R E S E A R C H
Open Access
On the spectral norms of some circulant
matrices with the trigonometric functions
Baijuan Shi
1**Correspondence:
1School of Mathematics, Northwest
University, Xi’an, P.R. China
Abstract
In this paper, we use the properties of anr-circulant matrix and a geometric circulant matrix to study the spectral norms of ther-circulant matrix and the geometric circulant matrix involving trigonometric functions by some algebra methods. Keywords: Geometric circulant matrix;r-circulant matrix; Trigonometric functions; Spectral norms; Euclidean norms
1 Introduction
In 1885, circulant matrix was first proposed by Muir, and he did some basic research. Until 1950–1955, Good et al. began to study the inverse, determinants and characteristic values of circulant matrices; these efforts have opened the door to study circulant matri-ces. A circulant matrix is a kind of matrix with a special structure, which has been widely used in algebra, geometry, signal processing and coding theory. In recent years, the cir-culant matrix is still a topic of focus in the research of matrix theory. Especially, some scholars studied the norms ofr-circulant matrices and geometric circulant matrices with some famous numbers and polynomials, for example, on the spectral norms of circulant matrices,r-circulant matrices, geometric circulant matrices with Fibonacci number, Lu-cas number, generalized Fibonacci and LuLu-cas numbers, generalizedk-Horadam numbers, the biperiodic Fibonacci and Lucas numbers have been studied [1–13]. To the best of our knowledge, no one has studied the upper and lower estimate problems for the spectral norms involving trigonometric functionscos(kπ
n),sin( kπ
n) yet by using exponential sum.
An×n r-circulant matrixCris defined by [8]
Cr=
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
c0 c1 c2 · · · cn–2 cn–1
rcn–1 c0 c1 · · · cn–3 cn–2
rcn–2 rcn–1 c0 · · · cn–4 cn–3
..
. ... ... ... ...
rc1 rc2 rc3 · · · rcn–1 c0
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
n×n
.
Kızılateş and Tuglu [9] defined geometric circulant matrices by the form
Cr∗=
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
c0 c1 c2 · · · cn–2 cn–1
rcn–1 c0 c1 · · · cn–3 cn–2
r2cn–2 rcn–1 c0 · · · cn–4 cn–3
..
. ... ... ... ...
rn–1c
1 rn–2c2 rn–3c3 · · · rcn–1 c0
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
n×n
.
Obviously, when the parameter satisfiesr= 1, we can get the classical circulant matrix. Inspired by [7], in this paper, we shall use identities of the trigonometric functions and power sums ofcos(knπ),sin(knπ) to study the norms of ther-circulant matrices
A= Circr
cos0·π n ,cos
1·π n ,cos
2·π n , . . . ,cos
(n– 1)·π
n ,
B= Circr
sin0·π n ,sin
1·π n ,sin
2·π n , . . . ,sin
(n– 1)·π
n ,
and then we obtain the norms of geometric circulant matrices
Pr∗= Circr∗
cos0·π n ,cos
1·π n ,cos
2·π n , . . . ,cos
(n– 1)·π
n ,
Rr∗= Circr∗
sin0·π n ,sin
1·π n ,sin
2·π n , . . . ,sin
(n– 1)·π
n .
Then we get some interesting and concise results which are stated by the following theo-rems.
Theorem 1 Let A=Cr(cos0n·π,cos1·nπ,cos2·nπ, . . . ,cos(n–1)n·π)be an r-circulant matrix, then we have
|r| ≥1,
√
2
2 ≤ A2≤
n
2
(n– 1)|r|2+ 1;
|r|< 1,
√
2
2 |r| ≤ A2≤
√
2 2 n.
Theorem 2 Let B=Cr(sin0n·π,sin1·nπ,sin2·nπ, . . . ,sin(n–1)n·π)be an r-circulant matrix,then we have
|r| ≥1,
√
2
2 ≤ B2≤ |r|
n(n– 1) 2 ;
|r|< 1,
√
2
2 |r| ≤ B2≤
n(n– 1) 2 .
Theorem 3 Let Pr∗=Cr∗(cos0·nπ,cos1n·π,cos2n·π, . . . ,cos(n–1)n·π)be a geometric circulant matrix,we have
|r|> 1,
√
2
2 ≤ Pr∗2≤
n
2
1 –|r|2n
|r|< 1, |r|nN1≤ Pr∗2≤
√
2 2 n,
where N1=1–r
–2–r–2n+2
4 +
1–r–2n
2(1–r–2).
Theorem 4 Let Rr∗ =Cr∗(sin0n·π,sin1n·π,sin2·nπ, . . . ,sin(n–1)n·π) be a geometric circulant matrix,we have
|r|> 1,
√
2
2 ≤ Rr∗2≤
n
2
|r|2–|r|2n
1 –|r|2 ;
|r|< 1, |r|nN2≤ Rr∗2≤
n(n– 1) 2 ,
where N2= 1–r
–2n
2(1–r–2)–1–r –2–r–2n+2
4 .
2 Preliminaries
Definition 1([9]) Let any matrixA= (aij)∈Mm×n(C), the spectral norm and the
Eu-clidean norm of matrixAare defined by
A2=
max
1≤i≤nλi
AHA, A E=
m
i=1
n
j=1
|aij|2
1 2
, respectively,
where theλi(AHA) are the eigenvalues of matricesAHAandAHis the conjugate transpose
ofA.
The following important inequalities hold between the Euclidean norm and spectral norm:
1
√
nAE≤ A2≤ AE. (1)
Definition 2([9]) Let bothA= (aij) andB= (bij) bem×nmatrices, then the Hadamard
product ofAandBis them×nmatrix of elementwise products, namelyA◦B= (aijbij).
Then we have the following inequalities:
A◦B2≤r1(A)C1(B), (2)
r1(A) = max 1≤i≤m
n
j=1
|aij|2, C1(B) =max 1≤j≤n
m
i=1
|bij|2.
Lemma 1([7]) For any positive integer n≥2,we have
n–1
k=0
cos2
kπ
n =
n–1
k=0
sin2
kπ n =
n
2.
Lemma 2 For any positive integer n≥2,we can get
n–1
k=0
r–2kcos2
kπ
n =
1 –r–2n
2(1 –r–2)+
1 –r–2–r–2n+2
n–1
k=0
r–2ksin2
kπ
n =
1 –r–2n
2(1 –r–2)–
1 –r–2–r–2n+2
4 =N2.
Proof By the properties ofcos2θ= 2cos2θ– 1 = 1 – 2sin2θ,eiθ=cosθ+isinθ, we can
easily getcosθ=eiθ+2e–iθ; lete(x) =e2πix, note thate(1) =e(–1) = 1, using the properties of
the trigonometric sumsnk–1=0e(kn) = 0. Hence,
n–1
k=0
r–2kcos2
kπ
n =
n–1
k=0
r–2k1 +cos(
2kπ
n )
2
= 1 –r
–2n
2(1 –r–2)+
1 4
n–1
k=0
r–2k
e
k n +e
–k n
.
Taking
S1=
n–1
k=0
r–2ke
k n
=r–2·01 +r–2·1e
1
n +r
–2·2e
2
n +· · ·+r
–2·(n–2)e
n– 2
n +r
–2·(n–1)e
n– 1
n , e
1
n S1=r
–2·0e
1
n +r
–2·1e
2
n +r
–2·2e
3
n +· · ·
+r–2·(n–2)e
n– 1
n +r
–2·(n–1)e(1).
Therefore,
1 –e
1
n
S1= 1 +r–2
n–1
k=1
e
k n –r
–2n+2= 1 –r–2–r–2n+2,
that isS1=
n–1
k=0r–2ke(nk) =
1–r–2–r–2n+2
1–e(1n) , as the same time,
n–1
k=0r–2ke(–nk) =
1–r–2–r–2n+2
1–e(–1n) .
So,
n–1
k=0
r–2kcos2
kπ
n =
1 –r–2n
2(1 –r–2)+
1 –r–2–r–2n+2
4
1 1 –e(1n)+
1 1 –e(–1n) = 1 –r
–2n
2(1 –r–2)+
1 –r–2–r–2n+2
4 =N1. Using the same methods, note that
S2=
n–1
k=0
r–2ksin2
kπ n =
n–1
k=0
r–2k1 –cos(
2kπ
n )
2
=1 2
n–1
k=0
r–2k–
n–1
k=0
r–2kcos
2kπ n
= 1 –r
–2n
2(1 –r–2)–
1 –r–2–r–2n+2
3 Proofs of theorems
Proof of Theorem1 The matrixA=Cr(cos0·nπ,cos1n·π,cos2·nπ, . . . ,cos(n–1)n·π) is of the
fol-lowing form:
A=
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
cos0·nπ cos1·nπ cos2n·π · · · cos(n–1)n·π rcos(n–1)n·π cos0·nπ cos1n·π · · · cos(n–2)n·π rcos(n–2)n·π rcos(n–1)n·π cos0·π
n · · · cos
(n–3)·π
n
..
. ... ... . .. ...
rcos1·nπ rcos2·nπ rcos3n·π · · · cos0n·π
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
n×n
.
(i) From|r| ≥1, using the definition of Euclidean norm and Lemma1, we have
A2
E= n–1
k=0
(n–k)cos2
k·π
n +
n–1
k=1
k|r|2cos2
k·π n
≥
n–1
k=0
(n–k)cos2
k·π
n +
n–1
k=1
kcos2
k·π n
=n n–1
k=0
cos2
k·π
n =
n2
2 , by (1), that is to say,
A2≥
1
√
nAE≥
√
2 2 .
Moreover, let the matricesEandFbe defined by
E=
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
1 1 1 · · · 1
r 1 1 · · · 1
r r 1 · · · 1 ..
. ... ... ...
r r r · · · 1
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
n×n
and
F=
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
cos0·nπ cos1·nπ cos2n·π · · · cos(n–1)n·π cos(n–1)n·π cos0·nπ cos1n·π · · · cos(n–2)n·π cos(n–2)n·π cos(n–1)n·π cos0·π
n · · · cos
(n–3)·π
n
..
. ... ... . .. ...
cos1·nπ cos2·nπ cos3n·π · · · cos0·nπ
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
n×n
,
thenA=E◦F. SoA2=E◦F2≤r1(E)C1(F),
r1(E) = max 1≤i≤n
n
j=1
|eij|2=
c1(F) =max 1≤j≤n
n
i=1
|fij|2=
n–1
k=0
cos2k·π
n =
n
2.
Therefore, we have
A2≤
(n– 1)r2+ 1
n
2. Thus, we can obtain the inequality
√
2
2 ≤ A2≤
n
2
(n– 1)|r|2+ 1.
(ii) From|r|< 1,
A2E=
n–1
k=0
(n–k)cos2
k·π
n +
n–1
k=1
k|r|2cos2
k·π n
≥
n–1
k=0
(n–k)|r|2cos2
k·π
n +
n–1
k=1
k|r|2cos2
k·π n
=n|r|2 n–1
k=0
cos2
k·π
n =
|r|2n2
2 , we can get
A2≥
1
√
nAE≥
√
2 2 |r|.
Moreover, for the matricesEandFas mentioned above,A=E◦F. SoA2=E◦F2≤
r1(E)C1(F) =
√
2 2 n.
Therefore, we have
√
2
2 |r| ≤ A2≤
√
2 2 n.
This proves Theorem1. Now we prove Theorem2.
Proof
B=
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
sin0·nπ sin1·nπ sin2·nπ · · · sin(n–1)n·π rsin(n–1)·π
n sin
0·π
n sin
1·π
n · · · sin
(n–2)·π
n rsin(n–2)·π
n rsin
(n–1)·π
n sin
0·π
n · · · sin
(n–3)·π
n
..
. ... ... . .. ...
rsin1·nπ rsin2·nπ rsin3·nπ · · · sin0n·π
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
n×n
.
(i) From|r| ≥1, using the definition of Euclidean norm and Lemma1, we have
B2E=
n–1
k=0
(n–k)sin2
k·π
n +
n–1
k=1
k|r|2sin2
≥
n–1
k=0
(n–k)sin2
k·π
n +
n–1
k=1
ksin2
k·π n
=n n–1
k=0
sin2
k·π
n =
n2
2, that is,
B2≥
1
√
nBE≥
√
2 2 .
Moreover, let the matricesCandDbe defined by
C=
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
sin0·nπ 1 1 · · · 1
r sin0·π
n 1 · · · 1
r r sin0·nπ · · · 1
..
. ... ... ...
r r r · · · sin0·nπ
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
n×n
and
D=
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
sin0n·π sin1·nπ sin2·nπ · · · sin(n–1)n·π sin(n–1)n·π sin0·nπ sin1·nπ · · · sin(n–2)n·π sin(n–2)·π
n sin
(n–1)·π
n sin
0·π
n · · · sin
(n–3)·π
n
..
. ... ... . .. ...
sin1n·π sin2·nπ sin3·nπ · · · sin0·nπ
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
n×n
,
thenB=C◦D. SoB2=C◦D2≤r1(C)C1(D),
r1(C) =max 1≤i≤n
n
j=1
|cij|2=
(n– 1)r2;
c1(D) = max 1≤j≤n
n
i=1
|dij|2=
n–1
k=0
sin2k·π n =
n
2.
Therefore, we have
B2≤ |r|
n(n– 1) 2 . Thus, we can obtain
√
2
2 ≤ B2≤ |r|
n(n– 1) 2 . (ii) From|r|< 1,
B2E=
n–1
k=0
(n–k)sin2
k·π
n +
n–1
k=1
k|r|2sin2
≥
n–1
k=0
(n–k)|r|2sin2
k·π
n +
n–1
k=1
k|r|2sin2
k·π n
=n|r|2 n–1
k=0
sin2
k·π
n =
|r|2n2
2 , we can get
B2≥
1
√
nBE≥
√
2 2 |r|.
On the other hand, for the matricesCandDas mentioned above,B=C◦D. SoB2=
C◦D2≤r1(C)C1(D) =
n(n–1) 2 .
Therefore, we have
√
2
2 |r| ≤ B2≤
n(n–1) 2 .
This proves Theorem2. Now we prove Theorem3and Theorem4.
Proof
Pr∗=
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
cos0·nπ cos1n·π cos2·nπ · · · cos(n–1)n·π rcos(n–1)n·π cos0n·π cos1·nπ · · · cos(n–2)n·π r2cos(n–2)·π
n rcos
(n–1)·π
n cos
0·π
n · · · cos
(n–3)·π
n
..
. ... ... . .. ...
rn–1cos1·π
n r
n–2cos2·π
n r
n–3cos3·π
n · · · cos
0·π
n
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
n×n
.
(i) On the one hand,|r|> 1 and by using the definition of Euclidean norm, we can obtain
Pr∗2E= n–1
k=0
(n–k)cos2
k·π
n +
n–1
k=1
irn–k2cos2
k·π n
≥
n–1
k=0
(n–k)cos2
k·π
n +
n–1
k=1
kcos2
k·π n
=n n–1
k=0
cos2
k·π
n =
n2
2. That is,
Pr∗2≥
1
√
nPr∗E≥
√
2 2 n.
On the other hand, let the matricesSandQbe represented by
S=
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
1 1 1 · · · 1 1
r 1 1 · · · 1 1
r2 r 1 · · · 1 1
..
. ... ... ... ...
rn–1 rn–2 rn–3 · · · r 1
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
and
Q=
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
cos0·π
n cos
1·π
n cos
2·π
n · · · cos
(n–1)·π
n cos(n–1)·π
n cos
0·π
n cos
1·π
n · · · cos
(n–2)·π
n cos(n–2)n·π cos(n–1)n·π cos0·nπ · · · cos(n–3)n·π
..
. ... ... . .. ...
cos1·π
n cos
2·π
n cos
3·π
n · · · cos
0·π
n
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠
n×n
,
thenPr∗=S◦Q. SoPr∗2=S◦Q2≤r1(S)C1(Q),
r1(S) = max 1≤i≤n
n
j=1
|sij|2=
1 +|r|2+· · ·+rn–12=
1 –|r|2n
1 –|r|2,
c1(Q) =max 1≤j≤n
n
i=1
|qij|2=
n–1
k=0
cos2
k·π
n =
n
2.
Therefore,
Pr∗2≤r1(S)c1(Q) =
1 –|r|2n
1 –|r|2
n
2. (ii) For|r|< 1,
Pr∗2E= n–1
k=0
(n–k)cos2
k·π
n +
n–1
k=1
krn–k2cos2
k·π n
≥
n–1
k=0
(n–k)rn–k2cos2
k·π
n +
n–1
k=1
irn–k2cos2
k·π n
=n|r|2n n–1
k=0
|r|–2kcos2
k·π n =n|r|
2nN
1.
So
Pr∗2≥
1
√
nPr∗E≥ |r| nN
1,
whereN1= 1–r
–2n
2(1–r–2)+1–r –2–r–2n+2
4 .
Moreover, for the matricesSandQas mentioned above, in this case,Pr∗=S◦Q. So
Pr∗2=S◦Q2≤r1(S)C1(Q),
r1(S) = max 1≤i≤n
n
j=1
|sij|2=
√
n,
c1(Q) =max 1≤j≤n
n
i=1
|qij|2=
n–1
i=0
cos2
k·π
n =
n
2,
Pr∗2≤
√
Therefore, we have
|r|nN1≤ Pr∗2≤
√
2 2 n.
By the same methods, using Lemma2and Theorem2, we can get Theorem4.
This completes all of the theorems.
Remark Lemma2of this paper gave a new method to compute the power sums of the trigonometric functions.
4 Conclusion
By the same methods as of this paper, we can also get determinants and norms of some other special circulant matrices involving trigonometric functionscos(kπ
n),sin( kπ
n).
Acknowledgements
The author is grateful to anonymous referees and the associate editor for their careful reading, helpful comments, and constructive suggestions, which improved the presentation of the results.
Funding
This work is supported by N.S.F. (11771351).
Competing interests
The author declares to have no competing interests.
Authors’ contributions
The author contributed to each part of this work seriously and read and approved the final version of the manuscript.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Received: 11 December 2018 Accepted: 12 August 2019
References
1. Solak, S.: On the norms of circulant matrices with the Fibonacci and Lucas numbers. Appl. Math. Comput.160, 125–132 (2005)
2. Shen, S.Q., Cen, J.M.: On the bounds for the norms ofr-circulant matrices with Fibonacci and Lucas numbers. Appl. Math. Comput.216, 2891–2897 (2010)
3. Yazlik, Y., Taskara, N.: On the norms of anr-circulant matrix with the generalizedk-Horadam numbers. J. Inequal. Appl.
2013, 394 (2013)
4. Bahsi, M.: On the norms of circulant matrices with the generalized Fibonacci and Lucas numbers. TWMS J. Pure Appl. Math.6(1), 84–92 (2015)
5. Köme, C., Yazlik, Y.: On the spectral norms ofr-circulant matrices with the biperiodic Fibonacci and Lucas numbers. J. Inequal. Appl.2017, 192 (2017)
6. Yazlik, Y., Taskara, N.: Spectral norm, eigenvalues and determinant of circulant matrix involving the generalized
k-Horadam numbers. Ars Comb.104, 505–512 (2012)
7. Lv, X.X., Shen, S.M.: On Chebyshev polynomials and their applications. Adv. Differ. Equ.2017, 343 (2017) 8. Davis, P.J.: Circulant Matrices. Wiley, New York (1979)
9. Kizilates, C., Tuglu, N.: On the bounds for the spectral norms of geometric circulant matrices. J. Inequal. Appl.2016, 312 (2016)
10. Tuglu, N., Kizilates, C.: On the norms of circulant andr-circulant matrices with the hyperharmonic Fibonacci numbers. J. Inequal. Appl.2015, 253 (2015)
11. He, C., Ma, J., Zhang, K., Wang, Z.: The upper bound estimation on the spectral norm ofr-circulant natrices with the Fibonacci and Lucas numbers. J. Inequal. Appl.2015, 72 (2015)
12. Köme, C., Yazlik, Y.: On the determinants and inverses ofr-circulant matrices with biperiodic Fibonacci and Lucas numbers. Filomat32(10), 3637–3650 (2018)
