Abstract
The explicit formulae of spectral norms for circulant-type matrices are investigated; the matrices are circulant matrix, skew-circulant matrix, and -circulant matrix, respectively. The entries are products of binomial coefficients with harmonic numbers. Explicit identities for these spectral norms are obtained. Employing these approaches, some numerical tests are listed to verify the results.
1. Introduction
The classical hypergeometric summation theorems are exploited to derive several striking identities on harmonic numbers [1]. In numerical analysis, circulant matrices (named “premultipliers” in numerical methods) are important because they are diagonalized by a discrete Fourier transform, and hence linear equations that contain them may be quickly solved using a fast Fourier transform. Furthermore, circulant, skew-circulant, and -circulant matrices play important roles in various applications, such as image processing, coding, and engineering model. For more details, please refer to [2–13] and the references therein. The skew-circulant matrices were collected to construct preconditioners for LMF-based ODE codes; Hermitian and skew-Hermitian Toeplitz systems were considered in [14–17]; Lyness employed a skew-circulant matrix to construct -dimensional lattice rules in [18]. Recently, there are lots of research on the spectral distribution and norms of circulant-type matrices. In [19], the authors pointed out the processes based on the eigenvalue of circulant-type matrices and the convergence to a Poisson random measure in vague topology. There were discussions about the convergence in probability and distribution of the spectral norm of circulant-type matrices in [20]. The authors in [21] listed the limiting spectral distribution for a class of circulant-type matrices with heavy tailed input sequence. Ngondiep et al. showed that the singular values of -circulants in [22]. Solak established the lower and upper bounds for the spectral norms of circulant matrices with classical Fibonacci and Lucas numbers entries in [23]. İpek investigated an improved estimation for spectral norms in [24].
In this paper, we derive some explicit identities of spectral norms for some circulant-type matrices with product of binomial coefficients with harmonic numbers.
The outline of the paper is as follows. In Section 2, the definitions and preliminary results are listed. In Section 3, the spectral norms of some circulant matrices are studied. In Section 4, the formulae of spectral norms for skew-circulant matrices are established. Section 5 is devoted to investigate the explicit formulae for -circulant matrices. The numerical tests are given in Section 6.
2. Preliminaries
The binomial coefficients are defined by for all natural numbers at once by
Note that is the th binomial coefficient of . It is clear that , , and , for .
The generalized harmonic numbers are defined to be partial sums of the harmonic series [1]:
For in particular, they reduce to classical harmonic numbers:
We recall the following harmonic number identities [1]:
Definition 1 (see [6, 8]). A circulant matrix is an complex matrix with the following form:
The first row of is ; its th row is obtained by giving its th row a right circular shift by one position.
Equivalently, a circulant matrix can be described with polynomial as where Obviously, .
Now, we discuss the eigenvalues of . We declare that the eigenvalues of are the corresponding eigenvalues of with the function in (6), which is
Whereas , , then can be calculated by where .
Similarly, we recall a skew-circulant matrix.
Definition 2 (see [6, 8]). A skew-circulant matrix is an complex matrix with the following form:
Moreover, a skew-circulant matrix can be described with polynomial as where Obviously, .
Thus we have to calculate the eigenvalues of . For the same reason, we obtain that
Whereas , can be computed by where , .
Definition 3 (see [21, 25]). A -circulant matrix is an complex matrix with the following form: where is a nonnegative integer and each of the subscripts is understood to be reduced modulo .
The first row of is ; its th row is obtained by giving its th row a right circular shift by positions (equivalently, mod positions). Note that or yields the standard circulant matrix. If , then we obtain the so-called reverse circulant matrix [21].
Definition 4 (see [26]). 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. Spectral Norms of Some Circulant Matrices
Now, we will analyse spectral norms of some given circulant matrices, whose entries are binomial coefficients combined with harmonic numbers.
Our main results for those matrices are stated as follows.
Theorem 5. Let -circulant matrix is as in (5), and the first row of is where . Then one has
Proof. Since circulant matrix is normal, employing Definition 4, we claim that the spectral norm of is equal to its spectral radius. Furthermore, applying the irreducible and entrywise nonnegative properties, we claim that (i.e., its spectral norm) is equal to its Perron value. We select an -dimensional column vector ; then Obviously, is an eigenvalue of associated with , which is necessarily the Perron value of . Employing (4), we obtain This completes the proof.
Hence, employing the same approaches, we get the following corollary.
Corollary 6. Let -circulant matrix be as in (5), and the first row of is where . Then
Now, we investigate some even-order alternative as follows, where is odd (i.e., is even).
Theorem 7. Let -circulant matrix be as in (5), and the first row of is where . Then there holds the following identity:
Proof. Noticing (9) and (17), it is clear that the spectral norm of can be calculated by
where , and we employed that all circulant matrices are normal.
Note that, if is odd, then is even, and is an eigenvalue of , so
Combining (4) and (27) yields
This completes the proof.
Employing the same approaches, we get the following corollary.
Corollary 8. Let -circulant matrix be as in (5), and the first row of is where . Then one has the following identity:
Similarly, we set , .
Corollary 9. Let , be as above, respectively, and is odd. Then
4. Spectral Norms of Skew-Circulant Matrices
An odd-order alternative skew-circulant matrix is defined as follows, where is even.
Theorem 10. Let -circulant matrix be as in (10), and the first row of is where . Then one obtains
Proof. We employ (14) and (17) to calculate the spectral norm of as follows, for all :
where .
Since the skew-circulant matrix is normal, we deduce that
If is even, then is odd. We declare that is an eigenvalue of ; then we calculate the corresponding eigenvalue of as follows:
where we had employed (14).
Noticing (34), we claim that is the maximum of , which means
Thus, from (4) we obtain
This completes the proof.
Similarly, we can calculate the identity for .
Corollary 11. Let -circulant matrix be as in (10), and the first row of is where . Then there holds
Corollary 12. Let and , and is even. Then one has the identities for spectral norm
5. Spectral Norms of -Circulant Matrices
Inspired by the above propositions, we analyse spectral norms of some given -circulant matrices in this section.
Lemma 13 (see [25]). The matrix is unitary if and only if where is a -circulant matrix with the first row .
Lemma 14 (see [25]). is a -circulant matrix with the first row if and only if where
In the following part, we set .
Theorem 15. Let -circulant matrix be as in (15), and the first row of is where . Then
Proof. With the help of Lemmas 13 and 14, we know that the -circulant matrix is normal; then we claim that the spectral norm of is equal to its spectral radius. Furthermore, applying the irreducible and entrywise nonnegative properties, we claim that (i.e., its spectral norm) is equal to its Perron value. We select a -dimensional column vector ; then Obviously, is an eigenvalue of associated with , which is necessarily the Perron value of . Employing (4), we obtain This completes the proof.
Corollary 16. Let -circulant matrix be as in (15), and the first row of is where . Then one obtains
6. Numerical Examples
Example 1. In this example, we give the numerical results for and .
Comparing the data in Table 1, we declare that the identities of spectral norms for hold.
Example 2. In this example, we list the numerical results for , .
With the help of data in Table 2, it is clear that the identities of spectral norms for , hold.
Example 3. In this example, we reveal the numerical results for alternative skew-circulant matrices , .
Combining the data in Table 3, we deduce that the identities of spectral norms for , hold.
Example 4. In this example, we show numerical results for and .
Considering the data in Table 4, we deduce that the identities of spectral norms for hold.
The above results demonstrate that the identities of spectral norms for the given matrices hold.
7. Conclusion
This paper had discussed the explicit formulae for identical estimations of spectral norms for circulant, skew-circulant and -circulant matrices, whose entries are binomial coefficients combined with harmonic numbers. Furthermore, it is easy to take other entries to obtain more interesting identities, and the same approaches can be used to verify those identities. Furthermore, explicit formulas for both norms and help us to estimate the so-called condition number. It is an interesting problem to investigate the properties of , such as the explicit formulations for determinants and inverses, by just using the entries in the first row.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is supported by NFSC (no. 11201212), Promotive Research Fund for Excellent Young and Middle-Aged Scientists of Shandong Province (no. BS2012DX004), Shandong Province Higher Educational Science and Technology Program (J13LI11), AMEP, and the Special Funds for Doctoral Authorities of Linyi University.