arXiv:1902.07626v1 [math.CO] 20 Feb 2019 h hrceitcplnma fti arx(hti,det is, (that matrix this of polynomial characteristic The arx lokona lmn matrix, Clement as known also matrix, a arx nfc,setting fact, In matrix. Kac mn te aes stefudro h mrcnJunlo M of Journal Sylve American Joseph the James of mathematician founder the British as century facets, 19th other the among by [19], in is J uadPB hn rpsdi 1]tefloigconjecture. following the 1. [10] Conjecture in proposed Zhang P.B. and Hu n h matrix The e od n phrases. and words Key 2000 Date oieta ojcue1i qiaett tt htteeigenvalue the that state to equivalent is 1 Conjecture that Notice ut eety nodrt n omlsfrtedtriat fso of determinants the for formulas find to order in recently, Quite ( z 0 z , eray2,2019. 21, February : ahmtc ujc Classification. Subject Mathematics Abstract. eemnn facranSletrKctp arxadcnie a consider and matrix type Sylvester-Kac certain a of determinant 1 = ) NOSRAINO H EEMNN FA OF DETERMINANT THE ON OBSERVATION AN          J z n 0 h eemnn ftematrix the of determinant The ( z + ae nals-nw eut epoearcn ojcueconc conjecture recent a prove we result, less-known a on Based 0 z n z , nz 1 ± ALSM AFNEAADERHKILIC¸ EMRAH AND FONSECA DA M. CARLOS 1 a eesl dnie sa xeso ftes-aldSylvester- so-called the of extension an as identified easily be can ) YVSE-A YEMATRIX TYPE SYLVESTER-KAC yvse-a arx lmn arx eemnn,eigenvalues. determinant, matrix, Clement matrix, Sylvester-Kac ( n 0 − ( + z 2 k Y 1 n k        =0 n efidtecaatrsi arxo h Sylvester-Kac the of matrix characteristic the find we 0 = n ) − 1 q  − n 1 0 1. 1 z z 2) 1 2 0 h Conjecture The n 1 + − z 51,15A15. 15A18, − 1 2 0 ( n 0 1 , z − 0 . ( + 1 for 2 . . k ) . n 2 k q . ( . . n . − 0 1 . . 0 = z . . 1) + 1 4) 2 1 + n , z 1 1        , . . . , ×  . . ( 2 . . . . n . . J ⌊ n 1) + n/ ( z x, 0 xeso fit. of extension n 2 − ) a rtconjecture first was 0)) ⌋ teais n1878. in athematics, . n ( n − 1 of s eLeagba,Z. algebras, Lie me − 1 2) J rigthe erning trcelebrated, ster z n 1 (0 z , z 0 1 are ) − n nz 1          2 CARLOSM.DAFONSECAANDEMRAHKILIC¸

A fully comprehensive list of results on the different proofs for Sylvester’s conjecture and the eigenpairs of non-trivial extensions of the Sylvester-Kac matrix can be found in [1–9, 11–18, 20, 21]. The aim of this short note is to prove Conjecture 1 based on a result by W. Chu in [3]. We also provide a general result containing other particular known determinants.

2. An extension to Sylvester-Kac matrix In 2010, cleverly based on two generalized Fibonacci sequences, W. Chu proved the following theorem. Theorem 2.1 ([3]). The determinant of the matrix (n + 1) × (n + 1) x u  nv x + y 2u  ..  (n − 1)v x +2y .  Mn(x,y,u,v)=  . .   .. .. n − 1     2v x − (n − 1)y nu     v x + ny    is n ny n − 2k x + + y2 +4uv .  2 2  kY=0 p Of course, the formula for the determinant in Theorem 2.1 can be rewritten as ⌊n/2⌋ ny 2 (n − 2k)2 x + − (y2 +4uv) .  2 4  kY=0   Now setting x = z0 + nz1, y = −2z1, and u = v = 1, we prove immediately Conjecture 1. Moreover, in the spirit of [1,8,9], using Theorem 2.1, we can also conclude the following theorem. Theorem 2.2. The eigenvalues of nar b  na ((n − 1)a ± b)r 2b  (n − 1)a ((n − 2)a ± 2b)r 3b ±  . .  Mn (a, b, r)=  .. ..  .  (n − 2)a   . .   .. .. nb     a ±nbr    are 1 nr(a ± b)+(n − 2k) 4ab + r2(a ∓ b)2 , 2  p  for k =0, 1,...,n....

References [1] R. Askey, Evaluation of Sylvester type determinants using orthogonal polynomials, In: H.G.W. Begehr et al. (eds.): Advances in analysis. Hackensack, NJ, World Scientific 2005, pp. 1-16. [2] T. Boros, P. R´ozsa, An explicit formula for singular values of the Sylvester-Kac matrix, Linear Algebra Appl. 421 (2007), 407-416. AN OBSERVATION ON THE DETERMINANT OF A SYLVESTER-KAC TYPE MATRIX 3

[3] W. Chu, Fibonacci polynomials and Sylvester determinant of tridiagonal matrix, Appl. Math. Com- put. 216 (2010), 1018-1023. [4] W. Chu, X. Wang, Eigenvectors of tridiagonal matrices of Sylvester type, Calcolo 45 (2008), 217-233. [5] P.A. Clement, A class of triple-diagonal matrices for test purposes, SIAM Rev. 1 (1959), 50-52. [6] A. Edelman, E. Kostlan, The road from Kac’s matrix to Kac’s random polynomials, in: J. Lewis (Ed.), Proc. of the Fifth SIAM Conf. on Applied Linear Algebra, SIAM, Philadelpia, 1994, pp. 503-507. [7] D.K. Faddeev, I.S. Sominskii, in: J.L. Brenner (Translator), Problems in Higher Algebra, Freeman, San Francisco, 1965. [8] C.M. da Fonseca, D.A. Mazilu, I. Mazilu, H.T. 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), 1206-1211. [9] O. Holtz, Evaluation of Sylvester type determinants using block-triangularization, In: H.G.W. Begehr et al. (eds.): Advances in analysis. Hackensack, NJ, World Scientific 2005, pp. 395-405. [10] Z. Hu, P.B. Zhang, Determinants and characteristic polynomials of Lie algebras, Linear Algebra Appl. 563 (2019), 426-439. [11] Kh.D. Ikramov, On a remarkable property of a matrix of Mark Kac, Math. Notes 72 (2002), 325-330. [12] M. Kac, Random walk and the theory of Brownian motion, Amer. Math. Monthly 54 (1947), 369-391. [13] E. Kılı¸c, Sylvester-tridiagonal matrix with alternating main diagonal entries and its spectra, Inter. J. Nonlinear Sci. Num. Simulation 14 (2013), 261-266. [14] E. Kılı¸c, T. Arıkan, Evaluation of spectrum of 2-periodic tridiagonal-Sylvester matrix, Turk. J. Math. 40 (2016), 80-89. [15] T. Muir, The Theory of Determinants in the Historical Order of Development, vol. II, Dover Publi- cations Inc., New York, 1960 (reprinted) [16] R. Oste, J. Van den Jeugt, Tridiagonal test matrices for eigenvalue computations: Two-parameter extensions of the Clement matrix, J. Comput. Appl. Math. 314 (2017), 30-39. [17] P. R´ozsa, Remarks on the spectral decomposition of a stochastic matrix, Magyar Tud. Akad. Mat. Fiz. Oszt. K¨ozl. 7 (1957), 199-206. [18] E. Schr¨odinger, Quantisierung als Eigenwertproblem III, Ann. Phys. 80 (1926), 437-490. [19] J.J. Sylvester, Th´eor`eme sur les d´eterminants de M. Sylvester, Nouvelles Ann. Math. 13 (1854), 305. [20] O. Taussky, J. Todd, Another look at a matrix of Mark Kac, Linear Algebra Appl. 150 (1991), 341-360. [21] I. Vincze, Uber¨ das Ehrenfestsche Modell der W¨arme¨ubertragung, Archi. Math XV (1964), 394-400.

Kuwait College of Science and Technology, Doha District, Block 4, P.O. Box 27235, Safat 13133, Kuwait E-mail address: [email protected] University of Primorska, FAMNIT, Glagoljsaˇska 8, 6000 Koper, Slovenia E-mail address: [email protected] TOBB University of Economics and Technology, Mathematics Department, 06560 Ankara, Turkey E-mail address: [email protected]