On the Norms of RFMLR-Circulant Matrices with the Exponential and Trigonometric Functions

Abstract

In this paper, based on combinatorial methods and the structure of RFMLR-circulant matrices, we study the spectral norms of RFMLR-circulant matrices involving exponential forms and trigonometric functions. Firstly, we give some properties of exponential forms and trigonometric functions. Furthermore, we study Frobenius norms, the lower and upper bounds for the spectral norms of RFMLR-circulant matrices involving exponential forms and trigonometric functions by some ingenious algebra methods, and then we obtain new refined results.

1. Introduction

Matrix analysis theory is a powerful tool to study modern communication systems; especially, matrix norm is very important for neural network-based adaptive tracking control for switched nonlinear systems [1]. Recently, studying the norms of special matrices has been a hot topic in matrix theory. Especially, some scholars studied the norms of -circulant matrices, geometric circulant matrices, and -Hankel and -Toeplitz matrices with some famous numbers and polynomials. For example, on the spectral norms of circulant matrices, -circulant matrices, geometric circulant matrices, and -Hankel and -Toeplitz matrices with Fibonacci number, Lucas number, generalized Fibonacci and Lucas numbers, and generalized -Horadam numbers have been studied [2–8]. We have obtained several results [9, 10] of the norms of matrices mentioned above with exponential forms and trigonometric functions and . There is a RFMLR-circulant matrix applied first by Jiang [11]. As far as we know, it seems that no one has studied the upper and lower estimate problems for the spectral norms of RFMLR-circulant matrices involving exponential forms and trigonometric functions and yet. Although some scholars at home and abroad have given algorithms about norms of circulant matrices, the computational complexity of these algorithms is amazing with the increase of matrix order. To overcome this defect, we have constructed matrix factorization and then we study the norms of RFMLR-circulant matrices involving exponential forms and trigonometric functions. Compared with the existing methods, the algorithm model of this study is easy to implement and then we use some ingenious methods to obtain new refined results. All results can be well applied in adaptive feedback control, and these have potential applications in neural network nonlinear system-based norms.

For , and then ; by , note that , and by the trigonometric sums, we have

Particularly, , . By the relationship between exponential forms and trigonometric functions and , we can obtain some power sums of these functions.

A row first-minus-last right- (RFMLR-) circulant matrix with the first row , denoted by , is defined by [11]

Obviously, the RFMLR-circulant matrix is determined by its first row, and we define as the basic RFMLR-circulant matrix with the first row , namely,

We can obtain . According to the structure of the power of the basic RFMLR-circulant matrix , it is clear that

Inspired by reference [11], based on preliminary work, in this paper, we shall use identities of exponential forms and trigonometric functions and and power sums of , , and to study the norms of RFMLR-circulant matrices:

We give Frobenius norms, the lower and upper bounds for the spectral norms of these matrices. Some interesting and concise results are stated by the following theorems.

Theorem 1. Letbe a RFMLR-circulant matrix; then, we have

Theorem 2. Letbe a RFMLR-circulant matrix; then, we have the following:
If is even,If is odd,where

Theorem 3. Letbe a RFMLR-circulant matrix, we havewhere

2. Preliminaries

Definition 1. (see [9]). Let any matrix ; the spectral norm and the Euclidean norm of matrix are defined byrespectively, where is the eigenvalues of matrices and is the conjugate transpose of .
The following important inequalities hold between the Frobenius norm and spectral norm:

Lemma 1. For exponential forms ,

Proof. Using the definition of , and , we have , namely, .
is a geometric sequence, the common ratio is , and , so we have

Lemma 2. For any positive integer , we have

Proof. By the relationship between exponential forms and trigonometric functions, , and using Lemma 1, we can obtain sums mentioned above.

Lemma 3. If is even,If is odd,

Proof. If is even,If is odd,

3. Proofs of Theorems

The following is the proof of Theorem 1.

Proof. The matrixis of the following form:whereUsing the definition of Frobenius norm and Lemma 1, , we haveand then ,that is to say,Using , we can obtain the lower boundIn another case, let the matrices , , and be defined byThen, we can obtain , by identities of matrix norms,SinceHence,and by , ; thus,This proves Theorem 1.
Now, we prove Theorem 2.

Proof. whereUsing the Frobenius norms and Theorem 1, as well as , we haveBy Lemma 1, . Using the identities ,we can obtainHence,and then, we can obtainTherefore,We can obtain , whereIn another case, using Theorem 1, and for the matrices , , and as mentioned above, we haveBy Lemma 3, if is even,Therefore, if is even, .
If is odd,Therefore, if is odd, .
This proves Theorem 2.
Now, we prove Theorem 3.

Proof. whereUsing the Frobenius norms and Theorem 1, as well as , we haveBy Lemma 1, . Using the identitiesand by Theorem 1, we can obtainTherefore,We can obtain , whereIn another case, using Theorem 1, and for the matrices , , and as mentioned above, by , . We haveThus, the result is obtained as follows:whereThis completes all of the theorems.

4. Conclusion

The spectral norms of RFMLR-circulant matrices involving exponential forms and trigonometric functions are investigated in this paper. The computation complexity of this paper is lower than the previous work. By using the algorithms of this paper, we can further study the identities of -circulant matrices, such as RFPrLrR-cieculant matrix and RFMLrR-circulant matrix. Simulation is very important for applications of our work, and these will be our further topics to study.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The author declares conflicts of interest.

Authors’ Contributions

The author contributed to each part of this work seriously and read and approved the final version of the manuscript.

Acknowledgments

This work was supported by NSF (11771351).