DRAZIN INVERSE OF SINGULAR ADJACENCY MATRICES OF DIRECTED WEIGHTED CYCLES

We present a necessary and sufficient condition for the singularity of circulant matrices associated with directed weighted cycles. This condition is simple and independent of the order of matrices from a complexity point of view. We give explicit and simple formulas for the Drazin inverse of these circulant matrices. We also provide a Bjerhammar-type condition for the Drazin inverse.


Introduction and preliminaries
Circulant matrices appear in many applications, for example to approximate the finite difference of elliptic equations with periodic boundary (see [8]) and to approximate periodic functions with splines (see [9]); they also play an important role in coding theory and in statistics. The standard references are [10] and [13].
In [15] a combinatorial description of the Drazin inverse of the adjacency matrix of a tree is exhibited. In the present work we study the Drazin inverse of the singular adjacency matrix of directed weighted cycles, as a first step in order to give a combinatorial description of the Drazin inverse of other graphs. The standard reference for generalized inverses is [4]. For all linear algebra-theoretic notions not defined here, the reader is referred to [17]; for all graph-theoretic notions not defined here, the reader is referred to [11].
This work is organized as follows. In Section 2 we give a Bjerhammar-type condition for the Drazin inverse. In the setting of Moore-Penrose theory, Bjerhammar, in [5], was the first to notice that if you have additional information about the fundamental spaces of the matrix and its candidate to generalized inverse you do not need to check all the Moore-Penrose conditions. In Section 3 we give some basic properties of circulant matrices. In Section 4 we characterize which circulant matrices associated to directed cycles are singular. In Sections 5, 6, and 7 we give the Drazin inverse of singular circulant matrices associated with cycles.
Throughout the paper, given a ∈ C we denote by a # the number Let A = (a ij ) be a matrix of order n. All the matrices in this work have real or complex coefficients. We associate with A a digraph D(A) with n vertices. The vertices of D(A) are denoted by 0, 1, 2, . . . , n − 1. There is an edge from vertex i to vertex j of weight a ij for each i, j = 0, 1, 2, . . . , n − 1. A linear subdigraph of D(A) is a spanning subdigraph of D(A) in which each vertex has indegree 1 and outdegree 1. We associate to A a second digraph D * (A) = D(A T ), where A T is the transpose of A. This digraph is called the Coates digraph of the matrix A. Definition 3.1. An n × n matrix A = (a ij ) is a circulant matrix if it has the form a ij = α j−i for some α 0 , α 1 , . . . , α n−1 , where the subscript j − i is taken modulo n. We denote A. M. ENCINAS, D. A. JAUME, C. PANELO, AND A. PASTINE Note that Circ(α 0 , α 1 , . . . , α n−1 ) T = Circ(α 0 , α n−1 , . . . , α 1 ). Many properties about circulant matrices are well known. For instance, if we consider A = Circ(α 0 , α 1 , . . . , α n−1 ), then A can be expressed in the form where P is the cyclic permutation matrix of order n; that is, where k ∈ {1, . . . , n − 1}, I n is the identity matrix of order n, P 0 = I n , and O m,n is an m × n zero-matrix (see [10]). In addition, circulant matrices of order n form an n-dimensional vector space and also a commutative algebra, since for any two given circulant matrices its product is also a circulant matrix, and moreover any two circulant matrices commute with each other. Therefore, any circulant matrix is normal, which implies that it has index 1, and therefore any circulant matrix has group inverse that coincides with its Moore-Penrose inverse. Although not every generalized inverse of a circulant matrix must be circulant itself, this property holds for the group inverse.
One of the main problems in the field of circulant matrices is to determine the Drazin (group) inverse of a circulant matrix, and moreover to know when the matrix is in fact invertible. This problem has been widely studied in the literature by using the primitive n-th root of unity and some polynomial associated with it (see [13] and [23]). Specifically, let ω = e 2π n i be the primitive n-th root of unity. In addition, define for each j = 0, . . . , n − 1, the vector t j = 1, ω j , . . . , ω j(n−1) T ∈ R n and for any a = (α 0 , . . . , α n−1 ) T ∈ R n the polynomial P a (x) = n−1 j=0 α j x j . Observe that t 0 = (1, . . . , 1) T . The following lemma provides a necessary and sufficient condition for the invertibility of Circ(α 0 , . . . , α n−1 ) and gives a formula for its group inverse (see [10]).
On one hand, let us notice that property (i) of the previous lemma implies that all circulant matrices of order n have the same eigenvectors but different eigenvalues. On the other hand, part (ii) in the above lemma establishes that the problem of finding the Drazin (group) inverse of a circulant matrix is completely solved. However, the computational complexity of formula (ii) for the determination of its group inverse grows with the order of the matrix, so it is not useful at all from the computational point of view. So, it is interesting to look for alternative expressions for the group inverse of specific classes of circulant matrices. Over the years, many papers have considered this topic specially for circulant matrices with few nonzero entries. In many of these cases, the special structure of the matrix is highly used and leads to the employment of alternative methods, for example solving linear difference equations (see for instance [7], [20], and [22]). The aim of this work is to provide formulas for the Drazin (group) inverse of a circulant matrix associated with directed weighted cycles.

Circulant matrices with two parameters
Circulant matrices of the form Circ(0, a, 0, . . . , 0, b) of order n are denoted by C n (a, b). The next theorem gives a simple and independent of the order, necessary and sufficient condition for the singularity of C n (a, b). We usually call the Coates digraph of C n (a, b) "cycle" or "directed cycle". Proof. We break the proof in three parts.
If n = 0 mod 4, then the Coates digraph D * (C 4k (a, b)), where k is some positive integer, looks like the digraph in Figure 1 and has four linear subdigraphs (see Figure 2). Hence, by Theorem 1.1, Therefore, det(C 4k (a, b)) = 0 if and only if |a| = |b|.
If n = 2k + 1 with k ∈ N, then the Coates digraph D * (C 2k+1 (a, b)) looks like in Figure 3. Therefore, the cycles of length 2k + 1 have two linear subdigraphs (see Figure 4). Hence, by Theorem 1.1, If n = 4k + 2 with k ∈ N, then the Coates digraph D * (C 4k+2 (a, b)) has four linear subdigraphs; they look similar to the ones in Figure 2. Hence, Our objective is to give formulas for the Drazin inverse of singular adjacency matrices of cycles. By Theorem 4.1, we need to study three cases: Case 1, when n = 0 mod 4; Case 2, when n = 2 mod 4; and Case 3, when n = 1 mod 2. This will be done in the next three sections, one for each case.
In this section we provide formulas for the Drazin inverse in each case. Our proof strategy is the following: We show that a given candidate satisfies the conditions in Definition 2.2. Note that, since we are working with circulant matrices, the commutative property always holds. Thus, we just need to prove Conditions 2 and 3. We prove Condition 2 directly, but Condition 3 is proved via Theorem 2.4. This is the main section of this work. Ideas developed here will be used in the next sections.
where δ j is the lowest nonnegative integer such that In order to prove that C D1 n (a, a) is the Drazin inverse of C n (a, a) we need the following results.  The next lemma follows directly from Definition 5.1. The next theorem allows us to prove Condition 3 of the definition of Drazin inverse via Theorem 2.4. Note that both D (C 4k (a, a)) and D(C D1  4k (a, a)) are bipartite, with partition into even and odd vertices. This means that there are no directed edges between odd vertices, nor between even vertices.  N (C 4k (a, a) 4k (a, a)).  Proof. Assume a = 0. By the preceding discussion, writing C 4k (a, a) in circulant notation: C 4k (a, a) = Circ(c 0 , c 1 , . . . , c 4k−1 ).
Theorem 5.5. If n = 0 mod 4 and a ∈ R, then C D1 n (a, a) is the Drazin inverse of C n (a, a).
Proof. Assume a = 0. The proof is similar to the proof of Theorem 5.4, but using the vectors x and y of the proof of Theorem 5.9.
Theorem 6.4. If n = 2 mod 4 and a ∈ R, then C D2 n (a, −a) is the Drazin inverse of C n (a, −a).