The group inverse of circulant matrices depending on four parameters


 Explicit expressions for the coefficients of the group inverse of a circulant matrix depending on four complex parameters are analytically derived. The computation of the entries of the group inverse are now reduced to the evaluation of a polynomial. Moreover, our methodology applies to both the invertible and the singular case, the latter being computationally less expensive. The techniques we use are related to the solution of boundary value problems associated with second order linear difference equations.


Introduction and Preliminaries
The problem of solving a linear system with circulant coe cient matrices appears in many problems related to the periodicity of that problem. This kind of system occurs in many applications: time series analysis, image processing, spline approximation, di erence solutions of partial di erential equations or the nite di erence method to approximate elliptic equations with periodic boundary conditions, see for instance [4].
Even if the problem of computing the inverse, or the group inverse, of a circulant matrix can be considered solved from a theoretical point of view, the computational cost of nding the solution is very high, even for low dimensions.
Di erent approches to solve the problem have focused on special classes of circulant matrices. For instance, S. R. Searle in [12], provided a method for obtaining analytic expressions for the coe cients of the inverse matrix of a family of three-element circulant matrices; O. Rojo in [11] gives the solution of a linear system having a symmetric circulant tridiagonal matrix, and L. Fuyong, [8], obtained the elements of the inverse matrix as functions of zero points of the characteristic polynomial of the circulant matrix. With the application of the FFT, M. Chen [5] gave algorithms to solve circulant systems in O(n log n) operations instead of the O(n ) arithmetic operations required; and the properly election of circulant preconditioners, see [4], reduced the problem from O(n ) to O(n).
We study circulant matrices of type Circ(a, b, c, . . . , c, d) in full generality and we compute their group inverse. We give necessary and su cient conditions for the invertibility of circulant matrices of type Circ(a, b, c, . . . , c, d) and we obtain analytical expressions for the coe cients of their inverse or group inverse. This means that, by just checking some relations between the four coe cients, we can explicitly com-pute the coe cients of the inverse. In particular, we obtain the group inverse of any circulant matrix of order four. The results here obtained encompass those in [2], when the corresponding matrix is non singular, and the ones in [3] where we obtained the group inverse of singular circulant matrices depending on three parameters.
Moreover, if the inverse or the group inverse of a circulant matrix can be easily computed, we can slightly modify it by introducing new parameters in such a way that the group inverse of the new matrix is still computable with a reasonable number of new operations. We are able to obtain the corresponding explicit expression according to the values of the coe cients of the matrix. For instance, Corollary 4.4 is a generalization of the result given in [11].
The paper is organized as follows. Section 2 is devoted to notation and some results, both old and new, on circulant matrices. In Section 3 we focus on second order di erence equations and the tools to tackle the goal of the present work. Section 4 contains the main theorem and some additional results.
In order to preserve the development of our results and not break it with additional and most likely known explanations we have added a last section, Section 4 named Appendix, at the end of the paper that refers to known results related to second order di erence equations and Chebyshev polynomials.
We refer the reader to [2,3] for common notation with the present paper, although here we recall some. The real part of the complex number z is denoted by (z). Given z ∈ C, we de ne z # as z − if z ≠ and otherwise.
or equivalently, a ij = a +(j−i)(mod n) . Given a = (a , . . . , an) ∈ C n , we denote Circ(a) = Circ(a , . . . , an) as the circulant matrix with parameters a , . . . , an . The vector e ∈ C n is such that e = and e i = , j = , . . . , n. In addition, N(a) and R(a) denote the null space and the range of Circ(a) respectively, whereas Pa is the orthogonal projection onto N(a).
One of the main problems in this setting is to determine the group inverse of a circulant matrix and, moreover, to know when the matrix is invertible. This problem has been widely studied in the literature and solved using Rn, see [9]. We next give a short account of these results, since it will be useful in the rest of this paper.
For any z ∈ C, we consider the vector f(z) and the matrix J(z) de ned respectively as Clearly f( ) = e and f( ) = . Moreover,f(z) = f(z) and when r ∈ Rn, then f(r)τ =f(r). In addition, √ n f(r) : r ∈ Rn} is an orthonormal basis of C n .

Remark:
. . , f(ω n− ) is called Fourier matrix and for any a ∈ C n , Fa = √ n n− k= a k+ f(ω k ) is the discrete Fourier transform of a.
The following lemma provides a necessary and su cient condition for the invertibility of Circ(a) and gives a formula for its group inverse, see [5,6,12]. First, notice that property (i) implies that all circulant matrices of order n are diagonalizable and have the same eigenvectors but di erent eigenvalues. In particular, any two circulant matrices commute each other and the conjugate transpose of a circulant matrix is also circulant. Moreover, any circulant matrix is normal, which in turns implies that it is range-hermitian with index . On the other hand, part (ii) in the above Lemma establishes that the problem of nding the group inverse of a circulant matrix is completely solved. Although its computation is not straightforward in most cases, even in low dimensions the formulae of Lemma 2.1 for g(a) can involve a great number of computations, see Corollaries 3.5 and 3.6 in [3] for the cases n = and n = with d = c, respectively.
The complexity of the formula in Lemma 2.1 for the determination of the group inverse of a circulant matrix grows with the order of the matrix, so it is not useful at all from the computational point of view, even when the matrix is invertible. So, it is reasonable to focus on special classes of circulant matrices and/or to look for alternative methods to compute their group inverses. Many papers have considered this topic, specially for circulant matrices depending on few parameters. In many of these cases, the special structure of the matrix is highly used and leads to the use of alternative methods, such as solving linear di erence equations, either directly, see for instance [2,3,12], or through special LU decompositions, see [5,11].
In addition, if the group inverse of a given circulant matrix can be easily computed, we can perturb it by introducing new parameters in such a way that the group inverse of the perturbed matrix is still computable with few new operations.  Proof. Since √ n f(r) r∈Rn is an orthonormal system, given r ∈ Rn, then â, f(r) = a, f(r) + The formula for the determinant is a consequence of part (i) of Lemma 2.1.
Of course, when |S| is big, close to n, the computational complexity of the expression for g(â) in the above result is similar to that in the general case. Therefore, in practice, we must achieve a compromise between the computation of g(a), that we assume to be easy and with low cost, and the number of perturbations of a. For instance, if S = {s}, the perturbation only adds the computation of a, f(s) + nbs # − a, f(s) # f(s).
In particular, when s = , there is nothing new to compute, since a, = a + · · · + an consists only of sums, and hence we only need to add the value n a + · · · + an + nb # − (a + · · · + an) # to the entries of Circ(a) # .
The following results are straightforward consequences of Theorem 2.2 that are interesting of themselves. Corollary 2.4. Given a, b ∈ C and r ∈ Rn, then det (aI + bJ(r)) = a n− (a + nb). Therefore when n ≥ , aI + bJ(r) is singular i a(a + nb) = and moreover, Sometimes, for a xed vector a, it happens that Circ(a) # or even aCirc(a) # + s∈S bsJ(s) for some a, bs ∈ C determines a structured family of matrices depending on the parameters a, bs. So, we can use Theorem 2.2 to establish a sort of converse to compute the group inverse of this new class of matrices. We take into account that g(a), f(r) = a, f(r) # for any r ∈ Rn.
Among all the possible generalized inverses of Circ(a) some of them are circulants, see [3, Proposition 2.5]. In particular, we are interested in the group inverse Circ(g(a)). The next result gives the algebraic characterization of the vector g(a).

Theorem 2.7. ([3, Theorem 2.7])
Given a ∈ C n , there exists a unique g(a) ∈ N(ā) ⊥ such that Circ g(a) is the group inverse of Circ(a). Moreover, g(a)τ is characterized as the unique solution of the system Circ(a)z = e−Pa(e) belonging to N(a) ⊥ . In particular, Circ(a) is invertible i the system Circ(a)z = e is compatible.
In [2], we computed the inverse matrix of some circulant matrices of order n ≥ with three real parameters at most and in [3] we also considered complex parameters and we determined the expression for their group inverse. Indeed, we reduced signi cantly the computational cost of applying Lemma 2.1, since the key point for nding the mentioned inverse matrix consists in solving the system that provides g(a)τ by means of a non-homogeneous rst order linear di erence equation.
We aim here to extend the methodology to the computation of the group inverse of the matrices Circ(a, b, c, . . . , c, d). The main di culty in this objective is that we need to solve non-homogeneous second order linear di erence equations.

Second order di erence equations and Chebyshev polynomials
Since the computation of the group inverse of the circulant matrices here considered involves the solution of second order linear di erence equations with constant coe cients, we have enumerated some of the properties of this kind of di erence equations, as well as a review of Chebyshev polynomials in Section 4.
Given z ∈ C, ρ ∈ C * and w ∈ C n , we are interested in nding h ∈ C n such that In this case, h is named solution of the di erence equation. Clearly, the choice of h , h ∈ C uniquely determines h j , j = , . . . , n and hence the solution h.
In order to solve the second order di erence equation (5), for any ρ, z ∈ C where ρ ≠ , we consider the vectors t(ρ, z) and u(ρ, z) in C n whose components are de ned as Where T j (z) and U j− (z) stand for the rst and second kind Chebyshev polynomials, see Appendix (13). In addition, given w ∈ C n , we also consider the vector Ψ w (ρ, z) ∈ C n whose components are Notice that ψ w (ρ, z) = and moreover ψ w Given z ∈ C, ρ ≠ and w ∈ C n , then h ∈ C n satis es The characteristic polynomial for the Chebyshev recurrence −h j− + zh j − h j+ = suggests that Chebyshev polynomials must be related to the function φ de ned in (1) via the Binet Formula. Speci cally, the aimed relation is given by the following identities, see [10]: for any k ∈ Z. We have , the identity remains true for w = ± . Given ρ, r ∈ C * , we have the following identities for any k ∈ N As we can see below, the zeroes of Chebyshev polynomials Tn(z) − q, where q ∈ C, play a fundamental role in the computation of the group inverse of the circulant matrices here considered. Since q = φ(w) for some w ∈ C * , we can take ρ = w n and then q = Tn φ(ρ) . (ii) If ρ ∈ Rn and z ∈ Rn(ρ), then T m+ n (z) = ρ − n Tm(z) for any , m ∈ Z and hence, Tn(z) = ρ −n . (iii) If ρ ∈ Rn and z ∈ Rn(ρ) \ {± }, then U m+ n (z) = ρ − n Um(z) for any , m ∈ Z and hence, U n− (z) = .
Proof. It is straightforward taking into account identities (9).

Lemma 3.4.
Given n ∈ N and ρ ∈ C * such that ρ / ∈ Rn , for any r ∈ Rn we have We end this section with new identities about inner products, whose proofs newly use (15) Moreover, if ρ ∈ Rn, then whereas when ρ ∈ Rn, for any r ∈ Rn \ {±ρ},  Circ(a, b, c, . . . , c, d) Our aim in this section is the computation of Circ(a, b, c, . . . , c, d) # where a, b, c, d ∈ C, which requires matrices of order n ≥ . Actually, for n = , we obtain the group inverse of any circulant matrix. Moreover, we also get explicit expressions for the coe cients of the group inverse of any symmetric circulant matrix of this type, that corresponds to taking d = b . De ning a = (a, b, c, . . . , c, d) , since a, = a + b + d + (n − )c and a, f(r) = a + br + c(r + · · · +r n− )

The Group Inverse of Matrices
from Lemma 2.1 we get that det Circ (a, b, c, . . . , c, d) and moreover, if Except for very elemental cases, it is very di cult to reduce the expression for g (a, b, c, d)  However, in the remaining cases, the computation of g (a, b, c, d) is not so easy. For instance, reference [3] is entirely devoted to obtaining an explicit formula for g (a, b, c, c).
To achieve the objective, we use the same strategy as in our previous works on the topic, [2,3]. The main idea is to interpret a as a simple perturbation of a vector associated with a circulant matrix whose group inverse is known. So, the aimed result will come from applying Theorem 2.2. In fact, we can obtain the above result on g(a, c, c, c) from Corollary 2.4, taking into account that (a, c, . . . , c) = (a − c)e + c .
Since a = (a − c, b − c, , . . . , , d − c) + c , from Theorem 2.2 we know that and hence our aim can be reduced to the computation of The solution of this problem when d = c is given at [3,Theorem 3.4]). In addition, the case b = c is equivalent to the previous one, since g(a, c, c, d) = gτ (a, d, c, c).  that is, we can reduce our problem to compute the group inverse of the circulant matrices of the form Circ( zρ, − , , . . . , , −ρ ), for any z ∈ C and ρ ∈ C * . Moreover, since we also conclude that In addition, if we denote by N(ρ, z) the nullity of Circ( zρ, − , , . . . , , −ρ ) and by Pρ,z the projection on it, from Theorem 2.7 we know that it is satis ed that k( zρ, − , , −ρ ) = hτ, where h is the unique solution of the system Circ( zρ, − , , . . . , , −ρ )h = e − Pρ,z(e) belonging to N(ρ, z) ⊥ . As we mention above, our techniques come from di erence equations theory. Observe that, given w ∈ C n , h ∈ C n satis es Circ We remark that we could achieve the same conclusion computing directly the determinant of the circulant matrix Circ ( zρ, − , , . . . , , −ρ ): Since for any r ∈ Rn, In addition, from identities (9), for any r ∈ Rn, we have that When ρ / ∈ Rn we have that rank M ρ, φ(ρr) = for any r ∈ Rn and moreover  of k(a, b, c, d) when (c − b)(c − d) ≠ , that together with the results in [3], covers the computation of the group inverse of all matrices of the type Circ(a, b, c, . . . , c, d). , for any j = , . . . , n the j-th component of  k(a, b,
The case n = deserves special attention, since it describes the group inverse of any circulant matrix with this order. In the next result, we include together those obtained in [3,Theorem 3.4]) and Theorem 4.1.
, −c ± (b + d) and moreover, where vector k(a, b, c, d) is given by: To change the above identity into an explicit formula, we need to analyze the cases when a is equal or not to any of the values c ± ( c − b − d) and c ± i(b − d). This is precisely the analysis contained in Corollary 4.2 and our methodology allows us to do the same in higher dimensions, where easy identities for g(a, b, c, d) as the previous one are no longer available.
The circulant matrix Circ(a, b, c, . . . , c, d) is symmetric i d = b. We remark that the necessary and sufcient conditions for the invertibility of Circ(a, b, c, . . . , c, b) when a, b, c ∈ R, together with the expression for its inverse, were already obtained in [2, Theorem 3.5]. Next, we describe the expression for the group inverse. Corollary 4.3. Given a, b, c ∈ C, the matrix Circ(a, b, c, . . . , c, b) Moreover, where for any j = , . . . , n the j-th component of k(a, b, c, b) is given by: In particular, we next analyze the case c = . Notice that the group inverse of Circ(a, b, , . . . , , b) was obtained by O. Rojo in [11] under the hypothesis of strictly diagonally dominance, that is |a| > |b|, which implies a ≠ − b cos πk n ≠ for any k = , . . . , n− . Here we obtain Circ(a, b, , . . . , , b) # with no additional hypotheses, except b ≠ , of course.  k(a, b, , b) where For instance, in the case in which zρ, ρ ∈ Z, the homogeneous equation is related with many problems in Combinatorics and Enumerative Geometry.
The usual treatment of Equation (5) involves the so-called Binet Formula, that is based on the roots of the characteristic polynomial P(w) = −w + zρw−ρ ∈ C[w] and depends on the nature of such roots. However, it is more convenient to express the solutions of (5) in a more closed form that involves Chebyshev polynomials, see [7,Section 8] for a more complete explanation. This is the way followed in previous works by some of the authors, see [1,2].
First, remember that a Chebyshev equation or Chebyshev recurrence corresponds to Equation (5) when ρ = . Our methodology is based on the simple fact that Equation (5), and so any second order linear equation, is equivalent to a Chebyshev equation: h ∈ C n is a solution of Equation (5) i h j = ρ j k j , where k ∈ C n is a solution of the Chebyshev equation −k j− + zk j − k j+ = ρ −(j+ ) w j , j = , . . . , n − .
A Chebyshev sequence is a sequence of polynomials {Qn(z)} n∈Z ⊂ C[z] that satis es the recurrence Q n+ (z) = zQn(z) − Q n− (z), for each n ∈ Z. (13) Notice that {Qn(z)} n∈Z is a Chebyshev sequence i {Qn+m(z)} n∈Z is also a Chebyshev sequence for any m ∈ Z. In addition, recurrence (13) shows that any Chebyshev sequence is uniquely determined by the choice of polynomials of order zero and one, Q and Q respectively. In particular, the sequences {Tn} +∞ n=−∞ and {Un} +∞ n=−∞ denote the rst and second kind Chebyshev polynomials that are obtained when we choose T (z) = U (z) = , T (z) = z, U (z) = z.
The main role played by the rst and second kind Chebyshev polynomials is shown by taking into account that if {Qn} +∞ n=−∞ is any Chebyshev sequence, there exist α, β ∈ C such that Qn = αTn + βU n− , for any n ∈ Z. Besides, it is easy to prove that for any n ∈ Z we have T n ( ) = U n ( ) = (− ) n , T n+ ( ) = U n+ ( ) = ,