PAijpam.eu ON THE BOUNDS FOR THE NORMS OF R-CIRCULANT MATRICES WITH THE JACOBSTHAL AND JACOBSTHAL LUCAS NUMBERS Ş. Uygun 1, S.

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1 International Journal of Pure and Applied Mathematics Volume 11 No 1 017, 3-10 ISSN: (printed version); ISSN: (on-line version) url: doi: 10173/ijpamv11i17 PAijpameu ON THE BOUNDS FOR THE NORMS OF R-CIRCULANT MATRICES WITH THE JACOBSTHAL AND JACOBSTHAL LUCAS NUMBERS Ş Uygun 1, S Yaşamalı 1, Department of Mathematics Science and Art Faculty Gaziantep University Campus, 7310, Gaziantep, TURKEY Abstract: In this study, we have found upper and lower bounds for the spectral norms of circulant matrices in the forms A = C r(j 0,j 1,,j ) and B = C r(c 0,c 1,,c ) After that we obtain some bounds related to the spectral norms of Hadamard and Kronecker product of these matrices AMS Subject Classification: 15A36, 15A45, 15A60 Key Words: Jacobsthal number, Jacobsthal Lucas number, circulant matrix, norm 1 Introduction and Preliminaries The Jacobsthal {j n } n N, and the Jacobsthal Lucas {c n } n N sequences are defined recurrently by j n = j +j n, j 0 = 0, j 1 = 1, n, (1) c n = c +c n, c 0 =, c 1 = 1, n, () respectively There have been several papers on the norms of special matrices [7-10] Solak [8] has defined A = [a ij ] and B = [b ij ] as nxn circulant matrices, where a ij = F (mod(j i,n)) and b ij = L (mod(j i,n)), then he has given some bounds for the A and B matrices concerned with the spectral and Eu- Received: June 5, 016 Revised: January 11, 017 Published: January 3, 017 c 017 Academic Publications, Ltd url: wwwacadpubleu Correspondence author

2 4 Ş Uygun, S Yaşamalı clidean norms In () Shen, Cen found the bounds for the norms of r-circulant matrices with the Fibonocci and Lucas numbers Shen and Cen [10] have given upper and lower bounds for the spectral norms of r- circulant matrices A = C r (F (k, 1) 0,F (k, 1) 1,,F (k, 1) ) and B = C r (L (k, 1) 0,L (k, 1) 1,,L (k, 1) ) In addition, they also have obtained some bounds for the spectral norms of Hadamard and Kronecker products of these matrices In (13) the authors gave the relations among k Fibonacci, k Lucas and generalized k Fibonacci numbers and the spectral norms of the matrices of involving these numbers, In this paper we give lower and upper bounds for the spectral norms of the r circulant matrices A = C r (j 0,j 1,,j ) and B = C r (c 0,c 1,,c ) After that we obtain some bounds related to the spectral norms of Hadamard and Kronecker product of these matrices Recurrences (1) and () involve the characteristic equation x x = 0, with roots α =, β = 1 Their Binet s formulas are defined by j n = αn β n α β, c n = α n +β n (3) A matrix C = [c ij ] M m,n (C) is called a circulant matrix if it is of the form { cj i, j i c ij = (4) rc n+j i, j < i For any A = [a ij ] M m,n (C) The Frobenious( or Euclidean) norm of matrix A is 1 m n A F = a ij (5) and the spectral norm of matrix A is A = where λ i (A H A) is eigenvalue of A H A max 1 i n λ i(a H A), (6)

3 ON THE BOUNDS FOR THE NORMS OF R-CIRCULANT 5 Lemma 1 For any A,B M m,n (C), the Hadamard product of A,B is entrywise product and defined by (see [5,6]) and have the following properties AoB = (a ij b ij ) AoB A B, (7) AoB r 1 (A)c 1 (B) (8) Lemma LetA M m,n (C), B M p,q (C)begiven, thenthekronecker product of A,B is defined by and have the following property [11] a 11 B a 1n B A B = a m1 B a mn B A B = A B () Lemma 3 Let A M m,n (C) be given, then the inequality is hold (see [4]) 1 n A F A A F (10) Lemma 4 The sum of squares of the first n elements of Jacobsthal and Jacobsthal Lucas sequences are given as jk = j n +( 1) n j n +n, (11) c k = j n ( 1) n j n +n (1)

4 6 Ş Uygun, S Yaşamalı Lower and Upper Bounds of r-circulant Matrices Involving Jacobsthal and Jacobsthal-Lucas Numbers Theorem 5 Let A = C r (j 0,j 1,,j ) be r-circulant matrix, then we obtain where r C (i) If r 1, then jn +( 1) n j n +n A r j n +( 1) n j n +n (13) (ii) If r 1, then jn +( 1) n j n +n ()(jn +( 1) n j n +n) r A (14) Proof The matrix A is of the form j 0 j 1 j j rj j 0 j 1 j n A = rj n rj j 0 j n 3 rj 1 rj rj 3 j 0 For r 1, by using (4), (11) we have From (10), A F = (n k)jk + k r jk (n k)jk + kjk = n = n [j n +( 1) n j n +n] 1 j n +( 1) n j n +n 1 A 3 n F A On the other hand, let A = BoC where as rj rj rj B = rj n rj rj 0 1, C = rj 1 rj rj 3 rj 0 j k j 0 j 1 j j 1 j 0 j 1 j n 1 1 j 0 j n j 0

5 ON THE BOUNDS FOR THE NORMS OF R-CIRCULANT 7 then r 1 (B) = = c 1 (C) = = max 1 i n r n b ij = n b nj jk = r 3 jn +( 1) n j n +n, n n c ij = c in max 1 j n jk = 3 1 jn +( 1) n j n +n By using (8) we obtain A r 1 (B)c 1 (C) = r [j n +( 1) n j n +n] The proof is completed for the first part (ii) For r 1 by using (11) we have, From (10), A F = (n k)jk + k r jk (n k) r jk + k r jk = n r = n r [j n +( 1) n j n +n] r j n +( 1) n j n +n 1 A 3 n F A On the other hand, let A = BoC, where B = j r j r r j 0 1 r r r j 0, C = j k j 0 j 1 j j j j 0 j 1 j n j n j j 0 j n 3 j 1 j j 3 j 0,

6 8 Ş Uygun, S Yaşamalı n r 1 (B) = max b ij = 1 i n j 0 +() = (), c 1 (C) = max n c ij = j 1 j n k = 1 j n +( 1) n j n +n 3 By using (8) we obtain ()(jn +( 1) n j n +n) A r 1 (B)c 1 (C) = Therefore the proof is completed r 3 j n +( 1) n j n +n A 1 3 ()(j n +( 1) n j n +n) Theorem 6 Let A = C r (c 0,c 1,,c ) be r- circulant matrix, then we obtain: (i) If r 1, then j n ( 1) n j n +n A ( ) 4+ r (j n ( 1) n j n +n 4) (j n ( 1) n j n +n 3) (15) (ii) If r 1, then r j n ( 1) n j n +n A Proof The matrix A is of the form c 0 c 1 c c rc c 0 c 1 c n A = rc n rc c 0 c n 3 rc 1 rc rc 3 c 0 For r 1, by using (4), (1) we have n(j n ( 1) n j n +n) (16) A F = (n k)c k + k r c k

7 ON THE BOUNDS FOR THE NORMS OF R-CIRCULANT (n k)c k + kc k = n c k = n(j n ( 1) n j n +n) From (10), A 1 A n F j n ( 1) n j n +n On the other hand, let A = BoC where c c 1 c c rc c c 1 c n B = rc n rc c 0 1, C = c n 3 rc 1 rc rc 3 c n r 1 (B) =max b ij = b nj By using (8) we obtain 1 i n = c 0 + r c k = 4+ r = 4+ r (j n ( 1) n j n +n 4) c 1 (C) = max n c ij = n c in = 1+ 1 j n = j n ( 1) n j n +n 3 c k c k A = BoC ( ) 4+ r (j n ( 1) n j n +n 4) (j n ( 1) n j n +n 3) (ii) For r 1, A F (n k) r c k + k r c k = n r c k

8 100 Ş Uygun, S Yaşamalı From (10), A 1 n A F r c k = r (j n ( 1) n j n +n) On the other hand, let A = BoC where r r r 1 1 B = r r r 1, C = r r r r n r 1 (B) = max b ij = n, 1 i n c 0 c 1 c c c c 0 c 1 c n c n c c 0 c n 3 c 1 c c 3 c 0 c 1 (C) = max n c ij = c k j = n ( 1) n j n +n 1 j n By using (8) we obtain A = BoC r 1 (B)c 1 (C) n(j n ( 1) n j n +n), r j n ( 1) n j n +n A n(j n ( 1) n j n +n) So the proof is completed Corollary 7 Let A = C r (j 0,j 1,,j ) and A = C r (c 0,c 1,,c ) be r circulant matrices, where r C (i) If r 1, then AoB r j n +( 1) n j n +n ( ) 4+ r [j n ( 1) n j n +n 4] (j n ( 1) n j n +n 3),

9 ON THE BOUNDS FOR THE NORMS OF R-CIRCULANT 101 (ii) If r 1, then AoB (()(jn +( 1) n j n +n) ) (n(j n ( 1) n j n +n)) Proof Since AoB A B,the proof is trivial by (7) Corollary 8 Let A = C r (j 0,j 1,,j ) and A = C r (c 0,c 1,,c ) be r circulant matrices, where r C (i) If r 1, then A B r j n +( 1) n j n +n ( ) 4+ r [j n ( 1) n j n +n 4] (j n ( 1) n j n +n 3), A B (ii) If r 1, then (jn +( 1) n j n +n ) (j n ( 1) n j n +n) ( ()(jn +( 1) n ) j n +n) A B n (j n ( 1) n j n +n), A B r (j n ( 1) n j n +n)(j n ( 1) n j n +n) Proof By using () and theorems 5 and 6, the proof is easily seen References [1] A F Horadam, Jacobsthal representation numbers, The Fibonacci Quarterly, 34, No 1 (16), [] T Koshy, Fibanacci and Lucas Numbers with Applications, John Wiley and Sons Inc, NY, 001 [3] NJA Sloane, The On-Line Encyclopedia of Integer Sequences, 006 [4] G Zielke, Some remarks on matrix norms, condition numbers and error estimates for linear equations, Linear Algebra and its Applications, 110 (188), -41 [5] R Mathias, The spectral norm of nonnegative matrix, Linear Algebra and its Applications, 131 (10), 6-84

10 10 Ş Uygun, S Yaşamalı [6] R Reams, Hadamard inverses square roots and products of almost semidefinite matrices, Linear Algebra and its Applications, 88 (1), [7] S Solak, D Bozkurt, On the spectral norms of Cauchy-Toeplitz and Cauchy-Henkel matrices, Appl Math Comput, 140 (003), [8] S Solak, On the norms of circulant matrices with the Fibonocci and Lucas numbers, Applied Mathematics and Computation, 160 (005), [] S Shen, J Cen, On the bounds for the norms of r-circulant matrices with the Fibonocci and Lucas numbers, Appl Math Comput, 16 (010), [10] S Shen, On the spectral norms of r-circulant matrices with the k-fibonocci and k-lucas numbers, Int J Contemp Math Sciences, 5, No 1 (010), [11] R A Horn, C R Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 11 [1] Ş Uygun, The (s, t)-jacobsthal and (s, t)-jacobsthal Lucas sequences, Applied Mathematical Sciences, 70, No (015), [13] K Uslu, N Taşkara, Ş Uygun, The relations among k Fibonacci, k Lucas and generalized k Fibonacci numbers and the spectral norms of the matrices of involving these numbers, Ars Combinatoria, 10 (011), 183-1