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ON THE NORMS OF SOME SPECIAL MATRICES WITH GENERALIZED FIBONACCI SEQUENCE
ON THE NORMS OF SOME SPECIAL MATRICES WITH GENERALIZED FIBONACCI SEQUENCE
Journal of Applied Mathematics & Informatics. 2015. Sep, 33(5_6): 593-605
Copyright © 2015, Korean Society of Computational and Applied Mathematics
  • Received : June 08, 2014
  • Accepted : June 22, 2015
  • Published : September 30, 2015
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About the Authors
ZAHID RAZA
MUHAMMAD ASIM ALI

Abstract
In this study, we define r −circulant, circulant, Hankel and Toeplitz matrices involving the integer sequence with recurrence relation Un = pU n-1 + U n-2 , with U 0 = a , U 1 = b . Moreover, we obtain special norms of above mentioned matrices. The results presented in this paper are generalizations of some of the results of [1 , 10 , 11] . AMS Mathematics Subject Classification : 15A45, 15A60, 15A36, 11B39.
Keywords
1. Introduction and Preliminaries
A lot of research papers on the norms of some special matrices have been writ-ten during the last decade [1 , 2 , 6 , 10 , 11] . Akbulak and Bozkurt [1] found lower and upper bounds for the spectral norms of Toeplitz matrices
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. Solak found bounds for the special norms of circulant matrices [10] . In [12] the authors determined the upper and lowers bounds for Cauchy-Toeplitz and Cauchy Hankel matrices. In [8] bounds of circulant, r −circulant, semi-circulant and Hankel matrices with tribonacci sequence obtained. In [6] , the author pre-sented some results about circulant, negacyclic and semi-circulant matrices with the modified Pell, Jacobsthal and Jacobsthal- Lucas numbers. Shen and Cen found the bounds of spectral norm of Fibonacci and Lucas numbers [11] . The generalized Fibonacci sequence is defined as:
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with initial conditions U 0 = a , U 1 = b , where a and b are positive integer.
It is clear that (1) can be written as:
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where Fn is called the n th term of p -Fibonacci sequence and defined by
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Generally from equation (3), we have
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Equation (2) can be written as:
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A matrix A = A r = (a ij ) ∈ Mn,n (ℂ) is called r −circulant on generalized se-quence, if it is of the form
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where r ∈ ℂ. If r =1, then matrix A is called circulant.
A matrix A = (a ij ) ∈ Mn,n (ℂ) is called semi-circulant on generalized Fi-bonacci sequence, if it is of the form
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A Hankel matrix on generalized Fibonacci sequence is defined as:
H = (h ij ) ∈ Mn,n (ℂ), where h ij = U i + j −1 . Similarly, a matrix A = (a ij ) ∈ Mn,n (ℂ) is Toeplitz matrix on generalized Fibonacci sequence (1), if it is of the form a ij = Ui−j . The p norm of a matrix A = (a ij ) ∈ Mn,n (ℂ) is defined by
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If
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. The Euclidean (Frobenius) norm of the matrix A is defined as:
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The spectral norm of the matrix A is given as:
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where γi are the eigenvalues of the matrix (Ā) t A.
The following inequality between Euclidean and spectral norm holds [13]
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Definition 1.1 ( [9] ). Let A = (a ij ) and B = (b ij ) be m × n matrices. Then, the Hadamard product of A and B is given by
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Definition 1.2 ( [10] ). The maximum column length norm c 1 (.) and maximum row length norm r 1 (.) for m × n matrix A = (a ij ) is defined as
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Theorem 1.3 ( [7] ). Let A = ( aij ), B = ( bij ) and C = ( cij ) be p × q matrices. If C = A B, then C 2 r 1 ( A ) c 1 ( B ).
The following lemmas describe the properties of p -Fibonacci sequence.
Lemma 1.4 ( [5] ). Let Fn be the n-th term of p-Fibonacci sequence then,
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Lemma 1.5 ( [5] ). The sum of square of first n terms of p-Fibonacci sequence is given by
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The following lemmas describes the properties of generalized Fibonacci se-quence Un .
Lemma 1.6. The sum of first n terms of generalized Fibonacci sequence Un is given as:
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Lemma 1.7 ( [5] ). The sum of square of first n terms of the sequence Un is given by:
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Lemma 1.8. Sum of product of consecutive terms of generalized Fibonacci se-quence is given as:
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where
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Proof . From equation (2), we have.
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By lemmas (1.4) and (1.5), we get
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Theorem 1.9. For all n ≥ 1
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where Rn and Sn are defined in lemma (1.4) and (1.5) respectively.
Proof . From Lemma (1.7) and (1.8), we have
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Lemma 1.10. For all n > 1
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Proof . From equation (5), we obtain
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On the other hand, from equation (3), we have
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Thus, we have
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2.r−circulant, circulant and semi-circulant
In this section, we shall give main results related to r −circulant, circulant and sem-circulant on generalized Fibonacci sequence Un .
Theorem 2.1. Let A = Ar ( U 0 , U 1 ,..., U n−1 ) be r−circulant matrix .
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Proof . The r −circulant matrix A on the sequence (1) is given as:
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and from the definition of Euclidean norm, we have
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Here we have two cases depending on r .
Case 1. If | r | ≥ 1, then from equation (8), we have
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and from lemma (1.7), we get.
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By inequality (7), we obtain
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On the other hand, let us define two new matrices C and D as :
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Then it is easy to see that A = C ◦ D, so from definition (1.2)
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Now using theorem (1.3), we obtain
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Combine inequalities (9) and (10), we get following inequality
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Case 2. If | r | ≤ 1, then we have
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By inequality (7), we get
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On the other hand, let the matrices C′ and D′ be defined as:
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such that A = C′ ◦ D′, then by definition (1.2), we obtain
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and
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Again by applying theorem (1.3)
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and combing inequality (11) and (12), we obtain the required result.
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Remark 2.2. The above theorem is the generalization of the result [11] . If put p = 1, U 0 = 0 and U 1 = 1 then Un = U n−1 + U n−2 , which is same as Fn = F n−1 + F n−2 with initial conditions F 0 = 0 and F 1 = 1.
Theorem 2.3. Let A be the circulant matrix on generalized Fibonacci sequence.
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Proof . Since by definition of circulant matrix, the matrix A is of the form
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and form the definition of Euclidean norm, one can get,
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By inequality (7), we get
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Let matrices B and C be defined as:
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Then the row norm and column norm of B and C are given as:
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Using theorem (1.3), we have
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Combine (14) and (15), we get
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Remark 2.4. Above result is the generalization of Solak 's work [10] , in which the author found the upper and lower bounds for the Euclidean and spectral norms of circulant matrices.
Theorem 2.5. Let A be an n × n semi-circulant matrix A = ( aij ) with the generalized Fibonacci numbers then,
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Proof . For the semi-circulant matrix A = (a ij ) with the Generalized Fibonacci sequence numbers we have
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From the definition of Euclidean norm, we have
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Using lemma (1.9), we get the required result
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3. Hankel and Toeplitz matrix norm
In this section, we have calculated the bounds of Hankel and Toeplitz matrix associated with generalized Fibonacci sequence.
Theorem 3.1. If A = ( aij ) is an n × n Hankel matrix with aij = U i + j −1 , then
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where Tn is defined in lemma (1.9).
Proof . From the definition of Hankel matrix, the matrix A is of the form
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So, we have
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Theorem 3.2. If A = ( aij ) is an n × n Hankel matrix with aij = U i + j −1 then, we have
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Proof . From theorem (3.1) and inequality (7), we have
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Let us define two new matrices
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It can be easily seen that A = M ◦ N. Thus we get
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Using the theorem (1.3), we have
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Theorem 3.3. If A = ( aij ) is an n × n Hankel matrix with aij = U i + j −1 . Then we have A 1 = ║ A = U 2n+1 U n+1 .
Proof . From the definition of the matrix A , we can write
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by lemma (1.6), we have
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Similarly, the row norm of the matrix A can be computed as:
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Theorem 3.4. The bounds of spectral norms of the Toeplitz matrix A are given as:
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and
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where T n−1 and T −(n−1) are defined in lemma (1.9) and (1.10) respectively.
Proof . The Toeplitz matrix A define by the sequence (1) is given as
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From the definition of Euclidean norm , we have
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From Lemma (1.9) and (1.10), we have
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Using inequality (7), we obtain
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On the other hand , let us consider the matrices.
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such that, A = C D . Then using definition (1.2)
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By theorem (1.3), we obtain the desired result
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Remark 3.5. Norms of Toeplitz matrix with Fibonacci and Lucas numbers have been calculated by Akbulak and Bozkurt [1] . Theorem (3.4) is a generalization of their paper.
BIO
Zahid Raza received M.Sc. from the University of Punjab and Ph.D from Abdus Salam School of Mathematical Sciences Government College University Lahore, Pakistan. Since 2010 he has been working at National University of Computer & Emerging Sciences, Lahore campus. Currently, he is working as an associate professor of mathematics in the faculty of Sciences and Humanities at Lahore Campus. His research interests include algebraic combinatorics and elementary number theory.
Department of Mathematics, National University of Computer & Emerging Sciences, La-hore campus B-Block Faisal Town, Lahore, Pakistan.
E-mail: zahid.raza@nu.edu.pk
Muhammad Msim Ali received MS mathematics from National University of Computer & Emerging Sciences. He is currently a faculty member at Punjab Group of Colleges, Lahore. His research interests are computational mathematics and combinatorial number theory.
Department of Mathematics, National University of Computer & Emerging Sciences, La-hore B-Block Faisal Town, Lahore, Pakistan.
E-mail: masimali99@gmail.com
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