Abstract
The well known circulant matrices are applied to solve networked systems. In this paper, circulant and left circulant matrices with the Fermat and Mersenne numbers are considered. The nonsingularity of these special matrices is discussed. Meanwhile, the exact determinants and inverse matrices of these special matrices are presented.
1. Introduction
Circulant matrices are an important tool in solving networked systems. In [1], the authors investigated the storage of binary cycles in Hopfield-type and other neural networks involving circulant matrix. In [2], the authors considered a special class of the feedback delay network using circulant matrices. Distributed differential space-time codes that work for networks with any number of relays using circulant matrices were proposed by Jing and Jafarkhani in [3]. Bašić [4] solved the question for when circulant quantum spin networks with nearest-neighbor couplings can give perfect state transfer. Wang et al. considered two-way transmission model ensured that circular convolution between two frequency selective channels in [5]. Li et al. [6] presented a low-complexity binary framewise network coding encoder design based on circulant matrix.
Circulant matrices have been applied to various disciplines including image processing, communications, signal processing, and encoding. Circulant type matrices have established the substantial basis with the work in [7–12] and so on.
Lately, some authors gave the explicit determinant and inverse of the circulant and skew-circulant involving famous numbers. For example, Yao and Jiang [13] presented the determinants, inverses, norm, and spread of skew circulant type matrices involving any continuous Lucas numbers. Jiang et al. [14] considered circulant type matrices with the -Fibonacci and -Lucas numbers and presented the explicit determinant and inverse matrix by constructing the transformation matrices. Dazheng [15] got the determinant of the Fibonacci-Lucas quasi-cyclic matrices. Determinants and inverses of circulant matrices with Jacobsthal and Jacobsthal-Lucas numbers were obtained by Bozkurt and Tam in [16].
For any integer , let be the th Fermat number. It is well known that is prime for , but there is no other m for which is known to prime. The Mersenne and Fermat sequences are defined by the following recurrence relations [17, 18], respectively:with the initial condition , , , , for .
Let and be the roots of the characteristic equation ; then the Binet formulas of the sequences and have the form
Lemma 1. Let be the th Mersenne number and let be the th Fermat number; then (1), , , .(2), , , .
We define a Fermat circulant matrix which is an matrix with the following form:
A Mersenne circulant matrix which is an matrix is defined with the following form:
Besides, a Fermat left circulant matrix is given by
A Mersenne left circulant matrix is given by
The main content of this paper is to obtain the results for the exact determinants and inverses of Fermat and Mersenne circulant matrix. In this paper, let be a nonnegative integer, , and .
2. Determinant and Inverse of Fermat Circulant Matrix
In this section, let be a Fermat circulant matrix. Firstly, we obtain the exact form determinant of the matrix . Afterwards, we find the exact form inverse of the matrix .
Theorem 2. Let be a Fermat circulant matrix. Then one has where , , , , and is the th Fermat number. Moreover, is singular if and only if and , for , , where , .
Proof. It is clear that det + satisfies (7). In the following, letbe two matrices; we have whereWe obtainwhile We haveNext, we discuss the singularity of the matrix .
The roots of polynomial are , where . We have By Lemma 1 in [14], the matrix is nonsingular if and only if ; that is, when , is nonsingular if and only if ; when , we obtain or .
Let ; then the eigenvalue of is for , , , , , so is nonsingular. The arguments for are similar. Thus, the proof is completed.
Lemma 3. Let the matrix be of the formThen the inverse of the matrix is equal to
Proof. Let . Distinctly, for . In the case , we obtain For , we get We check on , where is identity matrix. Similarly, we can verify . Thus, the proof is completed.
Theorem 4. Let be a Fermat circulant matrix. Then one acquires , where
Proof. Let whereWe have where is a diagonal matrix, and is the direct sum of and . If we denote , then we obtain Let . Since the last row elements of the matrix are , according to Lemma 3, then the last row elements of are given by the following equations:where , , according to Lemma 1; then we have (i),(ii).Hence, we obtainThus, the proof is completed.
3. Determinant and Inverse of Mersenne Circulant Matrix
In this section, let be a Mersenne circulant matrix. Firstly, we obtain the determinant of the matrix . Afterwards, we seek out the inverse of the matrix .
Theorem 5. Let be a Mersenne circulant matrix. Then one obtainswhere , , , , and is the th Mersenne number. Furthermore, is singular if and only if and , for , , where , .
Proof. Obviously, satisfies (27). In the following, letbe two matrices; then we have where We getbesides We have Now, we discuss the singularity of the matrix .
The roots of polynomial are , where . So we have By Lemma 1 in [14], the matrix is nonsingular if and only if . That is when , is nonsingular if and only if , for , , . When , we obtain or . Let ; then the eigenvalue of is for , , , , , so is nonsingular. The arguments for are similar. Thus, the proof is completed.
Lemma 6. Let the matrix be of the form Then the inverse of the matrix is equal to
Proof. Let . Distinctly, for . When , we obtain For , we obtain We verify , where is identity matrix. Similarly, we check on . Thus, the proof is completed.
Theorem 7. Let be a Mersenne circulant matrix. Then one acquires where where
Proof. Let where We have where is a diagonal matrix, and is the direct sum of and . If we denote , then we obtain Let . Since the last row elements of the matrix are , according to Lemma 6, then the last row elements of are given by the following equations:where , according to Lemma 1; then we have (i),(ii).We get Thus, the proof is completed.
4. Determinants and Inverses of Fermat and Mersenne Left Circulant Matrix
In this section, let and be Mersenne and Fermat left circulant matrices, respectively. By using the obtained conclusions, we give a determinant formula for the matrix and . In addition, the inverse matrices of and are derived.
According to Lemma 2 in [14] and Theorems 2, 4, 5, and 7, we can obtain the following theorems.
Theorem 8. Let be a Fermat left circulant matrix; then one has where , , and is the th Fermat number. Moreover, is singular if and only if and , for , , where , .
Theorem 9. Let be a Fermat left circulant matrix; thenwhere were given by Theorem 4 and was given by Lemma 2 in [14].
Theorem 10. Let be a Mersenne left circulant matrix; then one has where , , and is the th Mersenne number. Furthermore, is singular if and only if and , for , , where , .
Theorem 11. Let be a Mersenne left circulant matrix; then one has where were given by Theorem 7 and was given by Lemma 2 in [14].
5. Conclusion
In this paper, we present the exact determinants and the inverse matrices of Fermat and Mersenne circulant matrix, respectively. Furthermore, we give the exact determinants and the inverse matrices of Fermat and Mersenne left circulant matrix. Meanwhile, the nonsingularity of these special matrices is discussed. On the basis of circulant matrices technology, we will develop solving the problems in [19–22].
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This work was supported by the GRRC Program of Gyeonggi Province ((GRRC SUWON 2014-B4), Development of Cloud Computing-Based Intelligent Video Security Surveillance System with Active Tracking Technology). Their support is gratefully acknowledged.