Nonnegative matrices each of whose positive diagonals has the same sum
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- by Mark Blondeau Hedrick PDF
- Proc. Amer. Math. Soc. 59 (1976), 399-403 Request permission
Abstract:
The author shows that if $A$ is a fully indecomposable nonnegative matrix each of whose positive diagonals has sum $M$ and when ${a_{ij}} = 0$, the sum of each positive diagonal in the submatrix of $A$ obtained by deleting the $i{\text {th}}$ row and $j{\text {th}}$ column is less than $M$, then there is a unique positive matrix $B$ such that its rank is at most two, each of its diagonals has sum $M$, and ${a_{ij}} = {b_{ij}}$ when ${a_{ij}} > 0$. The author then compares his results to those obtained by Sinkhorn and Knopp who carried out a similar analysis for positive diagonal products.References
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Darald Hartfiel, An inequality concerning a matrix function (oral communication).
- Marvin Marcus and Henryk Minc, A survey of matrix theory and matrix inequalities, Allyn and Bacon, Inc., Boston, Mass., 1964. MR 0162808
- Richard Sinkhorn and Paul Knopp, Problems involving diagonal products in nonnegative matrices, Trans. Amer. Math. Soc. 136 (1969), 67–75. MR 233830, DOI 10.1090/S0002-9947-1969-0233830-7
- Richard Sinkhorn and Paul Knopp, Concerning nonnegative matrices and doubly stochastic matrices, Pacific J. Math. 21 (1967), 343–348. MR 210731
Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 59 (1976), 399-403
- MSC: Primary 15A48
- DOI: https://doi.org/10.1090/S0002-9939-1976-0414595-1
- MathSciNet review: 0414595