Abstract
In this paper, using the properties of Hankel and Toeplitz matrices, combining the properties of exponential form, we shall study the spectral norms of Hankel and Toeplitz matrices involving exponential form .
1. Introduction
Circulant matrix is first proposed by Professor T. Muir in 1885, and he has carried on the preliminary study. Until 1950-1955, Good et al. began to study the determinants and inverses of circulant matrices, which have opened the curtain of the research on circulant matrices. Recently, circulant matrices have been one of the most important research subjects in the field of the computation and pure mathematics, which has been widely applied in the field of modern science and technology engineering, for example, signal processing, coding theory, image processing and electrodynamics, etc. Particularly, circulant, circulant, geometric circulant matrices, Hankel, and Toeplitz matrices occupy very important position in matrices theory. As we all know, circulant matrix is a special form of Toeplitz matrix. So studying the Toeplitz matrix is a very meaningful job. Hence, based on the special properties and structures of circulant matrices, many scholars at home and abroad have studied the determinants, norms, and inverses of above matrices with well-known number sequences and made great achievements [1–11]. For example, Solak, S has studied the norms of circulant matrices with the Fibonacci and Lucas numbers. C. Köme and Y. Yazlik [5, 10] have studied the determinants, inverses, and spectral norms of of circulant matrices with biperiodic Fibonacci and Lucas numbers. H. Gökba and R. Türkmen have studied the norms of Toeplitz matrices involving Fibonacci and Lucas numbers. The authors Y. Yazlik, N. Yilmaz, and N. Taskara [11] have studied the norms of Hankel matrices with the Jacobsthal and Jacobsthal Lucas numbers. In 2017, H. Gökba and H. Köse have studied the norms of Hankel matrices involving Pell and Pell-Lucas numbers. Considering all of references, we shall study some new problems. As far as we know, it actually seems that no one has studied the upper and lower estimate problems for the spectral norms involving exponential form yet, where we define .
For , , by , note that
, and by the trigonometric sums, we have A Hankel matrix is defined by [9] Namely, A matrix is called a Toeplitz matrix
Obviously, when the parameter satisfies , we can get the classical Hankel and Toeplitz matrices. In this paper, we shall study the norms of and whose entries are exponential form .
Then we get some interesting and concise results which are stated by the following theorems.
Theorem 1. For any positive number , let be an Hankel matrix, then we have
Theorem 2. For any positive number , let be an Toeplitz matrix, then we have
2. Preliminaries
Definition 3 (see [3]). Let any matrix , the spectral norm is defined by ,
and the Euclidean norm of matrix is given by ,
where is the eigenvalues of matrices and is the conjugate transpose of .
The following important inequalities hold between the Euclidean norm and spectral norm:
Definition 4 (see [3]). Let and be matrices, then the Hadamard product of and is
Then we have the following inequalities:,
Lemma 5. For any positive integer , we have
Proof. Taking and , by the trigonometric sums, we have Taking , we have and therefore,That is,
3. Proofs of Theorems
Proof of Theorem 1.
Proof. For the matrix (i) From , using the definition of Euclidean norm and properties of , we have by (8); that is to say, On the other hand, let the matrices and be defined by and then . So , Therefore, we have Thus, we can obtain the inequality (ii) From , we can getFor another, let the matrices and be as mentioned above, .
So
Therefore, we have
This proofs Theorem 1.
Now we proof Theorem 2.
Proof. For , , (i) From , by , we have that is,
On the other hand, let the matrices and be given by and then . So , Therefore, we have Thus, we can obtain (ii) From , we can get On the other hand, for the matrices and as mentioned above, .
So
Therefore, we have
This proofs Theorem 2.
This completes all of the theorems.
Data Availability
The data of this article compute by algebra method which is available.
Conflicts of Interest
The author declare that there are no conflicts of interest.
Authors’ Contributions
I contributed to each part of this work seriously and read and approved the final version of the manuscript.
Acknowledgments
This work is supported by the NSF of China (Grant no. 11771351).