The Identical Estimates of Spectral Norms for Circulant Matrices with Binomial Coefficients Combined with Fibonacci Numbers and Lucas Numbers Entries

Improved estimates for spectral norms of circulant matrices are investigated, and the entries are binomial coefficients combined with either Fibonacci numbers or Lucas numbers. Employing the properties of given circulant matrices, this paper improves the inequalities for their spectral norms, and gets corresponding identities of spectral norms.Moreover, by somewell-known identities, the explicit identities for spectral norms are obtained. Some numerical tests are listed to verify the results.


Introduction
Circulant matrices have connection to physics, signal and image processing, probability, statistics, numerical analysis, algebraic coding theory, and many other areas.There are lots of examples from statistical signal processing and information theory that illustrate the application of the circulant matrices, which emphasize how the asymptotic eigenvalue distribution theorem allows one to evaluate results for processes (for the details please refer to [1][2][3] and the reference therein).Meanwhile a real circulant stochastic process can be described with autocovariance matrices, which are subjected to a cyclical permutation.With the help of autocovariance circulant matrices, it is easy to provide derivations of some results that are central to the analysis of statistical periodograms and empirical spectral density functions (see [4]).
In past decades, the estimates for spectral norms of matrices have been investigated in lots of literatures.Moreover, the determinants and inverses of circulant matrices are stated in many articles.The norms of circulant matrices play an important role in analysing the process of statistics, numerical analysis, and many other problems (for more details, please refer to [3,[5][6][7][8][9][10] and the reference therein).Bryc and Sethuraman [11] investigated the maximum eigenvalue for circulant matrices.Solak [7] obtained lower and upper bounds for the spectral norm of circulant matrices, where the entries are classical Fibonacci numbers.İpek [8] establishes spectral norms of circulant matrices with Fibonacci and Lucas numbers.Furthermore, circulant matrices take up an important status in stochastic calculus, Meckes [12,13] gave some results on the spectral norm of a special random Toeplitz matrix and random circulant matrices, Mehta [14] made a deep discussion on random circulant matrices.
The outline of this paper is as follows.In Section 2, we state some preliminaries and recall some well-known results.In Section 3, we focus on the identities of estimations for spectral norms.In Section 4, we present various numerical examples to exhibit the accuracy and efficiency of our results.Finally, we summarise this paper and illustrate our future work.
There are lots of identities for Fibonacci numbers and Lucas numbers combined with Binomial coefficients (for more details please refer to [8,[16][17][18] and the reference therein).In this paper, we focus on the following identities: Furthermore, for all  ∈ Z + , there hold the following identities: Definition 1 (see [19]).A circulant matrix is an  ×  complex matrix with the following form: The first row of  is ( 0 ,  1 , . . .,  −1 ) and its ( + 1)th row is obtained by giving its th row a right circular shift by one positions.
Definition 2 (see [3]).The spectral norm ‖ ⋅ ‖ 2 of a matrix  with complex entries is the square root of the largest eigenvalue of the positive semidefinite matrix  * : where  * denotes the conjugate transpose of .Therefore if  is an  ×  real symmetric matrix or  is a normal matrix, then where  1 ,  2 , . . .,   are the eigenvalues of .

The Identities of Estimations for Spectral Norms
We give the main theorems of this paper in the following parts.
Proof.Combining with Definition 2, the spectral radius of  1 is equal to its spectral norm, where we used the fact that  1 is normal.Moreover, by the irreducible and entrywise nonnegative properties, we deduce that ‖ 1 ‖ 2 is equal to its Perron value.Denote by V = (1, 1, . . ., 1)  an -dimensional column vector.There holds Obviously, ∑  =0 (   )   is an eigenvalue of  1 associated with the positive eigenvector V, which is the Perron value of  1 .Employing the first identity in (6), we have This completes the proof.
With the same approach, we obtain the following corollary.
Theorem 5. Let  3 be with the form as (9).For all  ∈ Z + , if the first row of  3 is ((  0 )   , (  1 )  1+ , . . ., (   )  + ), then one obtains Proof.Following the same techniques of the above theorem and combining with the fact that  3 is irreducible and entrywise nonnegative, we declare that the spectral norm of  3 is equal to its Perron value.Let V  = (1, 1, . . ., 1) 1× .Then Obviously, we declare that ∑  =0 (   )  + is an eigenvalue of  3 associated with V.With simple analysis, we obtain that ∑  =0 (   )  + is equal to the Perron value of  3 .Combining with the third identity of binomial coefficients and Fibonacci numbers in (6), we obtain which completes the proof.
Similarly, there holds the following corollary.
Corollary 6.Let  4 be as the matrix in (9).For all Now, we are at the point to recall the following lemma to verify the identities of spectral norms with other approaches.

Numerical Examples
In this section, we give some examples to verify our identities in the above theorems and corollaries.
Example 10.In this example, we give the numerical results for   ( = 1, 2, 3, 4) in Table 1.Example 11.For simplicity, let  = 1.We give the numerical results for  5 and  6 in Table 2.
With the data in Tables 1 and 2, we declare that the identity for the spectral norm of   ( = 1, . . ., 6) holds.

Conclusion
This paper had discussed the identical estimates of spectral norms for some circulant matrices, which are listed by explicit formulations.In the future, we are going to investigate the determinants, inverses of circulant matrices with certain entries, and, inspired by [6], we will investigate the properties of -circulant matrices.Particularly worth mentioning is the fact that, for the -circulant matrix, we had some numerical results to prove the fact that the same identical estimates hold precisely, and we will concern on the theoretical confirmation in part of the future work.