On the spectral norms of some circulant matrices with the trigonometric functions

In this paper, we use the properties of an r-circulant matrix and a geometric circulant matrix to study the spectral norms of the r-circulant matrix and the geometric circulant matrix involving trigonometric functions by some algebra methods.

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 matrices. 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 circulant matrix is still a topic of focus in the research of matrix theory. Especially, some scholars studied the norms of r-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, Lucas number, generalized Fibonacci and Lucas numbers, generalized k-Horadam numbers, the biperiodic Fibonacci and Lucas numbers have been studied [1,2,3,4,5,6,7,8,9,10,11,12,13]. To the best of our knowledge, no one has studied the upper and lower estimate problems for the spectral norms involving trigonometric functions \(\cos (\frac{k \pi }{n} )\), \(\sin (\frac{k\pi }{n} )\) yet by using exponential sum.

A \(n\times n\) r-circulant matrix \(C_{r}\) is defined by [8]

Kızılateş and Tuglu [9] defined geometric circulant matrices by the form

Obviously, when the parameter satisfies \(r=1\), we can get the classical circulant matrix. Inspired by [7], in this paper, we shall use identities of the trigonometric functions and power sums of \(\cos (\frac{k\pi }{n} )\), \(\sin (\frac{k\pi }{n} )\) to study the norms of the r-circulant matrices

and then we obtain the norms of geometric circulant matrices

Then we get some interesting and concise results which are stated by the following theorems.

Theorem 1

Let \(A=C_{r} (\cos \frac{0\cdot \pi }{n}, \cos \frac{1\cdot \pi }{n},\cos \frac{2\cdot \pi }{n},\ldots, \cos \frac{(n-1) \cdot \pi }{n} )\) be an r-circulant matrix, then we have

Theorem 2

Let \(B=C_{r} (\sin \frac{0\cdot \pi }{n}, \sin \frac{1\cdot \pi }{n},\sin \frac{2\cdot \pi }{n},\ldots, \sin \frac{(n-1) \cdot \pi }{n} ) \) be an r-circulant matrix, then we have

Theorem 3

Let \(P_{r^{*}}=C_{r^{*}} (\cos \frac{0 \cdot \pi }{n},\cos \frac{1\cdot \pi }{n},\cos \frac{2\cdot \pi }{n}, \ldots, \cos \frac{(n-1)\cdot \pi }{n} )\) be a geometric circulant matrix, we have

where \(N_{1}=\frac{1-r^{-2}-r^{-2n+2}}{4}+ \frac{1-r^{-2n}}{2(1-r^{-2})}\).

Theorem 4

Let \(R_{r^{*}}=C_{r^{*}} (\sin \frac{0 \cdot \pi }{n},\sin \frac{1\cdot \pi }{n},\sin \frac{2\cdot \pi }{n}, \ldots, \sin \frac{(n-1)\cdot \pi }{n} )\) be a geometric circulant matrix, we have

where \(N_{2}=\frac{1-r^{-2n}}{2(1-r^{-2})}- \frac{1-r^{-2}-r^{-2n+2}}{4}\).

Definition 1

([9])

Let any matrix \(A=(a_{ij})\in M_{m \times n}(C)\), the spectral norm and the Euclidean norm of matrix A are defined by

where the \(\lambda _{i}(A^{H}A)\) are the eigenvalues of matrices \(A^{H}A\) and \(A^{H}\) is the conjugate transpose of A.

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

Definition 2

([9])

Let both \(A=(a_{ij})\) and \(B=(b_{ij})\) be \(m \times n\) matrices, then the Hadamard product of A and B is the \(m \times n\) matrix of elementwise products, namely \(A\circ B=(a_{ij}b _{ij})\).

Then we have the following inequalities:

Lemma 1

([7])

For any positive integer \(n\geq 2\), we have

Lemma 2

For any positive integer \(n\geq 2\), we can get

Proof

By the properties of \(\cos 2\theta =2\cos ^{2}\theta -1=1-2 \sin ^{2}\theta \), \(e^{i\theta }=\cos \theta +i\sin \theta \), we can easily get \(\cos \theta =\frac{e^{i\theta }+e^{-i\theta }}{2}\); let \(e(x)=e^{2\pi ix}\), note that \(e(1)=e(-1)=1\), using the properties of the trigonometric sums \(\sum^{n-1}_{k=0}e (\frac{k}{n} )=0 \). Hence,

Taking

Therefore,

that is \(S_{1}= \sum^{n-1}_{k=0}r^{-2k}e (\frac{k}{n} )=\frac{1-r^{-2}-r ^{-2n+2}}{1-e (\frac{1}{n} )}\), as the same time, \(\sum^{n-1}_{k=0}r^{-2k}e (\frac{-k}{n} )=\frac{1-r^{-2}-r ^{-2n+2}}{1-e (\frac{-1}{n} )}\).

So,

Using the same methods, note that

 □

Proof of Theorem 1

The matrix \(A=C_{r} (\cos \frac{0\cdot \pi }{n}, \cos \frac{1\cdot \pi }{n},\cos \frac{2\cdot \pi }{n},\ldots, \cos \frac{(n-1) \cdot \pi }{n} )\) is of the following form:

(i) From \(|r|\geq 1\), using the definition of Euclidean norm and Lemma 1, we have

by (1), that is to say,

Moreover, let the matrices E and F be defined by

and

then \(A=E\circ F\). So \(\|A\|_{2}=\|E\circ F\|_{2}\leq r_{1}(E)C_{1}(F)\),

Therefore, we have

Thus, we can obtain the inequality

(ii) From \(|r|<1\),

we can get

Moreover, for the matrices E and F as mentioned above, \(A=E\circ F\). So \(\|A\|_{2}=\|E\circ F\|_{2}\leq r_{1}(E)C_{1}(F)= \frac{\sqrt{2}}{2}n\).

Therefore, we have \(\frac{\sqrt{2}}{2}|r|\leq \|A\|_{2}\leq \frac{ \sqrt{2}}{2}n\).

This proves Theorem 1. □

Now we prove Theorem 2.

Proof

(i) From \(|r|\geq 1\), using the definition of Euclidean norm and Lemma 1, we have

that is,

Moreover, let the matrices C and D be defined by

and

then \(B=C\circ D\). So \(\|B\|_{2}=\|C\circ D\|_{2}\leq r_{1}(C)C_{1}(D)\),

Therefore, we have

Thus, we can obtain

(ii) From \(|r|<1\),

we can get

On the other hand, for the matrices C and D as mentioned above, \(B=C\circ D\). So \(\|B\|_{2}=\|C\circ D\|_{2}\leq r_{1}(C)C_{1}(D)=\sqrt{ \frac{n(n-1)}{2}}\).

Therefore, we have \(\frac{\sqrt{2}}{2}|r|\leq \|B\|_{2}\leq \sqrt{ \frac{n(n-1)}{2}}\).

This proves Theorem 2. □

Now we prove Theorem 3 and Theorem 4.

Proof

(i) On the one hand, \(|r|>1\) and by using the definition of Euclidean norm, we can obtain

That is,

On the other hand, let the matrices S and Q be represented by

and

then \(P_{r^{*}}=S\circ Q\). So \(\|P_{r^{*}}\|_{2}=\|S\circ Q\|_{2} \leq r_{1}(S)C_{1}(Q)\),

Therefore,

(ii) For \(|r|<1\),

So

where \(N_{1}=\frac{1-r^{-2n}}{2(1-r^{-2})}+ \frac{1-r^{-2}-r^{-2n+2}}{4}\).

Moreover, for the matrices S and Q as mentioned above, in this case, \(P_{r^{*}}=S\circ Q\). So \(\|P_{r^{*}}\|_{2}=\|S\circ Q\|_{2}\leq r _{1}(S)C_{1}(Q)\),

\(\|P_{r^{*}}\|_{2}\leq \frac{\sqrt{2}}{2}n\).

Therefore, we have

By the same methods, using Lemma 2 and Theorem 2, we can get Theorem 4.

This completes all of the theorems. □

Remark

Lemma 2 of this paper gave a new method to compute the power sums of the trigonometric functions.

By the same methods as of this paper, we can also get determinants and norms of some other special circulant matrices involving trigonometric functions \(\cos (\frac{k\pi }{n} )\), \(\sin (\frac{k \pi }{n} )\).

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).

Affiliations

1. School of Mathematics, Northwest University, Xi’an, P.R. China

   • Baijuan Shi

Contributions

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

Corresponding author

Correspondence to Baijuan Shi.

Competing interests

The author declares to have no competing interests.

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