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BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access February 22, 2016

On the cardinality of complex matrix scalings

  • George Hutchinson
From the journal Special Matrices


We disprove a conjecture made by Rajesh Pereira and Joanna Boneng regarding the upper bound on the number of doubly quasi-stochastic scalings of an n × n positive definite matrix. In doing so, we arrive at the true upper bound for 3 × 3 real matrices, and demonstrate that there is no such bound when n ≥ 4.


[1] R. Sinkhorn. A relationship between arbitrary positive matrices and doubly stochastic matrices. Annals of Mathematical Statistics, 35(2):876–879, 1964. 10.1214/aoms/1177703591Search in Google Scholar

[2] A.W. Marshall and I. Olkin. Scaling of matrices to achieve specified row and column sums. Numer. Math., 12(1):83–90, 1968. 10.1007/BF02170999Search in Google Scholar

[3] M. V. Menon. Reduction of amatrix with positive elements to a doubly stochasticmatrix. Proc. Amer.Math. Soc., 18:244–247, 1967. 10.1090/S0002-9939-1967-0215873-6Search in Google Scholar

[4] R. Brualdi, S. Parter, and H. Schneider. The diagonal equivalence of a non-negative matrix to a stochastic matrix. J. Math. Anal. Appl., 16:31–50, 1966. 10.1016/0022-247X(66)90184-3Search in Google Scholar

[5] C. R. Johnson and R. Reams. Scaling of symmetric matrices by positive diagonal congruence. Linear Multilinear Algebra, 57:123–140, 2009. 10.1080/03081080600872327Search in Google Scholar

[6] R. Pereira and J. Boneng. The theory and applications of complex matrix scalings, Spec. Matrices, 2: 68-77, 2014 10.2478/spma-2014-0007Search in Google Scholar

[7] P. J. Davis. Circulant Matrices. John Wiley & Sons, 1979. Search in Google Scholar

[8] D. P. O’Leary. Scaling symmetric positive definite matrices to prescribed row sums. Linear Algebra Appl., pages 185–191, 2003. 10.1016/S0024-3795(03)00387-2Search in Google Scholar

Received: 2015-11-13
Accepted: 2016-1-30
Published Online: 2016-2-22

©2016 George Hutchinson

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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