An update on a few permanent conjectures

• Autorzy: Fuzhen Zhang
• Języki publikacji: EN

Abstrakty

EN

We review and update on a few conjectures concerning matrix permanent that are easily stated, understood, and accessible to general math audience. They are: Soules permanent-on-top conjecture†, Lieb permanent dominance conjecture, Bapat and Sunder conjecture† on Hadamard product and diagonal entries, Chollet conjecture on Hadamard product, Marcus conjecture on permanent of permanents, and several other conjectures. Some of these conjectures are recently settled; some are still open.We also raise a few new questions for future study. (†conjectures have been recently settled negatively.)

Słowa kluczowe

EN

Bapat-Sunder conjecture; Chollet conjecture; Drury conjecture; Foregger conjecture; Liang-So-Zhang conjecture; Lie conjecture; Marcus conjecture; Marcus-Minc conjecture; permanent; permanent dominanceconjecture; permanent-on-top conjecture; Schur power matrix

• Wydawca: De Gruyter Open
• Czasopismo: Special Matrices
• Rocznik: 2016
• Tom: 4
• Numer: 1
• Strony: 305-316

Daty

otrzymano

2016-03-18

zaakceptowano

2016-07-27

online

2016-08-26

Twórcy

autor

Fuzhen Zhang

• Department of Mathematics, Halmos College, Nova Southeastern University, 3301 College Ave, Fort Lauderdale, FL 33314, USA

Bibliografia

• [1] T. Ando, Inequalities for permanents, Hokkaido Math. J. 10 (1981), Special Issue, 18–36.
• [2] R.B. Bapat, Recent developments and open problems in the theory of permanents,Math. Student 76 (2007), no. 1-4, 55–69.
• [3] R.B. Bapat and V.S. Sunder, On majorization and Schur products, Linear Algebra Appl. 72 (1985) 107–117.
• [4] R.B. Bapat and V.S. Sunder, An extremal property of the permanent and the determinant, Linear Algebra Appl. 76 (1986) 153–163.
• [5] L.B. Beasley, An inequality on permanents of Hadamard products, Bull. Korean Math. Soc. 37 (2000), no. 3, 633–639.
• [6] R. Brualdi, Permanent of the direct product of matrices, Paciffic J. Math. 16 (1966) 471–482.
• [7] D.K. Chang, A note on a conjecture of T.H. Foregger, Linear Multilinear Algebra 15 (1984), no. 3-4, 341–344.
• [8] D.K. Chang, On two permanental conjectures, Linear Multilinear Algebra 26 (1990), no. 3, 207–213.
• [9] G.-S. Cheon and I.-M. Wanless, An update on Minc’s survey of open problems involving permanents, Linear Algebra Appl. 403 (2005) 314–342.
• [10] J. Chollet, Unsolved Problems: Is There a Permanental Analogue to Oppenheim’s Inequality? Amer.Math. Monthly 89 (1982), no. 1, 57–58.
• [11] D.Z. Djokovic, Simple proofs of a theorem on permanents, Glasgow Math. J. 10 (1969) 52–54.
• [12] J. Drew and C. Johnson, The maximum permanent of a 3-by-3 positive semideffinite matrix, given the eigenvalues, Linear Multilinear Algebra 25 (1989), no. 3, 243–251.
• [13] J. Drew and C. Johnson, Counterexample to a conjecture of Mehta regarding permanental maximization, Linear Multilinear Algebra 25 (1989), no. 3, 253–254.
• [14] S. Drury, A counterexample to a question of Bapat and Sunder, Electronic Journal of Linear Algebra, Volume 31, pp. 69-70 (2016).
• [15] S. Drury, A real counterexample to two inequalties involving permanents, Mathematical Inequalities & Applications, in press.
• [16] S. Drury, Two open problems on permanents, a private communication, May 30, 2016.
• [17] R.J. Gregorac and I.R. Hentzel, A note on the analogue of Oppenheim’s inequality for permanents, Linear Algebra Appl. 94 (1987) 109–112.
• [18] R. Grone, An inequality for the second immanant, Linear Multilinear Algebra 18 (1985), no. 2, 147–152.
• [19] R. Grone, C. Johnson, E. de S? and H. Wolkowicz, A note onmaximizing the permanent of a positive deffinite Hermitianmatrix, given the eigenvalues, Linear Multilinear Algebra 19 (1986), no. 4, 389–393.
• [20] R. Grone and R. Merris, Conjectures on permanents, Linear Multilinear Algebra 21 (1987), no. 4, 419–427.
• [21] R. Grone, S. Pierce and W. Watkins, Extremal correlation matrices, Linear Algebra Appl. 134 (1990) 63–70.
• [22] P. Heyfron, Immanant dominance orderings for hook partitions, Linear Multilinear Algebra 24 (1988), no. 1, 65–78.
• [23] R.A. Horn, The Hadamard product, Proceedings of Symposia in AppliedMathematics, Vol. 40, edited by C.R. Johnson, Amer. Math. Soc., Providence, RI, pp. 87–169, 1990.
• [24] G. James, Immanants, Linear Multilinear Algebra 32 (1992), no. 3-4, 197–210.
• [25] G. James and M. Liebeck, Permanents and immanants of Hermitian matrices, Proc. London Math. Soc. (3) 55 (1987), no. 2, 243–265.
• [26] C.R. Johnson, The permanent-on-top conjecture: a status report, in Current Trends in Matrix Theory, edited by F. Uhlig and R. Grone, Elsevier Science Publishing Co., New York, pp. 167–174, 1987.
• [27] C.R. Johnson and F. Zhang, Erratum: The Robertson-Taussky Inequality Revisted, Linear Multilinear Algebra 38 (1995) 281– 282.
• [28] E.H. Lieb, Proofs of some conjectures on permanents, J. Math. and Mech. 16 (1966) 127–134.
• [29] M. Marcus, Permanents of direct products, Proc. Amer. Math. Soc. 17 (1966) 226–231.
• [30] M. Marcus, Finite Dimensional Multilinear Algebra. Part I. (Pure and applied mathematics 23), Marcel Dekker, New York, 1973.
• [31] M. Marcus and H. Minc, Inequalities for general matrix functions, Bull. Amer. Math. Soc. 70 (1964) 308–313.
• [32] M. Marcus and H. Minc, Permanents, Amer. Math. Monthly 72 (1965) 577–591.
• [33] M. Marcus and M. Sandy, Bessel’s inequality in tensor space, Linear Multilinear Algebra 23 (1988), no. 3, 233–249.
• [34] R. Merris, Extensions of the Hadamard determinant theorem, Israel J. Math. 46 (1983), no. 4, 301–304.
• [35] R. Merris, The permanental dominance conjecture, Current trends in matrix theory : proceedings of the Third Auburn Matrix Theory Conference, March 19-22, 1986 at Auburn University, Auburn, Alabama, U.S.A., editors, Frank Uhlig, Robert Grone.
• [36] R. Merris, The permanental dominance conjecture, in Current Trends in Matrix Theory, edited by F. Uhlig and R. Grone, Elsevier Science Publishing Co., New York, pp. 213–223, 1987.
• [37] R. Merris, Multilinear Algebra, Gordon & Breach, Amsterdam, 1997.
• [38] R. Merris and W. Watkins, Inequalities and identities for generalized matrix functions, Linear Algebra Appl. 64 (1985) 223– 242.
• [39] H. Minc, Permanents, Addison-Wesley, New York, 1978.
• [40] H. Minc, Theory of permanents 1978-1981, Linear Multilinear Algebra 12 (1983) 227–263.
• [41] H. Minc, Theory of permanents 1982-1985, Linear Multilinear Algebra 21 (1987) 109–148.
• [42] A. Oppenheim, Inequalities Connected with Deffinite Hermitian Forms, J. London Math. Soc. S1-5, no. 2, 114–119.
• [43] T.H. Pate, An extension of an inequality involving symmetric products with an application to permanents, Linear Multilinear Algebra 10 (1981), no. 2, 103–105.
• [44] T.H. Pate, Inequalities relating groups of diagonal products in a Gram matrix, Linear Multilinear Algebra 11 (1982), no. 1, 1–17.
• [45] T.H. Pate, An inequality involving permanents of certain direct products, Linear Algebra Appl. 57 (1984) 147–155.
• [46] T.H. Pate, Permanental dominance and the Soules conjecture for certain right ideals in the group algebra, LinearMultilinear Algebra 24 (1989), no. 2, 135–149.
• [47] T.H. Pate, Partitions, irreducible characters, and inequalities for generalized matrix functions, Trans. Amer. Math. Soc. 325 (1991), no. 2, 875–894.
• [48] T.H. Pate, Rowappendingmaps, ζ -functions, and immanant inequalities for Hermitian positive semi-deffinitematrices, Proc. London Math. Soc. (3) 76 (1998), no. 2, 307–358.
• [49] T.H. Pate, Tensor inequalities, ζ -functions and inequalities involving immanants, Linear Algebra Appl. 295 (1999), no. 1-3, 31–59.
• [50] S. Pierce, Permanents of correlation matrices, in Current Trends in Matrix Theory, edited by F. Uhlig and R. Grone, Elsevier Science Publishing Co., New York, pp. 247–249, 1987.
• [51] I. Schur, Über endliche Gruppen und Hermitesche Formen (German), Math. Z. 1 (1918), no. 2-3, 184–207.
• [52] V.S. Shchesnovich, The permanent-on-top conjecture is false, Linear Algebra Appl. 490 (2016) 196–201.
• [53] G. Soules,Matrix functions and the Laplace expension theorem, Ph.D. Dissertation, University of California - Santa Barbara, July, 1966.
• [54] G. Soules, An approach to the permanental-dominance conjecture, Linear Algebra Appl. 201 (1994) 211–229.
• [55] R.C. Thompson, A determinantal inequality, Canad. Math. Bull., vol.4, no.1, Jan. 1961, pp. 57–62.
• [56] J.H. van Lint and R.M. Wilson, A Course in Combinatorics, Cambridge University Press, 2001.
• [57] F. Zhang, Notes on Hadamard products of matrices, Linear Multilinear Algebra 25 (1989) 237–242.
• [58] F. Zhang, An analytic approach to a permanent conjecture, Linear Algebra Appl. 438 (2013) 1570–1579.
• [59] F. Zhang, J. Liang, and W. So, Two conjectures on permanents in Report on Second Conference of the International Linear Algebra Society (Lisbon, 1992) by J.A.Dias da Silva, Linear Algebra Appl. 197/198 (1994) 791–844.

Identyfikatory

DOI

10.1515/spma-2016-0030