The Fibonacci sequence (or sequence of the Fibonacci numbers) (Fn)nâ¥0 is the sequence of non-negative integers satisfying the recurrence Fn+2 = Fn+1 +.
International Journal of Pure and Applied Mathematics Volume 102 No. 3 2015, 527-532 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: http://dx.doi.org/10.12732/ijpam.v102i3.10
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ON A SEQUENCE OF TRIDIAGONAL MATRICES WHOSE DETERMINANTS ARE FIBONACCI NUMBERS Fn+1 Pavel Trojovsk´ y Department of Mathematics Faculty of Science University of Hradec Kr´alov´e Rokitansk´eho 62 50003 Hradec Kr´ alov´e, CZECH REPUBLIC
Abstract: In this paper, we generalize two previous individual results on connection special tridiagonal matrices to Fibonacci numbers, as we found a sequence of tridiagonal matrices which are equal to Fibonacci numbers. AMS Subject Classification: 11C20, 65F40, 15A15 Key Words: Fibonacci number, tridiagonal matrix, determinant, recurrence
1. Introduction The Fibonacci sequence (or sequence of the Fibonacci numbers) (Fn )n≥0 is the sequence of non-negative integers satisfying the recurrence Fn+2 = Fn+1 + Fn with the initial conditions F0 = 0 and F1 = 1. The Fibonacci sequence has many surprising properties (see e. g. [5]), but this paper deals with its connections to determinants of matrices only. Strang [10] include, probably the first example of determinant of n × n matrix, which is equal to the Fibonacci number, as he showed that the following holds Received:
April 3, 2015
c 2015 Academic Publications, Ltd.
url: www.acadpubl.eu
528
P. Trojovsk´ y
det
1 −1 0 · · · 0 1 1 −1 0 · · · 0 1 1 −1 0 .. .. . . 0 1 1 .. .. .. .. . . . 0 . 0 0 ··· 0 1 0 0 0 ··· 0
0 0 ··· .. .
0 0 0 .. .
= Fn+1 −1 0 1 −1 1 1
(1)
for any n ≥ 1. Cahill et al. [1] showed that the following holds
det
1
i
0
i
1
i
··· .. . .. . .. .
··· .. . .. .
0 .. . .. .
= Fn+1 0 i 1
0 i 1 (2) .. . . .. . . i . .. . . .. . . i 1 . 0 ··· ··· 0 i √ for any n ≥ 1 (where i = −1). Matrices in (1) and (2) are the special cases of a tridiagonal matrix, what is a square matrix A = (ajk ) of the order n, with entries ajk = 0 for |k − j| > 1 and 1 ≤ j, k ≤ n, i. e.
a1,1 a1,2
0
··· .. . .. . .. .
0 .. .
a2,1 a2,2 a2,3 A(n) = 0 a3,2 a3,3 0 .. .. .. . . . an−1,n 0 ··· 0 an,n−1 an,n
.
Many authors derived the similar types of matrices which determinants are related to Fibonacci numbers or different kinds of their generalizations, e. g. k-generalized Fibonacci numbers, see [2], [4], [7], [6], [3], [8], [9] and [11]. Now we turn our attention to the relation of determinants of special tridiagonal matrices with Fibonacci numbers. We show that matrix in (1) can be easily changed into a matrix, whose determinant is equal to Fibonacci numbers too.
ON A SEQUENCE OF TRIDIAGONAL MATRICES WHOSE...
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2. Preliminary Results Cahill et al. [1] proved the following lemma, which can be easily used for finding the recurrence relation for determinants of a sequence of tridiagonal matrices. Lemma 1. (Lemma 1 of [1]) Let {H(n), n = 1, 2, . . . } be a sequence of tridiagonal matrices of the form
h1,1 h1,2
0
··· .. . .. . .. .
0 .. .
h2,1 h2,2 h2,3 H(n) = 0 h3,2 h3,3 0 .. .. .. . . . hn−1,n 0 ··· 0 hn,n−1 hn,n
.
Then the successive determinants of H(n) are given by recursive formula
det H(1) = h1,1 ;
(3)
det H(2) = h1,1 h2,2 − h1,2 h2,1 ;
det H(n) = hn,n det H(n − 1) − hn−1,n hn,n−1 det H(n − 2).
3. Main Results We formulate theorem which generalize identities (1) and (2). Theorem 2. Let (εn )n≥0 , (δn )n≥0 be any sequences of complex numbers, with property εk δk = −1 for any k, 1 ≤ k ≤ n. Let {B(n), n = 1, 2, 3, . . . } be a sequence of tridiagonal matrices in the form
bjk
1, j=k; εj , k = j + 1 ; = δ , k = j − 1; j 0, otherwise
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P. Trojovsk´ y
B(n) =
1
ε1
0
δ1
1
ε2
0 .. . .. . 0
δ2 .. . .. . ···
1 .. . .. . ···
··· .. . .. . .. . δ1 0
··· .. . .. .
0 .. . .. .
ε1
0
1 εn−1 δn−1 1
.
Then det B(n) = Fn+1 . Proof. We use the mathematical induction on n. The assertion holds for n = 1 and n = 2 as
det B(1) = 1 = F2 , 1 ε1 det B(2) = det = 1 − ε1 δ1 = 2 = F3 . δ1 1 Suppose that the assertion holds for every k, 3 ≤ k ≤ n. Then we have to show that the assertion is true for n + 1. We use recurrence (3) det B(n + 1) = bn+1,n+1 det B(n) − bn,n+1 bn+1,n det B(n − 1) = 1 · det B(n) − εn−1 δn−1 det B(n − 1)
= det B(n) + det B(n − 1) = Fn+1 + Fn = Fn+2 .
Corollary 3. Setting εk = −1, δk = 1 and εk = δk = i = 2, for 1 ≤ k ≤ n, we directly obtain (1) and (2) respectively.
√
−1 in Theorem
Similarly we can obtain infinitely many interesting n−square matrices, whose determinants are equal to the Fibonacci number Fn+1 , using Theorem 2, but there are integer matrices of this type only for entries εk = ±1, δk = − εk , where 1 ≤ k ≤ n. For example, we obtain the following sequence of integer matrices.
ON A SEQUENCE OF TRIDIAGONAL MATRICES WHOSE...
531
Corollary 4. Let {C(n) = (cjk )1≤j,k≤n , n = 1, 2, 3, . . . } be a sequence of tridiagonal matrices in the form j = k; 1, cjk = (−1)j , j = k ± 1; 0, otherwise, i. e.
C(n) =
1 −1
0
···
1
1
0
1
0 .. . .. . .. .
··· .. . .. . .. .
0 .. . .. . .. .
0 −1 1 −1 .. . . . . . . . . . . .. .. . . 1 (−1)n−2 0 0 .. .. n−1 . (−1) 1 (−1)n−1 . 0 ··· 0 ··· 0 (−1)n 1
.
Then det C(n) = Fn+1 .
(4)
Acknowledgements The author thanks to Specific research PˇrFUHK2015 for financial support.
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[5] T. Koshy, Fibonacci and Lucas numbers with applications, John Wiley & Sons, 2011. [6] G. Y. Lee, S. G. Lee, A note on generalized Fibonacci numbers, Fib. Quart., 33, No. 3 (1995), 273–278. [7] G. Y. Lee, J. S. Kim, The linear algebra of the k-Fibonacci matrix, Linear Algebra Appl. 373 (2003), 75–87. [8] A. Nalli, H. Civciv, A generalization of tridiagonal matrix determinants, Fibonacci and Lucas numbers, Chaos Solitons Fractals 40, No. 1 (2009), 355–361. ˝ [9] A. A. Ocal, N. Tuglu, and E. Altini¸sik, On the representation of kgeneralized Fibonacci and Lucas numbers, Appl. Math. Comput. 170 (2005), 584–596. [10] G. Strang, Linear algebra and its applications, Brooks/Cole, 3rd edition, 1988. [11] F. Yılmaz, T. Sogabe, A note on symetric k-tridiagonal matrix family and the Fibonacci numbers, Int. J. Pure and Appl. Math., 96, No. 2 (2014), 289–298.
