Nonnegative matrices each of whose positive diagonals has the same sum

Author:
Mark Blondeau Hedrick

Journal:
Proc. Amer. Math. Soc. **59** (1976), 399-403

MSC:
Primary 15A48

MathSciNet review:
0414595

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Abstract: The author shows that if is a fully indecomposable nonnegative matrix each of whose positive diagonals has sum and when , the sum of each positive diagonal in the submatrix of obtained by deleting the row and column is less than , then there is a unique positive matrix such that its rank is at most two, each of its diagonals has sum , and when . The author then compares his results to those obtained by Sinkhorn and Knopp who carried out a similar analysis for positive diagonal products.

**[1]**Darald Hartfiel,*An inequality concerning a matrix function*(oral communication).**[2]**Marvin Marcus and Henryk Minc,*A survey of matrix theory and matrix inequalities*, Allyn and Bacon, Inc., Boston, Mass., 1964. MR**0162808****[3]**Richard Sinkhorn and Paul Knopp,*Problems involving diagonal products in nonnegative matrices*, Trans. Amer. Math. Soc.**136**(1969), 67–75. MR**0233830**, 10.1090/S0002-9947-1969-0233830-7**[4]**Richard Sinkhorn and Paul Knopp,*Concerning nonnegative matrices and doubly stochastic matrices*, Pacific J. Math.**21**(1967), 343–348. MR**0210731**

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1976-0414595-1

Keywords:
Fully indecomposable,
doubly stochastic pattern,
elementary symmetric function of a diagonal

Article copyright:
© Copyright 1976
American Mathematical Society