On the cardinality of complex matrix scalings

George Hutchinson

Special Matrices (2016)

  • Volume: 4, Issue: 1, page 141-150
  • ISSN: 2300-7451

Abstract

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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.

How to cite

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George Hutchinson. "On the cardinality of complex matrix scalings." Special Matrices 4.1 (2016): 141-150. <http://eudml.org/doc/276685>.

@article{GeorgeHutchinson2016,
abstract = {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.},
author = {George Hutchinson},
journal = {Special Matrices},
keywords = {Diagonal Matrix Scalings; Positive Definite Matrices; Circulant Matrices; Doubly Stochastic Matrices; diagonal matrix scalings; positive definite matrices; circulant matrices; doubly stochastic matrices},
language = {eng},
number = {1},
pages = {141-150},
title = {On the cardinality of complex matrix scalings},
url = {http://eudml.org/doc/276685},
volume = {4},
year = {2016},
}

TY - JOUR
AU - George Hutchinson
TI - On the cardinality of complex matrix scalings
JO - Special Matrices
PY - 2016
VL - 4
IS - 1
SP - 141
EP - 150
AB - 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.
LA - eng
KW - Diagonal Matrix Scalings; Positive Definite Matrices; Circulant Matrices; Doubly Stochastic Matrices; diagonal matrix scalings; positive definite matrices; circulant matrices; doubly stochastic matrices
UR - http://eudml.org/doc/276685
ER -

References

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  1. [1] R. Sinkhorn. A relationship between arbitrary positive matrices and doubly stochastic matrices. Annals of Mathematical Statistics, 35(2):876–879, 1964. [Crossref] Zbl0134.25302
  2. [2] A.W. Marshall and I. Olkin. Scaling of matrices to achieve specified row and column sums. Numer. Math., 12(1):83–90, 1968. [Crossref] Zbl0165.17401
  3. [3] M. V. Menon. Reduction of amatrix with positive elements to a doubly stochasticmatrix. Proc. Amer.Math. Soc., 18:244–247, 1967. [Crossref] Zbl0153.05301
  4. [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. [Crossref] Zbl0231.15017
  5. [5] C. R. Johnson and R. Reams. Scaling of symmetric matrices by positive diagonal congruence. Linear Multilinear Algebra, 57:123–140, 2009. [Crossref] Zbl1166.15011
  6. [6] R. Pereira and J. Boneng. The theory and applications of complex matrix scalings, Spec. Matrices, 2: 68-77, 2014  Zbl1291.15080
  7. [7] P. J. Davis. Circulant Matrices. John Wiley & Sons, 1979.  Zbl0418.15017
  8. [8] D. P. O’Leary. Scaling symmetric positive definite matrices to prescribed row sums. Linear Algebra Appl., pages 185–191, 2003.  Zbl1038.65040

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