The DAD theorem for arbitrary row sums
Author:
Richard A. Brualdi
Journal:
Proc. Amer. Math. Soc. 45 (1974), 189194
MSC:
Primary 15A48
MathSciNet review:
0354737
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Given an symmetric nonnegative matrix and a positive vector , necessary and sufficient conditions are obtained in order that there exist a diagonal matrix with positive main diagonal such that DAD has row sum vector .
 [1]
Richard
A. Brualdi, Seymour
V. Parter, and Hans
Schneider, The diagonal equivalence of a nonnegative matrix to a
stochastic matrix, J. Math. Anal. Appl. 16 (1966),
31–50. MR
0206019 (34 #5844)
 [2]
R.
A. Brualdi, Convex sets of nonnegative matrices, Canad. J.
Math. 20 (1968), 144–157. MR 0219556
(36 #2636)
 [3]
, Combinatorial properties of symmetric nonnegative matrices, Proc. Internat. Conf. 'Combinatorial Theories' held in Rome, Sept. 315, 1973 (to appear).
 [4]
J.
Csima and B.
N. Datta, The 𝐷𝐴𝐷 theorem for symmetric
nonnegative matrices, J. Combinatorial Theory Ser. A
12 (1972), 147–152. MR 0289538
(44 #6726)
 [5]
Marvin
Marcus and Henryk
Minc, A survey of matrix theory and matrix inequalities, Allyn
and Bacon, Inc., Boston, Mass., 1964. MR 0162808
(29 #112)
 [6]
Albert
W. Marshall and Ingram
Olkin, Scaling of matrices to achieve specified row and column
sums, Numer. Math. 12 (1968), 83–90. MR 0238875
(39 #235)
 [7]
M.
V. Menon, Matrix links, an extremization problem, and the reduction
of a nonnegative matrix to one with prescribed row and column sums,
Canad. J. Math. 20 (1968), 225–232. MR 0220752
(36 #3804)
 [8]
M.
V. Menon and Hans
Schneider, The spectrum of a nonlinear operator associated with a
matrix, Linear Algebra and Appl. 2 (1969),
321–334. MR 0246893
(40 #162)
 [9]
Hazel
Perfect and L.
Mirsky, The distribution of positive elements in doublystochastic
matrices, J. London Math. Soc. 40 (1965),
689–698. MR 0183021
(32 #503)
 [10]
Richard
Sinkhorn and Paul
Knopp, Concerning nonnegative matrices and doubly stochastic
matrices, Pacific J. Math. 21 (1967), 343–348.
MR
0210731 (35 #1617)
 [11]
Richard
Sinkhorn, Diagonal equivalence to matrices with
prescribed row and column sums. II, Proc. Amer.
Math. Soc. 45
(1974), 195–198. MR 0357434
(50 #9902), http://dx.doi.org/10.1090/S00029939197403574348
 [1]
 R. A. Brualdi, S. V. Parter and H. Schneider, The diagonal equivalence of a nonnegative matrix to a stochastic matrix, J. Math. Anal. Appl. 16 (1966), 3150. MR 34 #5844. MR 0206019 (34:5844)
 [2]
 R. A. Brualdi, Convex sets of nonnegative matrices, Canad. J. Math. 20 (1968), 144157. MR 36 #2636. MR 0219556 (36:2636)
 [3]
 , Combinatorial properties of symmetric nonnegative matrices, Proc. Internat. Conf. 'Combinatorial Theories' held in Rome, Sept. 315, 1973 (to appear).
 [4]
 J. Csima and B. N. Datta, The DAD theorem for symmetric nonnegative matrices, J. Combinatorial Theory Ser. A 12 (1972), 147152. MR 44 #6726. MR 0289538 (44:6726)
 [5]
 M. Marcus and H. Minc, A survey of matrix theory and matrix inequalities, Allyn and Bacon, Boston, Mass., 1964, p. 130. MR 29 #112. MR 0162808 (29:112)
 [6]
 A. W. Marshall and I. Olkin, Scaling of matrices to achieve specified row and column sums, Numer. Math. 12 ( 1968), 8390. MR 39 #235. MR 0238875 (39:235)
 [7]
 M. V. Menon, Matrix links, an extremization problem, and the reduction of a nonnegative matrix to one with prescribed row and column sums, Canad. J. Math. 20 (1968), 225232. MR 36 #3804. MR 0220752 (36:3804)
 [8]
 M. V. Menon and H. Schneider, The spectrum of a nonlinear operator associated with a matrix, Linear Algebra and Appl. 2 (1969), 321334. MR 40 #162. MR 0246893 (40:162)
 [9]
 H. Perfect and L. Mirsky, The distribution of positive elements in doublystochastic matrices, J. London Math. Soc. 40 (1965), 689698. MR 32 #503. MR 0183021 (32:503)
 [10]
 R. Sinkhorn and P. Knopp, Concerning nonnegative matrices and doubly stochastic matrices, Pacific J. Math. 21 (1967), 343348. MR 35 #1617. MR 0210731 (35:1617)
 [11]
 R. Sinkhorn, Diagonal equivalence to matrices with prescribed row and column sums. II, Proc. Amer. Math. Soc. 45 (1974), 195198. MR 0357434 (50:9902)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC:
15A48
Retrieve articles in all journals
with MSC:
15A48
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939197403547378
PII:
S 00029939(1974)03547378
Keywords:
Nonnegative matrix,
diagonal matrix,
symmetric matrix,
completely reducible,
completely decomposable,
row sum vector
Article copyright:
© Copyright 1974
American Mathematical Society
