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Convex Sets of Non-Negative Matrices

  • R. A. Brualdi (a1)

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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.

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References

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1. Berge, C., The theory of graphs (New York, 1964).
2. Birkhoff, G. D., Lattice theory (Amer. Math. Soc. Colloq. PubL, Vol. xxv, rev. éd., 1948).
3. Ford, L. R. and Fulkerson, D. R., Flows in networks (Princeton, 1962).
4. Fulkerson, D. R., Hitchcock transportation problem, Rand Corp. Report, P-890 (July 1956).
5. Harary, F., Norman, R. Z., and Cartwright, D., Structural models (New York, 1965).
6. Horn, A., Doubly stochastic matrices and the diagonal of a rotation matrix, Amer. J. Math., 76 (1954), 620630.
7. Jurkat, W. B. and H.J. Ryser, Term ranks and permanents of nonnegative matrices (to appear).
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).
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