On the Explicit Formula for Eigenvalues, Determinant, and Inverse of Circulant Matrices

ABSTRACT

numbers, these are with the k-Fibonacci and k-Lucas numbers. The investigation is also about their invertibility. Then, (D. Bozkurt & T.-Y. Tam, 2016) establish some useful formulas for the determinants and inverses of circulant matrices using the nice properties of the number sequences, (X. Jiang & K. Hong, 2015) concerned with explicit inverse matrices of Tribonacci skew circulant type matrices, (Jiteng Jia & Sumei Li, 2015) found the formulation of the inverse and determinant of general bordered tridiagonal matrices, and (Ercan Altinisik et al., 2015) formulated the determinants and inverses of circulant matrices with complex Fibonacci numbers.
Inspired by all the above beautiful references, in this current paper we derive explicit formulas for determining eigenvalues, determinants, and inverses of circulant matrices with entry in general formation of numbers sequence, instead of a specific numbers sequence or defined by recurrence relation as we can see in the above references. The basic methods of the formulations are mainly by exploiting cyclic group properties which induced from the definition of the circulant matrix.

B. METHODS
In Section C.1, firstly we review the notion of nth root of unity in the system of complex numbers. Then, we derive a group cyclic notion that comes from the set of all nth roots of unity in the system of complex numbers. This group cyclic notion will become the basic theory of the subsequent sections which concern the formulations of eigenvalues, determinants, and inverses for circulant matrices of general type of entry.
In Section C.2, firstly we give an overview of how to get an explicit formula of eigenvalues for a general circulant matrix as presented in Theorem 1 whose proof is mainly based on the basic theory of the cyclic group explained in Section C.1. Then, the formulation of the determinant is easily derived from the above eigenvalues formulation using the spectral theory of the circulant matrix. As a corollary of the theorem, we also explain its relationship with the determinant of the left circulant matrix based on the theory of elementary row operations on a matrix.
In Section C.3, we derive an explicit formula of the inverse of circulant matrices presented in Theorem 2. To prove this theorem, we need two lemmas which we derive by exploiting cyclic group properties explained in Section C.1 and the proof based on the theory of elementary row operations on a matrix. We also present computation remarks in some specific topics of the discussion in connection with fast Fourier transform algorithm. At the end of this paper, we give a calculation illustration of all the results and close the paper by concluding remark.

C. RESULT AND DISCUSSION 1. Review th Roots of Unity in Complex Numbers
We denote ℂ be the field of complex numbers. For a positive integer , th roots of unity over ℂ we mean as the solution of the equation − 1 = 0. A set of those roots is = { ∈ ℂ| = 1} which is in fact a subgroup of the multiplication group ℂ * = ℂ\{0}. To formulate the elements in based on arithmetic of ℂ, firstly we will use the Euler's formula which states that for any ∈ ℝ we have = cos + sin . Afterwards, we apply the theorem of De Moivre which states that (cos + sin ) = cos + sin . Thus, by taking = 2 , we have 2 = (cos 2 + sin 2 ) = cos 2 + sin 2 = 1 (1) Since for = 1,2, ⋯ , − 1, = (cos 2 + sin 2 ) = cos 2 + sin 2 ≠ 1, we may conclude that is a solution of − 1 = 0, and hence we can rewrite = { 0 , 1 , ⋯ , −1 }, then finally we may also conclude that S is a cyclic group of order n. Therefore, again we can rewrite as where = cos 2 + sin 2 for some gcd ( , ) = 1, and we call as a primitive (generator) of .

An Overview on Explicit Formula for the Eigenvalues and Determinant
Given any sequence 0 , 1 , ⋯ , −2 , −1 of complex numbers, we use the usual notation from the references to define the × circulant matrix as Below is the well-known theorem about the eigenvalues of the above circulant matrix. Here, we give a detail proof for the sake of subsequently discussions. Theorem 1. Suppose we have = Circ( 0 , 1 , 2 , … , −2 , −1 ) ∈ ℂ × . For every = 0,1,2, … , − 1, let be the eigenvalue of A corresponding to be an eigenvector of A, then for some primitive of .
Proof. For all = 0,1,2, … , − 1 it is clear = ∈ . In this proof, we will show that vector = (1, , 2 , … , −1 ) ∈ ℂ is an eigenvector of : For this purpose, since is a cyclic group, consider that if and only if and so on until we have −1 = 0 −1 . Based on this fact, if we denote = 0 , then it is clear that = . Furthermore, since = for all = 1,2, … , − 1, then can be stated as of which the formulation of and given in the theorem.  Computation aspect of Theorem 1 is given in the following remark. Remark 1. (A computation note for eigenvalues) Formulation of eigenvalues in Equation 5 can be calculated using matrix multiplication = written as which is in fact a kind of discrete Fourier transform, so those eigenvalues can be computed efficiently using fast Fourier transform algorithm. Also, we note that the column vectors of P are the eigenvectors which can be computed efficiently by exploiting the recursive properties of the cyclic group S. We present the explicit formula for the determinant of those circulant matrices as the first corollary of Theorem 1.
Corollary 1. Given the matrix A in Theorem 1 and suppose that the eigenvalues of A has been computed efficiently, then the determinant of A is formulated as Proof. The matrix P, defined in Remark 1, is in fact a Vandermonde matrix, so it is easy to verify that det( ) = ∏ ( − ) ≠ 0 < which means that all n column vectors of P (i.e. all eigenvectors of A) are linearly independent, then we conclude that A is a simple matrix, and hence we can write = −1 . Now, we have Remark 2. (A computation note for determinant) From Corollary 1, It is clear that the computation efficiency for det ( ) depends on the efficiency of computing the eigenvalues of A, see Remark 1.
The second corollary of Theorem 1 is about the invertibility of A asserted as follows which the proof is very clear.
In this process, the number of exchanges are = ⌊ − 1 2 ⁄ ⌋ which is the number of row permutations. 

An Explicit Formula for the Inverse
The following lemma will be used to prove the subsequent lemma. Lemma 1. Recall the cyclic group = 〈 〉 = {1, , 2 , … , −1 }. If d is a positive integer and | , then we have where t is the positive integer such that n = td.
Proof. From the structure of P, we have the first fact that P is a symmetric matrix. The second fact, for every = 1,2, … , , let be the i-th row of P, then it is easy to see that the set of all entries of is a cyclic subgroup of order some integer positive | of the cyclic group = {1, , 2 , … , −1 }. From those facts, and by applying Lemma 1, then we have = where T be the permutation matrix resulting from elementary column operations on the identity matrix by interchanging column j and ( − + 2) of for every = 2,3, … , ⌊ + 1 2 ⁄ ⌋.  By those lemmas and all the previous discussion results, here we are at the last theorem of this paper. Theorem 2. For integer ≥ 3, given the matrix A in Theorem 1 and suppose that all the eigenvalues of A has been computed efficiently, then the inverse of A is formulated as where P and T are the matrices that have been asserted in Lemma 2.
Proof. Recall from the proof of Corollary 1, we already have that = −1 , so by Lemma 2, we have Furthermore, it is easy to verify that the second, third, ..., n-th rows of −1 is rotating 1 one step to the right recursively. Thus, we may conclude that −1 is circulant, and so is −1 .
Below is the computation remark of the above theorem.
Assuming that we have already computed by considering Remark 1, in the next step is to compute −1 which is just a fast computation way in arithmetic of complex numbers, then followed by computing which is definitely very fast because of just permuting −1 by T: Finally, to compute e can be done efficiently by applying fast Fourier transform algorithm, again see Remark 1.
In this process, we have  Before we close the paper by concluding remark, below we give an illustration connected to all formulations that have been discussed previously. To determine the eigenvalues, determinants, and inverses for general matrices can be done using simply methods which can be found in any standard books of linear algebra. But, that is a computationally hard work, especially when the size of the matrix is large enough. It is because the determinant calculation based on recursive method, and the calculation of inverse and eigenvalues depends on its determinant. When the matrix has a special structure, such as circulant, and more specific also having special type of entry, such as Fibonacci sequence, then the calculation can be made much easier by giving their explicit formulations.
For the case of circulant matrices with general type of entry, one of the explicit formulations we have derived and discussed in the above theorems, their proofs, and their corollaries in this article as the results of our research. If we compare these results to the previous results in the references is that the previous results using more specific type of entry of the matrix and more focus on the problems of mostly determining the determinant and the inverse. It means that our results more general which the explicit formulation can be applied to circulant matrices with any type of sequence of numbers, and of course more complex calculation.

D. CONCLUSIONS AND SUGGESTIONS
A direct method to get the eigenvalues, determinants, and, inverses for general matrices can be done using simply methods which can be found in any standard books of linear algebra. However, for the circulant matrices, one can make it faster by applying numerical methods such as using fast Fourier transform algorithm. Now in this paper, we present a different approach to get that explicit formulation of the inverse is just by matrix multiplication. Then, the computation of this matrix multiplication can also be accelerated by applying fast Fourier transform algorithm. The most important method of that formulation is based exploitation of cyclic group properties which could be explored further to other cases such as for either skew circulant matrix or circulant matrices over finite fields. Thus, the results of this paper still need to be continued, there are at least three subjects that would become our ongoing and nearly future works. (1) To find a subgroup cyclic of the group ℂ * that can be used to derive explicit formulas for eigenvalue, determinant, and inverse for general skew circulant matrices, (2) Exploring the methods of this paper for the case of circulant and skew circulant matrices over finite fields, (3) Studying and exploring fast Fourier transform algorithm especially for the case of circulant and skew circulant matrices over finite fields.