On The Circulant Matrices with Ducci Sequences and Fibonacci Numbers

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1 Filomat 3:15 (018), htts://doi.org/10.98/fil s Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: htt:// On The Circulant Matrices with Ducci Sequences Fibonacci Numbers Süleyman Solak a, Mustafa Bahşi b, Osman Kan c a N.E. University, A.K Education Faculty, Konya-TURKEY b Aksaray University, Education Faculty, Aksaray-TURKEY (Corresonding Author) c Karatay Belediyesi Religious Vocational Secondary School, Konya-TURKEY Abstract. A Ducci sequence generated by A = (a 1, a,..., a n ) Z n is the sequence { A, DA, D A,... } where the Ducci ma D : Z n Z n is defined by D(A) = D(a 1, a,..., a n ) = ( a a 1, a 3 a,..., a n a n 1, a n a 1 ). In this study, we examine some roerties of the matrices C n, DC n, D C n, where C n =Circ(c 0, c 1,..., c n 1 ) is a circulant matrix whose entries consist of Fibonacci numbers. 1. Introduction 1.1. Ducci Sequences Let A = (a 1, a,..., a n ) be an n tule of integers D : Z n Z n be a ma defined by D(a 1, a,..., a n ) := ( a a 1, a 3 a,..., a n a n 1, a n a 1 ). The ma D is called the Ducci ma the sequence { A, DA, D A,... } is called a Ducci sequence. Professor E.Ducci made some observations on the ma D in the 1800 s [6], so when Ducci sequences were first introduced in 1937 [7], their name was attributed to E. Ducci. Under the Ducci ma, the behavior of the starting vector A = (a 1, a,..., a n ) is interesting has been examined in a number of aers [ 6, 9, 16, 1, ].When n = k for some ositive integer k, every starting vector ends in a tule having all comonents equal [7] converges to the zero vector [3, 4]. Under the Ducci ma, every starting vector converges to a eriodic orbit [9] reaches to (x 1, x,..., x n ) when n k, where x j {0, m} m is a ositive constant [4, 6]. Also, there are integers s r with 0 s < r such that D s A = D r A. Thus, we say that the Ducci sequence { A, DA, D A,... } has eriod r s, when r s are as small as ossible []. 010 Mathematics Subject Classification. Primary 11B83; Secondary 15A15, 15A60 Keywords. Ducci Sequence; Circulant Matrix; Norm, Received: 6 June 018; Revised: 05 October 018; Acceted: 0 December 018 Communicated by Dijana Mosić addresses: ssolak4@yahoo.com (Süleyman Solak), mhvbahsi@yahoo.com (Mustafa Bahşi), matrixinequality@gmail.com (Osman Kan)

2 S. Solak et al. / Filomat 3:15 (018), Ducci matrix sequences are closely related to the set of rational irrational numbers [11, 17]. Thus, the researches on Ducci matrix sequences have increased in recent years [3, 11, 17, 0]. For examle, Solak Bahşi [0] have established relationshis between the sectral norm, Frobenius norm, l norm, determinant eigenvalues of circulant matrix Circ(A) = Circ(a 1, a,..., a n ) its image under the Ducci ma. Let us consider the following starting vector F = (F 1, F,..., F n ), where F s is the sth Fibonacci number defined by (6) in Section. Then, its consecutive images under the Ducci ma are DF = (F F 1, F 3 F,..., F n F n 1, F n F 1 ) = (F 0, F 1,..., F n, F n 1) D F = (F 1 F 0, F F 1,..., F n F n 3, F n 1 F n, F n 1 F 0 ) = (F 1, F 0, F 1,..., F n 4, F n 1 1, F n 1) = (1, F 0, F 1,..., F n 4, F n 1 1, F n 1), where F 1 := 1 (see (7) in Section for the Fibonacci numbers with negative indices). Thus, the above starting vector its consecutive images under the Ducci ma, for n 3, yield the following n n matrices: Circ(F) := Circ(DF) := Circ(D F) := F 1 F F n F n F 1 F n 1... F F 3 F 1 F 0 F 1 F n F n 1 F n 1 F 0 F n 3 F n.... F 1 F F n 1 F 0 1 F 0 F n 4 F n 1 1 F n 1 F n 1 1 F n 5 F n 4 F n F 0 F 1 F n 1 1 F n 1 1, (1) (). (3) Now that we have seen the matrices Circ(F), Circ(DF) Circ(D F), we would like to determine if there is any relationshi between the norms, determinants eigenvalues of them. Because they are all circulant matrices, we will begin our study with some roerties of circulant matrices. 1.. Circulant Matrices Let (c 0, c 1,..., c n 1 ) be an n tule of comlex numbers. Then, the matrix

3 S. Solak et al. / Filomat 3:15 (018), C := c 0 c 1 c n 1 c n 1 c 0 c n c n c n 1... c n 3... c 1 c c 0 is called a circulant matrix associated with n tule (c 0, c 1,..., c n 1 ). Thus, we denote the circulant matrix C by C =Circ(c 0, c 1,..., c n 1 ). By means of [ 8, Chater 3 ] the roerties of circulant matrices are well known. Some of them are: 1. Let C be an n n matrix. Then, C is a circulant matrix if only if CP = PC, where P is the n n matrix P =Circ(0, 1, 0,..., 0).. Circ(c 0, c 1,..., c n 1 ) = c 0 I + c 1 P c n 1 P n 1, where I denotes the n n identity matrix P is the n n matrix P =Circ(0, 1, 0,..., 0). 3. All circulant matrices of the same order commute. If C is a circulant matrix so is C, where C denotes conjugate transose of C. Hence C C commute therefore all circulant matrices are normal matrices. 4. The eigenvalues of matrix C =Circ(c 0, c 1,..., c n 1 ) are n 1 λ j = c k w jk, for 0 j n 1, where w = e πi n i = 1. There are many studies on general secial circulant matrices [1, 10, 13, 14, 3]. In recent years, researchers have been esecially attracted to the study of norms determinants of the circulant matrices with Fibonacci Lucas numbers as entries [13, 18 0]. The Ducci ma was first alied to circulant matrices by Solak Bahşi [0]. By alying the Ducci ma to each row of the circulant matrix Circ(A) = Circ(a 1, a,..., a n ) (4) associated with A = (a 1, a,..., a n ) Z n, they have defined the circulant matrix Circ(DA) = Circ( a a 1, a 3 a,..., a n a n 1, a n a 1 ) (5) associated with D(A) = ( a a 1, a 3 a,..., a n a n 1, a n a 1 ). Then, they have established some relationshis between sectral norm, Frobenius norm, l norm, determinant eigenvalues of the matrices Circ(A) Circ(DA). This aer is organized as follows: In Section we introduce some definitions lemmas related to our study. In Section 3 we rove some equalities inequalities involving norms, determinants eigenvalues of circulant matrices Circ(F), Circ(DF) Circ(D F). Throughout this study matrices Circ(F), Circ(DF) Circ(D F) are the n n circulant matrices defined by (1), () (3).. Preliminaries The Fibonacci numbers are defined by the second order linear recurrence relation: F n+1 := F n + F n 1 for n 1 F 0 := 0 F 1 := 1. (6)

4 S. Solak et al. / Filomat 3:15 (018), Also, we have the backwards rule F n := ( 1) n+1 F n, (7) where n > 0. The Fibonacci numbers have many interesting identities [ 15, Chater 5 ] such as F n+1 + F n = F n+1, (8) F n+1 F n = F n 1 F n+ (9) In addition, one can see easily that n 1 F s = F n 1 F n. (10) s=0 F n F n 3 = F n. (11) Definition.1. [1,. 91] Let A = ( ) ( ) a jk be any m n matrix. The l 1 < < norm of A is m n A := a ij j=1 k=1 In the articular case =, this is the Frobenius norm of A we denote it by 1. A F := m n a ij. j=1 k=1 Definition.. [1,. 95] Let A = ( a jk ) be any m n matrix. The sectral norm of A is A := max λ s (A A), where λ s (A A) are eigenvalues of A A A is conjugate transose of A. In the next three lemmas, matrices Circ(A) Circ(DA) denote the circulant matrices in (4) (5), resectively. Lemma.3. [0, Theorem 5] Let µ j λ j (j = 0, 1,..., n 1) be eigenvalues of the matrices Circ(DA) Circ(A), resectively. If a 1 a... a n, then where w = e πi n i = 1. µ j = ( λ j + a n a 1 ) w j λ j, for 0 j n 1 Lemma.4. [0, Theorem 6] The sectral norm of the matrix Circ(DA) with a 1 a... a n satisfies Circ(DA) = (a n a 1 ). Lemma.5. [0, Theorem 4] The determinant of the matrix Circ(DA) satisfies det Circ(DA) 1 n n Circ(DA) n E.

5 S. Solak et al. / Filomat 3:15 (018), Main Results Theorem 3.1. For the Frobenius norms of the n n matrices Circ(D F) Circ(DF), for n 3, we have Circ(D F) F Circ(DF) F = n (F n 1) (F n 3 1). Proof. By the definition of the Frobenius norm, we have n Circ(DF) F = n F k + (F n 1) Circ(D F) F n 4 = n F k + (F n 1 1) + (F n 1) k= 1 n = n 1 + F k + (F n 1 1) + (F n 1) F n F n 3 n = n F k + (F n 1) + n [ F n 1 F n 1 F n F n 3 + ] = Circ(DF) F + n [ F n 1 F n 1 F n F n 3 + ]. (1) By (9) (11), we have This comletes the roof. F n 1 F n 1 F n F n 3 + = F n 1 F n 3 + F n 3F n + = F n F n 3 + F n 3 (F n F n 3 ) + = F n F n 3 + F n F n 3 + = (F n 1) + F n 3 (F n 1) = (F n 1) (F n 3 1). Theorem 3.. For the Frobenius norms of the n n matrices Circ(D F) Circ(F), for n 3, we have Proof. By [ 0, Examle ], we have: By (8), (1) (13), we have Circ(D F) F which comletes the roof of the theorem. Circ(D F) F Circ(F) F = n (F n+1 + F n 5 3). Circ(F) F Circ(DF) F = n ( F n 1 + F n 1 ). (13) = Circ(F) F n [ F n 1 + F n 1 F n 1 + F n 1 + F n + F n 3 ] = Circ(F) F n [F n+1 + F n 5 3],

6 S. Solak et al. / Filomat 3:15 (018), Theorem 3.3. For the l norms of the n n matrices Circ(F), Circ(DF) Circ(D F), for n 3, we have Circ(D F) Circ(DF) = n [ (F n 1 1) F n F n 3 + 1] (14) Circ(F) Circ(D F) = n [ F n + F n 1 + F n + F n 3 (F n 1) (F n 1 1) 1 ]. (15) Proof. By the definition of l norm, we have Thus Circ(D F) Circ(D F) n 4 = n F k + (F n 1 1) + (F n 1), k= 1 n Circ(DF) = n F k + (F n 1) (16) Circ(F) = n n F k. (17) k=1 n = n 1 + F k + (F n 1 1) + (F n 1) F n F n 3 n = n F k + (F n 1) + n [ (F n 1 1) F n F n 3 + 1] which yields formula (14). On the other h, Circ(D F) = Circ(DF) + n [ (F n 1 1) F n F n 3 + 1], [ ] = n 1 + n F k + (F n 1 1) + (F n 1) F n F n 1 F n F n 3 = n n F k + n [ (F n 1 1) + (F n 1) F n F n 1 F n F n ] = Circ(F) + n [ (F n 1 1) + (F n 1) F n F n 1 F n F n ], from which formula (15) follows, the roof is comleted. We remark that the next equality, which is an immediate consequence of (16) (17), already aeared in [0, Examle 3]: Circ(F) Circ(DF) = n [ F n 1 + F n (F n 1) ]. Theorem 3.4. The determinant of the n n matrix Circ(D F) satisfies det Circ(D F) (Fn 4 F n 3 + F n 1 F n+1 + 3) n, where n 4.

7 S. Solak et al. / Filomat 3:15 (018), Proof. If we take DF = A, then DA = D(DF) = D F. Thus, for the determinant of the matrix Circ(D F), Lemma (.5) the equations (8) (10) yield det Circ(D F) 1 ( Circ(D F) ) n E n n 1 n 4 1 = n 1 + F k + (F n 1 1) + (F n 1) Hence, the desired result is obtained. = n n 1 n n [ (n [ Fn 4 F n 3 + F n 1 + F n F n 1 F n + 3 ]) 1 = (F n 4 F n 3 + F n 1 F n+1 + 3) n. For determinants of matrices Circ(F) Circ(DF), the inequalities n ] n det Circ(F) (F n F n+1 ) n det Circ(DF) ( F n F n 1 + (F n 1) ) n are given in [ 0, Examle 4 ]. Theorem 3.5. Let η j, µ j λ j (j = 0, 1,..., n 1) be eigenvalues of the n n (n 3), matrices Circ(D F), Circ(DF) Circ(F), resectively. Then η j = ( µ j + F n ) w j µ j η j = [( λ j + F n ) w j λ j ] w j + λ j, where w = e πi n i = 1. Proof. Since F 0 F 1... F n 1 for n 3, Lemma (.3) immediately yields η j = ( µ j + F n ) w j µ j µ j = ( λ j + F n ) w j λ j, for 0 j n 1. Thus η j = [( λ j + F n ) w j λ j + F n ] w j [( λ j + F n ) ] w j λ j = [( λ j + F n ) ] w j λ j w j + λ j. Corollary 3.6. For the sectral norms of the n n matrices Circ(D F), for n 3, we have Circ(D F) = F n. Proof. Since F 0 F 1... F n 1 for n 3, we have from Lemma (.4) Circ(D F) = F n.

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