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On the norms of an r-circulant matrix with the generalized k-Horadam numbers
Journal of Inequalities and Applications volume 2013, Article number: 394 (2013)
In this paper, we present new upper and lower bounds for the spectral norm of an r-circulant matrix whose entries are the generalized k-Horadam numbers. Furthermore, we obtain new formulas to calculate the eigenvalues and determinant of the matrix H.
MSC:11B39, 15A60, 15A15.
1 Introduction and preliminaries
The r-circulant matrices [1–3] have been one of the most interesting research areas in the field of computation mathematics. It is known that these matrices have a wide range of applications in signal processing, coding theory, image processing, digital image disposal, linear forecast and design of self-regress.
Although there are many works about special matrices and their norms, we can mainly depict the following ones since they will be needed for our results. In , Solak defined the circulant matrices and , where and , and then investigated the upper and lower bounds for A and B, respectively. In , they defined circulant matrices involving k-Lucas and generalized k-Fibonacci numbers and also investigated the upper and lower bounds for the norms of these matrices. Later on, Shen and Cen in , found some other upper and lower bounds for the spectral norms of r-circulant matrices in the form , , where is the n th Fibonacci number and is the n th Lucas number. In , the authors found some other upper and lower bounds for the spectral norms of r-circulant matrices associated with the k-Fibonacci and k-Lucas numbers, respectively. It is clear that the same study about similar subject can also be done for different numbers. For instance, in , authors defined a circulant matrix whose entries are the generalized k-Horadam numbers and then computed the spectral norm, eigenvalues and determinant of the matrix.
At this point, we can keep going to present the fundamental material which is about the definition of the generalized k-Horadam sequence . In fact, by , it was defined as the form
where, , and . Obviously, if we choose suitable values on , , a and b for (1), then this sequence reduces to the special all second-order sequences in the literature. For example, by taking , and , the well-known Fibonacci sequence is obtained.
Binet’s formula allows us to express the generalized k-Horadam number in function of the roots α and β of the characteristic equation . Binet’s formula related to the sequence has the form
where and .
Lemma 1 
Let the entries of each matrix be the generalized k-Horadam numbers. For , we get
Definition 2 For any given , the r-circulant matrix is defined by
Let denote an r-circulant matrix.
It is obvious that the matrix turns into a classical circulant matrix for . Let us take any (which could be a circulant matrix as well). The well-known Frobenius (or Euclidean) norm of the matrix A is given by
and also the spectral norm of A is presented by
where are the eigenvalues of such that is the conjugate transpose of A. Then it is quite well known that
The following lemma will be needed in the proof of Theorem 5 below.
Lemma 3 
For any matrices and , we have
where is the Hadamard product and , .
Throughout this paper, the r-circulant matrix, whose entries are the generalized k-Horadam numbers, will be denoted by .
In this paper, we first give lower and upper bounds for the spectral norms of H. In particular, by taking , we obtain lower and upper bounds for the spectral norms of the circulant matrix associated with the generalized k-Horadam numbers. Afterwards, we also formulate the eigenvalues and determinant of the matrix H defined above.
2 Main results
Let us first consider the following lemma which states the sum square of the generalized k-Horadam numbers.
Lemma 4 For , we have
where , and .
Proof Let . By using (1), we have
By considering (1), we obtain
which is desired. □
The following theorem gives us the upper and lower bounds for the spectral norms of the matrix H.
Theorem 5 Let be an r-circulant matrix. Then we have
if , then
if , then
where , , .
Proof The matrix H is of the form
and, by the definition of Frobenius norm, we clearly have
If , then we obtain
Also, by considering Lemma 4, we can write
It follows that
where . Then by (1), we have
Similarly, for , we can write
Again, by considering Lemma 4 and (1), we get
It follows that
Now, for , we give the upper bound for the spectral norm of the matrix H as in the following. Let the matrices B and C be as
such that . Then we obtain
where , and . By considering Lemma 3, we can write
For , we give the upper bound for the spectral norm of the matrix H as in the following. On the other hand, if the matrices D and E are
such that , then we obtain
where , and . By Lemma 3, then we have
as required. □
Lemma 6 Let H be an r-circulant matrix. Then we have
where , , .
Considering Lemma 6, we obtain the following theorem which gives us the eigenvalues of the matrix in (3).
Theorem 7 Let be an r-circulant matrix. Then the eigenvalues of H are written by
where is the nth generalized k-Horadam number and , , .
Proof By Lemma 6, we have . Moreover, by (2), . Hence, we obtain
which completes the proof. □
Theorem 8 The determinant of is formulated by
Proof It is clear that . By Theorem 7, we get
Also, by considering the well-known identity , we can write
Thus the proof is completed, as desired. □
Remark 9 We should note that choosing suitable values on r, , , a and b in Theorems 5, 7 and 8, lower and upper bounds of the spectral norm, eigenvalues and determinant of r-circulant matrices for the special all second-order sequences are actually obtained.
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The authors thank the referees for their helpful comments and suggestions concerning the presentation of this paper. The authors are also thankful to TUBITAK.
The authors declare that they have no competing interests.
All authors completed the paper together. All authors read and approved the final manuscript.
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Yazlik, Y., Taskara, N. On the norms of an r-circulant matrix with the generalized k-Horadam numbers. J Inequal Appl 2013, 394 (2013). https://doi.org/10.1186/1029-242X-2013-394
- generalized k-Horadam sequence
- r-circulant matrix
- spectral norm