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

Authors: Baijuan Shi
Journal: Journal of Mathematics
 (2021)

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## 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.


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## Body

## 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 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 [2–8]. 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 [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 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.Forex, ex=e2πix and then ex=1; by eiθ=cosθ+isinθ, note that e0=e1=e−1=en=1, and by the trigonometric sums, we have(1)∑k=1nekmn=n,n|m,0,otherwise.Particularly,∑k=0n−1ek/n=0, cosθ=eiθ+e−iθ/2andsinθ=eiθ−e−iθ/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.An×n row first-minus-last right- (RFMLR-) circulant matrix with the first row a0,a1,…,an−1, denoted by A=RFMLRcircfra0,a1,…,an−1, is defined by [11](2)A=a0a1a2⋯an−2an−1an−1a0−an−1a1⋯an−3an−2an−2an−1−an−2a0−an−1⋯an−4an−3⋮⋮⋮⋮⋮a1a2−a1a3−a2⋯an−1−an−2a0−an−1n×n.Obviously, the RFMLR-circulant matrix is determined by its first row, and we defineΘ1,−1 as the basic RFMLR-circulant matrix with the first row 0,1,0,…,0, namely,(3)Θ1,−1=010⋯0001⋯0⋮⋮⋮⋮1−10⋯0n×n.We can obtainΘ1,−1n=In−Θ1,−1. According to the structure of the power of the basic RFMLR-circulant matrix Θ1,−1, it is clear that(4)A=RFMLRcircfra0,a1,…,an−1=∑i=0n−1aiΘ1,−1i.Inspired by reference [11], 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,…,en−1n,B=RFMLRcircfrcos0⋅πn,cos1⋅πn,cos2⋅πn,…,cosn−1⋅πn,C=RFMLRcircfrsin0⋅πn,sin1⋅πn,sin2⋅πn,…,sinn−1⋅πn.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,…,en−1n,be a n×n RFMLR-circulant matrix; then, we have(7)n+12+2n−1sin2πn≤A2≤2n−1.Theorem 2.
Let(8)B=RFMLRcircfrcos0⋅πn,cos1⋅πn,cos2⋅πn,…,cosn−1⋅πn,be a n×n RFMLR-circulant matrix; then, we have the following:
Ifn is even,(9)αn≤B2≤2cotπ2n−1.
Ifn is odd,(10)αn≤B2≤2cscπ2n−1,where(11)α=5n2−8n+44+14secπn−n2−6n+6cosπn.Theorem 3.
Let(12)C=RFMLRcircfrsin0⋅πn,sin1⋅πn,sin2⋅πn,…,sinn−1⋅πn,be a n×n RFMLR-circulant matrix, we have(13)βnC2≤2cotπ2n,where(14)β=2n2+3n−24−18secπn.


## 2. Preliminaries

Definition 1.
(see [9]). Let any matrix A=aij∈Mm×nC; the spectral norm and the Euclidean norm of matrix A are defined by(15)A2=max1≤i≤nλiAHA,AF=∑i=1m∑j=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)1nAF≤A2≤AF.Lemma 1.
For exponential formsek/n,(17)∑k=0n−1ekn=n,∑k=0n−1ekn=∑k=0n−1e−kn=∑k=0n−1e2kn=0.Proof.
Using the definition ofex, ex=e2πix and eiθ=cosθ+isinθ, we have ex=1, namely, ∑k=0n−1ek/n=n.
ek/n is a geometric sequence, the common ratio is e1/n, and exey=ex+y, so we have(18)∑k=0n−1ekn=∑k=0n−1e−kn=∑k=0n−1e2kn=0.Lemma 2.
For any positive integern≥2, we have(19)∑k=0n−1coskπn=1,∑k=0n−1sinkπn=cotπ2n,∑k=0n−1cos2kπn=∑k=0n−1sin2kπn=n2.Proof.
By the relationship between exponential forms and trigonometric functions,cosθ=eiθ+e−iθ/2andsinθ=eiθ−e−iθ/2, and using Lemma 1, we can obtain sums mentioned above.Lemma 3.
Ifn is even,(20)∑j=1n−1cosjπn=cotπ2n−1.
Ifn is odd,(21)∑j=1n−1cosjπn=cscπ2n−1.Proof.
Ifn is even,(22)∑j=0n−1cosjπn=2∑j=1n/2cosjπn+1=2∑j=0n/2−1cosjπn−1,∑j=0n/2−1cosjπn=∑j=0n/2−1ej/2n+e−j/2n2=121−e1/41−e1/2n+1−e−1/41−e−1/2n=122−e1/4−e−1/4−e1/2n−e−1/2n+e1/2n−1/4+e1/4−1/2n2−e1/2n−e−1/2n=121−cosπ/n+sinπ/n1−cosπ/n=121+cotπ2n.
Ifn is odd,(23)∑j=0n−1cosjπn=∑j=1n−1cosjπn+1=2∑j=1n−1/2cosjπn+1=2∑j=0n−1/2cosjπn−1=∑j=0n−1/2ej2n+e−j2n−1=1−cosπ/n−cosn+1π/2n+cosn−1π/2n1−cosπ/n−1=cosn−1π/2n−cosn−1π/2n1−cosπ/n=cscπ2n.


## 3. Proofs of Theorems

The following is the proof of Theorem1.Proof.
The matrix(24)A=RFMLRcircfre0n,e1n,e3n,…,en−1n,is of the following form:(25)A=e0ne1ne2n⋯en−2nen−1nen−1ne1e1n⋯en−3nen−2nen−2ne2e1⋯en−4nen−3n⋮⋮⋮⋮⋮e1ne2n−e1ne3n−e2n⋯e2e1n×n,where(26)e1=e0n−en−1n,e2=en−1n−en−2n.
Using the definition of Frobenius norm and Lemma1, ak=ek/n, we have(27)AF2=∑k=0n−1ak2+n−1a12+n−2a22+⋯+an−12+n−1a0−an−12+n−2an−1−an−22+⋯+a2−a12,and then ak=ek/n=1,(28)a0−an−12=e0n−en−1n2=e0n−e−1n2=e1n−12,an−1−an−22=en−1n−en−2n2=e−1n−e−2n2=e1n−12,e1n−12=cos2πn−1+isin2πn2=4sin2πn,that is to say,(29)AF2=nn+12+2nn−1sin2πn.
UsingA2≥1/nAF, we can obtain the lower bound(30)A2≥n+12+2n−1sin2πn.
In another case, let the matricesQ1, Q2, and Q3 be defined by(31)Q1=010⋯00001⋯00000⋯00⋮⋮⋮⋮⋮000⋯01100⋯00n×n,Q2=000⋯00000⋯00010⋯00⋮⋮⋮⋮⋮000⋯00000⋯10n×n,Q3=000⋯00010⋯00001⋯00⋮⋮⋮⋮⋮000⋯10100⋯01n×n.Then, we can obtain A=∑k=0n−1akQ1k−∑k=1n−2an−k−1Q2k−an−1Q3, by identities of matrix norms,(32)A2=∑k=0n−1akQ1k−∑k=1n−2an−k−1Q2k−an−1Q32≤∑k=0n−1akQ12k+∑k=1n−2an−k−1Q22k+an−1Q32.
Since(33)Q1HQ1=10⋯001⋯0⋮⋮⋮00⋯1n×n,Q2HQ2=00⋯0001⋯00⋮⋮⋮⋮00⋯1000⋯00n×n,Q3HQ3=00⋯001⋯0⋮⋮⋮00⋯1n×n.
Hence,(34)Q12=Q22=Q32=1,A2≤∑k=0n−1ak+∑k=1n−2an−k−1+an−1,and by ak=ek/n=1, A2≤2n−1; thus,(35)n+12+2n−1sin2πn≤A2≤2n−1.
This proves Theorem1.
Now, we prove Theorem2.Proof.
(36)B=cos0πncos1πn⋯cosn−2πncosn−1πncosn−1πnm1⋯cosn−3πncosn−2πncosn−2πnm2⋯cosn−4πncosn−3πn⋮⋮⋮⋮cos1πncos2πn−cos1πn⋯m2m1n×n,where(37)m1=cos0πn−cosn−1πn,m2=cosn−1πn−cosn−2πn.
Using the Frobenius norms and Theorem1, as well as bj=cosjπ/n, we have(38)BF2=n∑j=0n−1bj2+∑j=1n−1jbj2−2∑j=1n−2jbjbj+1−2n−1b0bn−1=n∑j=0n−1bj2+∑k=1n−1∑j=n−kn−1bj2−2∑k=1n−2∑j=n−k−1n−2bjbj+1−2n−1b0bn−1=n∑j=0n−1bj2+∑k=1n−1∑j=0n−1bj2−∑j=0n−k−1bj2−2∑k=1n−2∑j=0n−2bjbj+1−∑j=0n−k−2bjbj+1−2n−1b0bn−1.By Lemma 1, ∑j=0n−1cos2jπ/n=n/2. Using the identities cosθ=eiθ+e−iθ/2,(39)ex=e2πix,cos2j+1πn=e2j+1/2n+e−2j+1/2n2,we can obtain(40)∑j=0n−1cos2j+1πn=12∑j=0n−1e2j+12n+e−2j+12n=12e1/2n1−e11−e1/n+e1/2n1−e−11−e−1/n=0,∑j=0n−1cosjπncosj+1πn=12∑j=0n−1cos2j+1πn+n2cosπn=n2cosπn.
Hence,(41)∑j=0n−2cosjπncosj+1πn=n−22cosπn,and then, we can obtain(42)∑k=1n−2∑j=0n−k−2cosjπncosj+1πn=18secπn+n−1n−24.
Therefore,(43)BF2=5n2−8n+44+14secπn−n2−6n+6cosπn.
We can obtainB2≥1/nBF=α/n, where(44)α=5n2−8n+44+14secπn−n2−6n+6cosπn.
In another case, using Theorem1, and for the matrices Q1, Q2, and Q3 as mentioned above, we have(45)B=∑j=0n−1bjQ1j−∑j=1n−2bn−j−1Q2j−bn−1Q3,B2=∑j=0n−1bjQ1j−∑j=1n−2bn−j−1Q2j−bn−1Q32≤∑j=0n−1bjQ12j+∑j=1n−2bn−j−1Q22j+bn−1Q32=∑j=0n−1bj+∑j=1n−2bn−j−1+bn−1=∑j=0n−1cosjπn+∑j=1n−2cosn−j−1πn+cosn−1πn=∑j=0n−1cosjπn+∑j=1n−2cosj+1πn+cosπn=2∑j=0n−1cosjπn−1.
By Lemma3, if n is even,(46)∑j=0n−1cosjπn=cotπ2n.
Therefore, ifn is even, B2≤2cotπ/2n−1.
Ifn is odd,(47)∑j=0n−1cosjπn=cscπ2n.
Therefore, ifn is odd, B2≤2cscπ/2n−1.
This proves Theorem2.
Now, we prove Theorem3.Proof.
(48)C=sin0πnsin1πn⋯sinn−2πnsinn−1πnsinn−1πns1⋯sinn−3πnsinn−2πnsinn−2πns2⋯sinn−4πnsinn−3πn⋮⋮⋮⋮sin1πnsin2πn−sin1πn⋯s2s1n×n,where(49)s1=sin0πn−sinn−1πn,s2=sinn−1πn−sinn−2πn.
Using the Frobenius norms and Theorem1, as well as cj=sinjπ/n, we have(50)CF2=n∑j=0n−1cj2+∑j=1n−1jcj2−2∑j=1n−2jcjcj+1−2n−1c0cn−1=n∑j=0n−1cj2+∑k=1n−1∑j=n−kn−1cj2−2∑k=1n−2∑j=n−k−1n−2cjcj+1=n∑j=0n−1cj2+∑k=1n−1∑j=0n−1cj2−∑j=0n−k−1cj2−2∑k=1n−2∑j=0n−2cjcj+1−∑j=0n−k−2cjcj+1.By Lemma 1, ∑j=0n−1sin2jπ/n=n/2. Using the identities(51)sinθ=eiθ−e−iθ2,cosθ=eiθ+e−iθ2,ex=e2πix,∑j=0n−2sinjπnsinj+1πn=12∑j=0n−2cosπn−cos2j+1πn=n2cosπn,and by Theorem 1, we can obtain(52)∑k=1n−2∑j=0n−k−2sinjπnsinj+1πn=∑k=1n−2n−k−12cosπn−12∑k=1n−2∑j=0n−k−2cos2j+1πn=nn−22cosπn−116secπn−n−1n−28.
Therefore,(53)CF2=2n2+3n−24−18secπn.
We can obtainC2≥1/nCF=β/n, where(54)β=2n2+3n−24−18secπn.
In another case, using Theorem1, and for the matrices Q1, Q2, and Q3 as mentioned above, by cj=sinjπ/n≥0, Q12=Q22=Q32=1. We have(55)C=∑j=0n−1cjQ1j−∑j=1n−2cn−j−1Q2j−cn−1Q3,C2=∑j=0n−1cjQ1j−∑j=1n−2cn−j−1Q2j−cn−1Q32≤∑j=0n−1cjQ12j+∑j=1n−2cn−j−1Q22j+cn−1Q32=∑j=0n−1cj+∑j=1n−2cn−j−1+cn−1=∑j=0n−1sinjπn+∑j=1n−2sinn−j−1πn+sinn−1πn=2∑j=0n−1sinjπn=2cotπ2n.
Thus, the result is obtained as follows:(56)βn≤C2≤2cotπ2n,where(57)β=2n2+3n−24−18secπn.
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 offx-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.



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