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

top
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

top

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

top
  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. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.