Abstract
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 some well-known identities, the explicit identities for spectral norms are obtained. Some numerical tests are listed to verify the results.
1. 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 [1–3] 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, 5–10] 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.
2. Preliminaries
The Fibonacci and Lucas sequences and are defined by the recurrence relations: with , , , and , respectively.
Obviously, the Fibonacci and Lucas sequences are listed in the following sequence: and their corresponding Binet forms are (see [15])
Now, we recall that, for there hold the following estimates: For the details please refer to [7].
There are lots of identities for Fibonacci numbers and Lucas numbers combined with Binomial coefficients (for more details please refer to [8, 16–18] and the reference therein). In this paper, we focus on the following identities: Furthermore, for all , 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 and its th row is obtained by giving its th row a right circular shift by one positions.
Definition 2 (see [3]). The spectral norm 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 are the eigenvalues of .
3. The Identities of Estimations for Spectral Norms
We give the main theorems of this paper in the following parts.
Theorem 3. Let be as the matrix in (9), and let the first row of be . Then one has
Proof. Combining with Definition 2, the spectral radius of is equal to its spectral norm, where we used the fact that is normal. Moreover, by the irreducible and entrywise nonnegative properties, we deduce that is equal to its Perron value. Denote by an -dimensional column vector. There holds
Obviously, is an eigenvalue of associated with the positive eigenvector , which is the Perron value of . Employing the first identity in (6), we have
This completes the proof.
With the same approach, we obtain the following corollary.
Corollary 4. Let be as the matrix in (9), and let the first row of be . Then one has the following identity:
Theorem 5. Let be with the form as (9). For all , if the first row of is , then one obtains
Proof. Following the same techniques of the above theorem and combining with the fact that is irreducible and entrywise nonnegative, we declare that the spectral norm of is equal to its Perron value. Let . Then
Obviously, we declare that is an eigenvalue of associated with . With simple analysis, we obtain that is equal to the Perron value of . 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 be as the matrix in (9). For all , the first row of is ; then
Now, we are at the point to recall the following lemma to verify the identities of spectral norms with other approaches.
Lemma 7 (see [3]). Let be a nonnegative matrix. If the column sums of are equal, then where and denotes the maximum column sum matrix norm.
Theorem 8. Let be with the form as (9) and let the first row of be . Then one deduces the following identity: where .
Proof. Obviously, the circulant matrix is normal; with the results of Definition 2, we declare that the spectral radius of is equal to ; that is, . Furthermore, applying entrywise nonnegative properties and column sum of are certain constant , which is described in (7). By Lemma 7, we obtain
Employing the identities of Fibonacci numbers and Binomial coefficients in (7), we have
This completes the proof.
Furthermore, we give the following corollary without proofs, which can be proved with the same approaches as the above theorem.
Corollary 9. Let be as the matrix in (9). For all , the first row of is ; then we have the following identity:
4. Numerical Examples
In this section, we give some examples to verify our identities in the above theorems and corollaries.
Example 1. In this example, we give the numerical results for in Table 1.
Example 2. For simplicity, let . We give the numerical results for and in Table 2.
With the data in Tables 1 and 2, we declare that the identity for the spectral norm of holds.
5. 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.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The author thanks Professor Z. L. Jiang for valuable discussions and suggestions and wishes to express sincere thanks to referees for their useful suggestions and comments. This work is partly supported by National Natural Science Foundation of China (Grant no. 11201212), Promotive Research Fund for Excellent Young and Middle-Aged Scientists of Shandong Province (Grant no. BS2012DX004), and the AMEP of Linyi University.