Convex Sets of Non-Negative Matrices

R. A. Brualdi

doi:10.4153/CJM-1968-016-9

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

References

1. Berge, C., The theory of graphs (New York, 1964).Google Scholar

2. Birkhoff, G. D., Lattice theory (Amer. Math. Soc. Colloq. PubL, Vol. xxv, rev. éd., 1948).Google Scholar

3. Ford, L. R. and Fulkerson, D. R., Flows in networks (Princeton, 1962).Google Scholar

4. Fulkerson, D. R., Hitchcock transportation problem, Rand Corp. Report, P-890 (July 1956).Google Scholar

5. Harary, F., Norman, R. Z., and Cartwright, D., Structural models (New York, 1965).Google Scholar

6. Horn, A., Doubly stochastic matrices and the diagonal of a rotation matrix, Amer. J. Math., 76 (1954), 620–630.Google Scholar

7. Jurkat, W. B. and H.J. Ryser, Term ranks and permanents of nonnegative matrices (to appear).Google Scholar

8. Menon, M. V., Matrix links, an extremization problem and the reduction of a nonnegative matrix to one with prescribed row and column sums, M.R.C. Report No. 651 (May 1966).Google Scholar

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.

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

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. 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.

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

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.

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.

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

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

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

Hartfiel, D. J. 1987. Fully indecomposable and nearly decomposable matrices in Ω(r,s). Linear and Multilinear Algebra, Vol. 21, Issue. 3, p. 231.

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.

Grza̧ślewicz, Ryszard 1990. Approximation theorems for positive operators on Lp-spaces. Journal of Approximation Theory, Vol. 63, Issue. 2, p. 123.

Loewy, R. Shier, D.R. and Johnson, C.R. 1991. Perron eigenvectors and the symmetric transportation polytope. Linear Algebra and its Applications, Vol. 150, Issue. , p. 139.

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.

Download full list

Article contents

Convex Sets of Non-Negative Matrices

Extract

Footnotes

References

Save article to Kindle

Save article to Dropbox

Save article to Google Drive

Reply to: Submit a response

Your details

You have entered the maximum number of contributors

Conflicting interests