The theory and applications of complex matrix scalings

Rajesh Pereira; Joanna Boneng

Special Matrices (2014)

  • Volume: 2, Issue: 1, page 68-77
  • ISSN: 2300-7451

Abstract

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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.

How to cite

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Rajesh Pereira, and Joanna Boneng. "The theory and applications of complex matrix scalings." Special Matrices 2.1 (2014): 68-77. <http://eudml.org/doc/266983>.

@article{RajeshPereira2014,
abstract = {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.},
author = {Rajesh Pereira, Joanna Boneng},
journal = {Special Matrices},
keywords = {Diagonal Matrix Scalings; Positive Definite Matrices; Circulant Matrices; Tridiagonal Matrices; Doubly Stochastic Matrices; Permanent; Symmetric States; Geometric Measure of Entanglement; diagonal matrix scalings; positive definite matrices; circulant matrices; tridiagonal matrices; doubly stochastic matrices; permanent; symmetric states; geometric measure of entanglement},
language = {eng},
number = {1},
pages = {68-77},
title = {The theory and applications of complex matrix scalings},
url = {http://eudml.org/doc/266983},
volume = {2},
year = {2014},
}

TY - JOUR
AU - Rajesh Pereira
AU - Joanna Boneng
TI - The theory and applications of complex matrix scalings
JO - Special Matrices
PY - 2014
VL - 2
IS - 1
SP - 68
EP - 77
AB - 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.
LA - eng
KW - Diagonal Matrix Scalings; Positive Definite Matrices; Circulant Matrices; Tridiagonal Matrices; Doubly Stochastic Matrices; Permanent; Symmetric States; Geometric Measure of Entanglement; diagonal matrix scalings; positive definite matrices; circulant matrices; tridiagonal matrices; doubly stochastic matrices; permanent; symmetric states; geometric measure of entanglement
UR - http://eudml.org/doc/266983
ER -

References

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  1. [1] I. Bengtsson and K. Zyczkowski. Geometry of Quantum States. Cambridge University Press, 2006. Zbl1146.81004
  2. [2] P. J. Davis. Circulant Matrices. John Wiley & Sons, 1979. Zbl0418.15017
  3. [3] P. J. Davis and I. Najfeld. Equisum matrices and their permanence. Quart. Appl. Math., 58(1):151-169, 2000. Zbl1036.15018
  4. [4] G. P. Egorychev. The solution of van der Waerden’s problem for permanents. Adv. Math., 42:299-305, 1981. Zbl0478.15003
  5. [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. Zbl0475.15007
  6. [6] G. Hardy, J.E. Littlewood, and G. Polya. Inequalities. Cambridge Mathematical Library, 1952. 
  7. [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. [8] C. R. Johnson and R. Reams. Scaling of symmetric matrices by positive diagonal congruence. Linear Multilinear Algebra, 57:123-140, 2009. Zbl1166.15011
  9. [9] M. Marcus. Subpermanents. Amer. Math. Monthly, 76:530-533, 1969. 
  10. [10] A. W. Marshall and I. Olkin. Scaling of matrices to achieve specified row and column sums. Numer. Math., 12(1):83-90, 1968. Zbl0165.17401
  11. [11] H. Minc. Permanents. Addison-Wesley Publishing Co., 1978. 
  12. [12] R. Pereira. Differentiators and the geometry of polynomials. Journal of Mathematical Analysis and Applications, 285(1):336-348, 2003. Zbl1046.47002
  13. [13] A. Pinkus. Interpolation by matrices. Electron. J. Linear Algebra, 11:281-291, 2004. Zbl1069.15012
  14. [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. [15] R. Sinkhorn. A relationship between arbitrary positive matrices and doubly stochastic matrices. Annals of Mathematical Statistics, 35(2):876-879, 1964.[Crossref] Zbl0134.25302
  16. [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. 

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