𝑞-Deformation of the Kac-Sylvester tridiagonal matrix

van Diejen, J.

doi:10.1090/proc/15413

$q$-Deformation of the Kac-Sylvester tridiagonal matrix
HTML articles powered by AMS MathViewer

by J. F. van Diejen PDF

Proc. Amer. Math. Soc. 149 (2021), 2291-2304 Request permission

Abstract:

Upon replacing integers by $q$-integers, one arrives at a natural $q$-deformation of a remarkably ubiquitous tridiagonal integer matrix whose eigenvalues and eigenvectors were first computed by J.J. Sylvester and M. Kac, respectively. The present note computes the eigenvalues and eigenvectors of this novel $q$-deformed Kac-Sylvester matrix through a connection with Macdonald’s hyperoctahedral Hall-Littlewood polynomials. This yields the spectrum as a minimizer of an associated Morse function reminiscent of Stieltjes’ electrostatic potential for the roots of classical orthogonal polynomials.

References

E. A. Abdel-Rehim, From the Ehrenfest model to time-fractional stochastic processes, J. Comput. Appl. Math. 233 (2009), no. 2, 197–207. MR 2568517, DOI 10.1016/j.cam.2009.07.010
Richard Askey, Evaluation of Sylvester type determinants using orthogonal polynomials, Advances in analysis, World Sci. Publ., Hackensack, NJ, 2005, pp. 1–16. MR 2184920
Roberto Bevilacqua and Enrico Bozzo, The Sylvester-Kac matrix space, Linear Algebra Appl. 430 (2009), no. 11-12, 3131–3138. MR 2517864, DOI 10.1016/j.laa.2009.01.029
M. Bruschi, F. Calogero, and R. Droghei, Proof of certain Diophantine conjectures and identification of remarkable classes of orthogonal polynomials, J. Phys. A 40 (2007), no. 14, 3815–3829. MR 2325078, DOI 10.1088/1751-8113/40/14/005
Wenchang Chu, Fibonacci polynomials and Sylvester determinant of tridiagonal matrix, Appl. Math. Comput. 216 (2010), no. 3, 1018–1023. MR 2607010, DOI 10.1016/j.amc.2010.01.089
Wenchang Chu, Spectrum and eigenvectors for a class of tridiagonal matrices, Linear Algebra Appl. 582 (2019), 499–516. MR 3996361, DOI 10.1016/j.laa.2019.08.017
Wenchang Chu and Xiaoyuan Wang, Eigenvectors of tridiagonal matrices of Sylvester type, Calcolo 45 (2008), no. 4, 217–233. MR 2473872, DOI 10.1007/s10092-008-0153-4
Paul A. Clement, A class of triple-diagonal matrices for test purposes, SIAM Rev. 1 (1959), 50–52. MR 99750, DOI 10.1137/1001006
J. F. van Diejen, Gradient system for the roots of the Askey-Wilson polynomial, Proc. Amer. Math. Soc. 147 (2019), no. 12, 5239–5249. MR 4021083, DOI 10.1090/proc/14625
J. F. van Diejen, Deformation of Wess-Zumino-Witten fusion rules from open $q$-boson models with diagonal boundary conditions, J. Phys. A 53 (2020), no. 27, 274002, 20. MR 4117188, DOI 10.1088/1751-8121/ab977f
J. F. van Diejen and E. Emsiz, Solutions of convex Bethe ansatz equations and the zeros of (basic) hypergeometric orthogonal polynomials, Lett. Math. Phys. 109 (2019), no. 1, 89–112. MR 3897593, DOI 10.1007/s11005-018-1101-0
Jan Felipe van Diejen, Erdal Emsiz, and Ignacio Nahuel Zurrián, Completeness of the Bethe Ansatz for an open $q$-boson system with integrable boundary interactions, Ann. Henri Poincaré 19 (2018), no. 5, 1349–1384. MR 3784914, DOI 10.1007/s00023-018-0658-6
Odo Diekmann, Mats Gyllenberg, and Johan A. J. Metz, Finite dimensional state representation of physiologically structured populations, J. Math. Biol. 80 (2020), no. 1-2, 205–273. MR 4062819, DOI 10.1007/s00285-019-01454-0
Charles F. Dunkl, A Krawtchouk polynomial addition theorem and wreath products of symmetric groups, Indiana Univ. Math. J. 25 (1976), no. 4, 335–358. MR 407346, DOI 10.1512/iumj.1976.25.25030
A. Edelman and E. Kostlan, The road from Kac’s matrix to Kac’s random polynomials. In: Proceedings of the Fifth SIAM Conference on Applied Linear Algebra, J.G. Lewis (ed.), SIAM, Philadelphia, 1994, 503–507.
Alan Edelman and Eric Kostlan, How many zeros of a random polynomial are real?, Bull. Amer. Math. Soc. (N.S.) 32 (1995), no. 1, 1–37. MR 1290398, DOI 10.1090/S0273-0979-1995-00571-9
P. Feinsilver and R. Schott, Krawtchouk polynomials and finite probability theory, Probability measures on groups, X (Oberwolfach, 1990) Plenum, New York, 1991, pp. 129–135. MR 1178979
K. V. Fernando, Computation of exact inertia and inclusions of eigenvalues (singular values) of tridiagonal (bidiagonal) matrices, Linear Algebra Appl. 422 (2007), no. 1, 77–99. MR 2298997, DOI 10.1016/j.laa.2006.09.008
C. M. da Fonseca and E. Kılıç, A new type of Kac-Sylvester matrix and its spectrum, Linear Multilinear Algebra (2019), DOI: 10.1080/03081087.2019.1620673.
C. M. da Fonseca, Dan A. Mazilu, Irina Mazilu, and H. Thomas Williams, The eigenpairs of a Sylvester-Kac type matrix associated with a simple model for one-dimensional deposition and evaporation, Appl. Math. Lett. 26 (2013), no. 12, 1206–1211. MR 3250435, DOI 10.1016/j.aml.2013.06.006
R. Gordon, Harmonic oscillation in a spatially finite array waveguide, Optics Lett. 29 (2004), 2752–2754.
Olga Holtz, Evaluation of Sylvester type determinants using block-triangularization, Advances in analysis, World Sci. Publ., Hackensack, NJ, 2005, pp. 395–405. MR 2184955
Mourad E. H. Ismail, Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications, vol. 98, Cambridge University Press, Cambridge, 2005. With two chapters by Walter Van Assche; With a foreword by Richard A. Askey. MR 2191786, DOI 10.1017/CBO9781107325982
Mark Kac, Random walk and the theory of Brownian motion, Amer. Math. Monthly 54 (1947), 369–391. MR 21262, DOI 10.2307/2304386
Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw, Hypergeometric orthogonal polynomials and their $q$-analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010. With a foreword by Tom H. Koornwinder. MR 2656096, DOI 10.1007/978-3-642-05014-5
I. G. Macdonald, Orthogonal polynomials associated with root systems, Sém. Lothar. Combin. 45 (2000/01), Art. B45a, 40. MR 1817334
Thomas Muir, A treatise on the theory of determinants, Dover Publications, Inc., New York, 1960. Revised and enlarged by William H. Metzler. MR 0114826
T. Muir, The Theory of Determinants in the Historical Order of Development, Vol. II, Dover Publications Inc., New York, 1960, Reprinted.
Roy Oste and Joris Van der Jeugt, Doubling (dual) Hahn polynomials: classification and applications, SIGMA Symmetry Integrability Geom. Methods Appl. 12 (2016), Paper No. 003, 27. MR 3440191, DOI 10.3842/SIGMA.2016.003
Roy Oste and Joris Van der Jeugt, Tridiagonal test matrices for eigenvalue computations: two-parameter extensions of the Clement matrix, J. Comput. Appl. Math. 314 (2017), 30–39. MR 3575547, DOI 10.1016/j.cam.2016.10.019
J. J. Sylvester, Nouvelles Annales de Mathématiques, vol. XIII, 1854, p. 305 (Reprinted in Collected Mathematical Papers by AMS Chelsea Publishing, American Mathematical Society, Providence Rhode Island, 2008, vol. II, p. 28).
Gábor Szegő, Orthogonal polynomials, 4th ed., American Mathematical Society Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, R.I., 1975. MR 0372517
Olga Taussky and John Todd, Another look at a matrix of Mark Kac, Proceedings of the First Conference of the International Linear Algebra Society (Provo, UT, 1989), 1991, pp. 341–360. MR 1102076, DOI 10.1016/0024-3795(91)90179-Z
C. N. Yang and C. P. Yang, Thermodynamics of a one-dimensional system of bosons with repulsive delta-function interaction, J. Mathematical Phys. 10 (1969), 1115–1122. MR 246614, DOI 10.1063/1.1664947