# The theory and applications of complex matrix scalings

Special Matrices (2014)

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

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topRajesh 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] I. Bengtsson and K. Zyczkowski. Geometry of Quantum States. Cambridge University Press, 2006. Zbl1146.81004
- [2] P. J. Davis. Circulant Matrices. John Wiley & Sons, 1979. Zbl0418.15017
- [3] P. J. Davis and I. Najfeld. Equisum matrices and their permanence. Quart. Appl. Math., 58(1):151-169, 2000. Zbl1036.15018
- [4] G. P. Egorychev. The solution of van der Waerden’s problem for permanents. Adv. Math., 42:299-305, 1981. Zbl0478.15003
- [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] 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. Zbl1166.15011
- [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. Zbl0165.17401
- [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. Zbl1046.47002
- [13] A. Pinkus. Interpolation by matrices. Electron. J. Linear Algebra, 11:281-291, 2004. Zbl1069.15012
- [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] Zbl0134.25302
- [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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