Vol. 70, Issue. , p. 89.

Li, Chi-Kwong 1985. On the extremal solutions for capacitated network problems. Linear and Multilinear Algebra, Vol. 18, Issue. 2, p. 117.

Brualdi, Richard A. Hartfiel, D. J. and Hwang, Suk-Geun 1986. On assignment functions. Linear and Multilinear Algebra, Vol. 19, Issue. 3, p. 203.

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Rothblum, Uriel G. and Schneider, Hans 1989. Scalings of matrices which have prespecified row sums and column sums via optimization. Linear Algebra and its Applications, Vol. 114-115, Issue. , p. 737.

Schneider, Michael H. 1989. Matrix scaling, entropy minimization, and conjugate duality. I. existence conditions. Linear Algebra and its Applications, Vol. 114-115, Issue. , p. 785.

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Rothblum, Uriel G. and Zenios, Stavros A. 1992. Scalings of matrices satisfying line-product constraints and generalizations. Linear Algebra and its Applications, Vol. 175, Issue. , p. 159.

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