JMATHJournal of Mathematics2314-47852314-4629Hindawi10.1155/2021/20791042079104Research ArticleOn the Norms of RFMLR-Circulant Matrices with the Exponential and Trigonometric Functionshttps://orcid.org/0000-0003-2145-8929ShiBaijuanShabbirGhulamSchool of ScienceXi’an University of Posts and TelecommunicationsXi’anShaanxiChinaxiyou.edu.cn202117620212021294202156202186202117620212021Copyright © 2021 Baijuan Shi.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

National Natural Science Foundation of China11771351
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 . Recently, studying the norms of special matrices has been a hot topic in matrix theory. Especially, some scholars studied the norms of r-circulant matrices, geometric circulant matrices, and r-Hankel and r-Toeplitz matrices with some famous numbers and polynomials. For example, on the spectral norms of circulant matrices, r-circulant matrices, geometric circulant matrices, and r-Hankel and r-Toeplitz matrices with Fibonacci number, Lucas number, generalized Fibonacci and Lucas numbers, and generalized k-Horadam numbers have been studied . We have obtained several results [9, 10] of the norms of matrices mentioned above with exponential forms ek/n and trigonometric functions coskπ/n and sinkπ/n. There is a RFMLR-circulant matrix applied first by Jiang . 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 ek/n and trigonometric functions coskπ/n and sinkπ/n 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 ex, ex=e2πix and then ex=1; by eiθ=cosθ+isinθ, note that e0=e1=e1=en=1, and by the trigonometric sums, we have(1)k=1nekmn=n,n|m,0,otherwise.

Particularly, k=0n1ek/n=0, cosθ=eiθ+eiθ/2and sinθ=eiθeiθ/2. By the relationship between exponential forms ek/n and trigonometric functions coskπ/n and sinkπ/n, we can obtain some power sums of these functions.

A n×n row first-minus-last right- (RFMLR-) circulant matrix with the first row a0,a1,,an1, denoted by A=RFMLRcircfra0,a1,,an1, is defined by (2)A=a0a1a2an2an1an1a0an1a1an3an2an2an1an2a0an1an4an3a1a2a1a3a2an1an2a0an1n×n.

Obviously, the RFMLR-circulant matrix is determined by its first row, and we define Θ1,1 as the basic RFMLR-circulant matrix with the first row 0,1,0,,0, namely,(3)Θ1,1=010000101100n×n.

We can obtain Θ1,1n=InΘ1,1. According to the structure of the power of the basic RFMLR-circulant matrix Θ1,1, it is clear that(4)A=RFMLRcircfra0,a1,,an1=i=0n1aiΘ1,1i.

Inspired by reference , based on preliminary work, in this paper, we shall use identities of exponential forms ek/n and trigonometric functions coskπ/n and sinkπ/n and power sums of ek/n, coskπ/n, and sinkπ/n to study the norms of RFMLR-circulant matrices:(5)A=RFMLRcircfre0n,e1n,e3n,,en1n,B=RFMLRcircfrcos0πn,cos1πn,cos2πn,,cosn1πn,C=RFMLRcircfrsin0πn,sin1πn,sin2πn,,sinn1πn.

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.

Let(6)A=RFMLRcircfre0n,e1n,e3n,,en1n,be a n×n RFMLR-circulant matrix; then, we have(7)n+12+2n1sin2πnA22n1.

Theorem 2.

Let(8)B=RFMLRcircfrcos0πn,cos1πn,cos2πn,,cosn1πn,be a n×n RFMLR-circulant matrix; then, we have the following:

If n is even,(9)αnB22cotπ2n1.

If n is odd,(10)αnB22cscπ2n1,where(11)α=5n28n+44+14secπnn26n+6cosπn.

Theorem 3.

Let(12)C=RFMLRcircfrsin0πn,sin1πn,sin2πn,,sinn1πn,be a n×n RFMLR-circulant matrix, we have(13)βnC22cotπ2n,where(14)β=2n2+3n2418secπn.

2. PreliminariesDefinition 1.

(see ). Let any matrix A=aijMm×nC; the spectral norm and the Euclidean norm of matrix A are defined by(15)A2=max1inλiAHA,AF=i=1mj=1naij21/2,respectively, where λiAHA is the eigenvalues of matrices AHA and AH is the conjugate transpose of A.

The following important inequalities hold between the Frobenius norm and spectral norm:(16)1nAFA2AF.

Lemma 1.

For exponential forms ek/n,(17)k=0n1ekn=n,k=0n1ekn=k=0n1ekn=k=0n1e2kn=0.

Proof.

Using the definition of ex, ex=e2πix and eiθ=cosθ+isinθ, we have ex=1, namely, k=0n1ek/n=n.

ek/n is a geometric sequence, the common ratio is e1/n, and exey=ex+y, so we have(18)k=0n1ekn=k=0n1ekn=k=0n1e2kn=0.

Lemma 2.

For any positive integer n2, we have(19)k=0n1coskπn=1,k=0n1sinkπn=cotπ2n,k=0n1cos2kπn=k=0n1sin2kπn=n2.

Proof.

By the relationship between exponential forms and trigonometric functions, cosθ=eiθ+eiθ/2 and sinθ=eiθeiθ/2, and using Lemma 1, we can obtain sums mentioned above.

Lemma 3.

If n is even,(20)j=1n1cosjπn=cotπ2n1.

If n is odd,(21)j=1n1cosjπn=cscπ2n1.

Proof.

If n is even,(22)j=0n1cosjπn=2j=1n/2cosjπn+1=2j=0n/21cosjπn1,j=0n/21cosjπn=j=0n/21ej/2n+ej/2n2=121e1/41e1/2n+1e1/41e1/2n=122e1/4e1/4e1/2ne1/2n+e1/2n1/4+e1/41/2n2e1/2ne1/2n=121cosπ/n+sinπ/n1cosπ/n=121+cotπ2n.

If n is odd,(23)j=0n1cosjπn=j=1n1cosjπn+1=2j=1n1/2cosjπn+1=2j=0n1/2cosjπn1=j=0n1/2ej2n+ej2n1=1cosπ/ncosn+1π/2n+cosn1π/2n1cosπ/n1=cosn1π/2ncosn1π/2n1cosπ/n=cscπ2n.

3. Proofs of Theorems

The following is the proof of Theorem 1.

Proof.

The matrix(24)A=RFMLRcircfre0n,e1n,e3n,,en1n,is of the following form:(25)A=e0ne1ne2nen2nen1nen1ne1e1nen3nen2nen2ne2e1en4nen3ne1ne2ne1ne3ne2ne2e1n×n,where(26)e1=e0nen1n,e2=en1nen2n.

Using the definition of Frobenius norm and Lemma 1, ak=ek/n, we have(27)AF2=k=0n1ak2+n1a12+n2a22++an12+n1a0an12+n2an1an22++a2a12,and then ak=ek/n=1,(28)a0an12=e0nen1n2=e0ne1n2=e1n12,an1an22=en1nen2n2=e1ne2n2=e1n12,e1n12=cos2πn1+isin2πn2=4sin2πn,that is to say,(29)AF2=nn+12+2nn1sin2πn.

Using A21/nAF, we can obtain the lower bound(30)A2n+12+2n1sin2πn.

In another case, let the matrices Q1, Q2, and Q3 be defined by(31)Q1=0100000100000000000110000n×n,Q2=0000000000010000000000010n×n,Q3=0000001000001000001010001n×n.Then, we can obtain A=k=0n1akQ1kk=1n2ank1Q2kan1Q3, by identities of matrix norms,(32)A2=k=0n1akQ1kk=1n2ank1Q2kan1Q32k=0n1akQ12k+k=1n2ank1Q22k+an1Q32.

Since(33)Q1HQ1=100010001n×n,Q2HQ2=0000010000100000n×n,Q3HQ3=000010001n×n.

Hence,(34)Q12=Q22=Q32=1,A2k=0n1ak+k=1n2ank1+an1,and by ak=ek/n=1, A22n1; thus,(35)n+12+2n1sin2πnA22n1.

This proves Theorem 1.

Now, we prove Theorem 2.

Proof.

(36)B=cos0πncos1πncosn2πncosn1πncosn1πnm1cosn3πncosn2πncosn2πnm2cosn4πncosn3πncos1πncos2πncos1πnm2m1n×n,where(37)m1=cos0πncosn1πn,m2=cosn1πncosn2πn.

Using the Frobenius norms and Theorem 1, as well as bj=cosjπ/n, we have(38)BF2=nj=0n1bj2+j=1n1jbj22j=1n2jbjbj+12n1b0bn1=nj=0n1bj2+k=1n1j=nkn1bj22k=1n2j=nk1n2bjbj+12n1b0bn1=nj=0n1bj2+k=1n1j=0n1bj2j=0nk1bj22k=1n2j=0n2bjbj+1j=0nk2bjbj+12n1b0bn1.By Lemma 1, j=0n1cos2jπ/n=n/2. Using the identities cosθ=eiθ+eiθ/2,(39)ex=e2πix,cos2j+1πn=e2j+1/2n+e2j+1/2n2,we can obtain(40)j=0n1cos2j+1πn=12j=0n1e2j+12n+e2j+12n=12e1/2n1e11e1/n+e1/2n1e11e1/n=0,j=0n1cosjπncosj+1πn=12j=0n1cos2j+1πn+n2cosπn=n2cosπn.

Hence,(41)j=0n2cosjπncosj+1πn=n22cosπn,and then, we can obtain(42)k=1n2j=0nk2cosjπncosj+1πn=18secπn+n1n24.

Therefore,(43)BF2=5n28n+44+14secπnn26n+6cosπn.

We can obtain B21/nBF=α/n, where(44)α=5n28n+44+14secπnn26n+6cosπn.

In another case, using Theorem 1, and for the matrices Q1, Q2, and Q3 as mentioned above, we have(45)B=j=0n1bjQ1jj=1n2bnj1Q2jbn1Q3,B2=j=0n1bjQ1jj=1n2bnj1Q2jbn1Q32j=0n1bjQ12j+j=1n2bnj1Q22j+bn1Q32=j=0n1bj+j=1n2bnj1+bn1=j=0n1cosjπn+j=1n2cosnj1πn+cosn1πn=j=0n1cosjπn+j=1n2cosj+1πn+cosπn=2j=0n1cosjπn1.

By Lemma 3, if n is even,(46)j=0n1cosjπn=cotπ2n.

Therefore, if n is even, B22cotπ/2n1.

If n is odd,(47)j=0n1cosjπn=cscπ2n.

Therefore, if n is odd, B22cscπ/2n1.

This proves Theorem 2.

Now, we prove Theorem 3.

Proof.

(48)C=sin0πnsin1πnsinn2πnsinn1πnsinn1πns1sinn3πnsinn2πnsinn2πns2sinn4πnsinn3πnsin1πnsin2πnsin1πns2s1n×n,where(49)s1=sin0πnsinn1πn,s2=sinn1πnsinn2πn.

Using the Frobenius norms and Theorem 1, as well as cj=sinjπ/n, we have(50)CF2=nj=0n1cj2+j=1n1jcj22j=1n2jcjcj+12n1c0cn1=nj=0n1cj2+k=1n1j=nkn1cj22k=1n2j=nk1n2cjcj+1=nj=0n1cj2+k=1n1j=0n1cj2j=0nk1cj22k=1n2j=0n2cjcj+1j=0nk2cjcj+1.By Lemma 1, j=0n1sin2jπ/n=n/2. Using the identities(51)sinθ=eiθeiθ2,cosθ=eiθ+eiθ2,ex=e2πix,j=0n2sinjπnsinj+1πn=12j=0n2cosπncos2j+1πn=n2cosπn,and by Theorem 1, we can obtain(52)k=1n2j=0nk2sinjπnsinj+1πn=k=1n2nk12cosπn12k=1n2j=0nk2cos2j+1πn=nn22cosπn116secπnn1n28.

Therefore,(53)CF2=2n2+3n2418secπn.

We can obtain C21/nCF=β/n, where(54)β=2n2+3n2418secπn.

In another case, using Theorem 1, and for the matrices Q1, Q2, and Q3 as mentioned above, by cj=sinjπ/n0, Q12=Q22=Q32=1. We have(55)C=j=0n1cjQ1jj=1n2cnj1Q2jcn1Q3,C2=j=0n1cjQ1jj=1n2cnj1Q2jcn1Q32j=0n1cjQ12j+j=1n2cnj1Q22j+cn1Q32=j=0n1cj+j=1n2cnj1+cn1=j=0n1sinjπn+j=1n2sinnj1πn+sinn1πn=2j=0n1sinjπn=2cotπ2n.

Thus, the result is obtained as follows:(56)βnC22cotπ2n,where(57)β=2n2+3n2418secπn.

This 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 fx-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).

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