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2014 | 2 | 1 | 68-77
Tytuł artykułu

The theory and applications of complex matrix scalings

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We generalize the theory of positive diagonal scalings of real positive definite matrices to complex diagonal scalings of complex positive definite matrices. A matrix A is a diagonal scaling of a positive definite matrix M if there exists an invertible complex diagonal matrix D such that A = D*MD and where every row and every column of A sums to one. We look at some of the key properties of complex diagonal scalings and we conjecture that every n by n positive definite matrix has at most 2n−1 scalings and prove this conjecture for certain special classes of matrices.We also use the theory of complex diagonal matrix scalings to formulate a van der Waerden type question on the permanent function; we show that the solution of this question would have applications to finding certain maximally entangled quantum states.
Wydawca
Czasopismo
Rocznik
Tom
2
Numer
1
Strony
68-77
Opis fizyczny
Daty
wydano
2014-01-01
otrzymano
2013-12-05
zaakceptowano
2014-04-13
online
2014-05-17
Twórcy
  • Department of Mathematics & Statistics, University of Guelph, Guelph, ON, Canada N1G 2W1, pereirar@uoguelph.ca
  • Department of Mathematics & Statistics, University of Guelph, Guelph, ON, Canada N1G 2W1
Bibliografia
  • [1] I. Bengtsson and K. Zyczkowski. Geometry of Quantum States. Cambridge University Press, 2006.
  • [2] P. J. Davis. Circulant Matrices. John Wiley & Sons, 1979.
  • [3] P. J. Davis and I. Najfeld. Equisum matrices and their permanence. Quart. Appl. Math., 58(1):151-169, 2000.
  • [4] G. P. Egorychev. The solution of van der Waerden’s problem for permanents. Adv. Math., 42:299-305, 1981.
  • [5] D. I. Falikman. A proof of the van der Waerden conjecture on the permanent of a doubly stochastic matrix. Mat. Zametki, 29:931-938, 1981.
  • [6] G. Hardy, J.E. Littlewood, and G. Polya. Inequalities. Cambridge Mathematical Library, 1952.
  • [7] R. Hubener, M. Kleinmann, T. Wei, C. Gonzalez-Guillen, and O. Guhne. Geometric measure of entanglement for symmetric states. Phys. Rev. A., 80:032324, 2009.
  • [8] C. R. Johnson and R. Reams. Scaling of symmetric matrices by positive diagonal congruence. Linear Multilinear Algebra, 57:123-140, 2009.
  • [9] M. Marcus. Subpermanents. Amer. Math. Monthly, 76:530-533, 1969.
  • [10] A. W. Marshall and I. Olkin. Scaling of matrices to achieve specified row and column sums. Numer. Math., 12(1):83-90, 1968.
  • [11] H. Minc. Permanents. Addison-Wesley Publishing Co., 1978.
  • [12] R. Pereira. Differentiators and the geometry of polynomials. Journal of Mathematical Analysis and Applications, 285(1):336-348, 2003.
  • [13] A. Pinkus. Interpolation by matrices. Electron. J. Linear Algebra, 11:281-291, 2004.
  • [14] A. Shimony. Degree of entanglement. In D.M. Greenberger and A. Zeilinger, editors, Fundamental problems in quantum theory. A conference held in honor of Professor John A. Wheeler. Proceedings of the conference held in Baltimore, MD, June 18-22, 1994, volume 755 of Annals of the New York Academy of Sciences, pages 675-679, New York, 1995. New York Academy of Sciences.
  • [15] R. Sinkhorn. A relationship between arbitrary positive matrices and doubly stochastic matrices. Annals of Mathematical Statistics, 35(2):876-879, 1964.[Crossref]
  • [16] T. Wei and P.M. Goldbart. Geometric measure of entanglement and applications to bipartite and multipartite quantum states. Phys Rev. A., 68:042307, 2003.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_spma-2014-0007
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