The theory and applications of complex matrix scalings

• Autorzy: Rajesh Pereira , Joanna Boneng
• 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.

Słowa kluczowe

EN

Diagonal Matrix Scalings; Positive Definite Matrices; Circulant Matrices; Tridiagonal Matrices; Doubly Stochastic Matrices; Permanent; Symmetric States; Geometric Measure of Entanglement

• Wydawca: De Gruyter Open
• Czasopismo: Special Matrices
• Rocznik: 2014
• Tom: 2
• Numer: 1
• Strony: 68-77

Daty

wydano

2014-01-01

otrzymano

2013-12-05

zaakceptowano

2014-04-13

online

2014-05-17

Twórcy

autor

Rajesh Pereira

• Department of Mathematics & Statistics, University of Guelph, Guelph, ON, Canada N1G 2W1,

autor

Joanna Boneng

• 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.

Identyfikatory

DOI

10.2478/spma-2014-0007