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.

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 [

For

Particularly,

A

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

We can obtain

Inspired by reference [

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.

Let

Let

If

If

Let

(see [

The following important inequalities hold between the Frobenius norm and spectral norm:

For exponential forms

Using the definition of

For any positive integer

By the relationship between exponential forms and trigonometric functions,

If

If

If

If

The following is the proof of Theorem

The matrix

Using the definition of Frobenius norm and Lemma

Using

In another case, let the matrices

Since

Hence,

This proves Theorem

Now, we prove Theorem

Using the Frobenius norms and Theorem

Hence,

Therefore,

We can obtain

In another case, using Theorem

By Lemma

Therefore, if

If

Therefore, if

This proves Theorem

Now, we prove Theorem

Using the Frobenius norms and Theorem

Therefore,

We can obtain

In another case, using Theorem

Thus, the result is obtained as follows:

This completes all of the theorems.

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

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

The author declares conflicts of interest.

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

This work was supported by NSF (11771351).