Using digrahs to compute determinat, permanent and Drazin (group) inverse of circulant matrices with two parameters
Andrés M. Encinas, Daniel A. Jaume, Cristian Panelo, Denis E. Videla
aa r X i v : . [ m a t h . G M ] A ug Using digrahs to compute determinat, permanent andDrazin (group) inverse of circulant matrices with twoparameters
Andr´es M. Encinas c , Daniel A. Jaume a, ˚ , Cristian Panelo a , Denis E.Videla b a Departamento de Matem´aticas. Facultad de Ciencias F´ısico-Matem´aticas y Naturales.Universidad Nacional de San Luis. b FaMAF - CIEM (CONICET), Universidad Nacional de C´ordoba. c Centro Escola d’Enginyeria de Barcelona Est (EEBE), Universitat Polit`ecnica deCatalunya.
Abstract
In this work we present closed formulas for the determinant, the permanent,the inverse and the Drazin inverse of circulant matrices with two non-zerocoefficients.
Keywords:
Circulant matrices, determinant, permanent, Drazin inverse,directed weighted cycles.
1. Introduction
Circulant matrices appear in many applications, for example, to approx-imate the finite difference of elliptic equations with periodic boundary, andto approximate periodic functions with splines. The circulant matrices playan important role in coding theory and in statistic. The standard referenceis [8].One of the main problems about circulant matrices is to determine in-vertibility conditions and to compute its group inverse. The problem hasbeen widely treated in the literature by using the primitive n –th root ofunity and some polynomial associated with it, see [8], [7], and [20]. Al-though there exists some classical and well-know results that enables us tosolve almost every thing we would raise about the inverse or Drazin (group)inverse of circulant matrices, in practice they are inapplicable in order toobtain manageable formulas when we treat with specific families of circulant ˚ Corresponding author: Daniel A. Jaume
Email addresses: [email protected] (Andr´es M. Encinas), [email protected] (Daniel A. Jaume), [email protected] (Cristian Panelo), [email protected] (Denis E.Videla)
Preprint submitted to Elsevier Received: date / Accepted: date atrices. Therefore, it has interest to look for alternative descriptions and infact, there exist many papers devoted to this question. The direct computa-tion for the inverse of some circulant matrices have been proposed in manyworks, see for example [10], [20], [25], [13], [7], and [16] (in chronologicalorder).Besides, the combinatorial structure of circulant matrices also has de-served attention. The graphs whose adjacency matrix is circulant had beenstudy in many works, specially those with integral spectrum, see for example[21], [3], [11], [15], and [19].In this work we delve into the combinatorial structure of circulatingmatrices with only two non-null generators, by considering the digraphsassociated with this kind of matrices. Therefore, we complete the previouswork of some of the authors, see [9], where only an specific class of thesematrices was considered.We use digraphs in the present work, for all of graph-theoretic notionsnot explicitly defined here, the reader is referred to [1].We warn that, as is natural for circulant matrices, our matrices indexesand permutations start at zero. Hence, permutations in this work are bijec-tions over t , . . . , n ´ u , and if A “ „ , then the set of indexes of A is t , u . So A “ A “
3. This isalso why in the present work r n s denotes the set t , . . . , n ´ u instead of t , . . . , n u .For permutations, we use the cyclic notation: p qp q is the per-mutation (in row notation) p q , see [17]. Given a permutation α of r n s , we denote by P α the n ˆ n matrix defined by p P α q α p j q ,j “ P α is known as the permutation matrix associated to α . It is well-known that P ´ α “ P Tα . The assignation α ÞÑ P α from the Sym-metric Group S n to the General Lineal Group GL p n q is a re-presentation of S n , i.e., P αβ “ P α P β where product is composition.The cycle type of a permutation α is a expression of the form p m , m , . . . , n m n q , where m k is the number of cycles of length k in α . It is well-known that theconjugacy classes of S n are determined by the cycle type, see page 3 of [18].Thus, α and β are conjugated (i.e. there exist a permutation σ such that σασ ´ “ β ) if and only if α and β have the same cycle type.We use the matrix associated to the permutation τ n “ p n ´ n ´ ¨ ¨ ¨ q P τ n we just write P n . Thus, P “ »——– fiffiffifl . Notice that P n “ I n , P kn “ P k mod nn “ P τ kn , det p P n q “ p´ q n ´ , and ` P kn ˘ ´ “ P n ´ kn . Notice that τ kn p i q “ i ´ k mod n for k P Z .The circulant matrix whose first row is c , . . . , c n ´ is denoted byCirc p c , . . . , c n ´ q : “ c I n ` ¨ ¨ ¨ ` c n ´ P n ´ n , the numbers c , . . . , c n ´ are called the parameters of the circulant matrix.Let A be an m ˆ n matrix and let S Ď t , . . . , m ´ u , T Ď t , . . . , n ´ u .According to [2], the submatrix of A obtained by deleting the rows in S andthe columns in T is denoted by A p S | T q .The paper is organized as follows. In Section 2, we present our main ideaabout how to work with circulant matrices with just two non-zero param-eters: to untangle the associated digraphs. In Section 3, we explicitly findthe matrices that untangle the digraphs associated with our matrices. InSection 4, we find explicit formulas for the determinant and the permanentof circulant matrices with two non-zero parameters. In Section 5, we give anexplicit formula for the inverse of non-singular circulant matrices with twonon-zero coefficients. Finally in Section 6, we give an explicit formula for theDrazin inverse of singular circulant matrices with two non-zero parameters.
2. Untangling the skein
Let n, s , s and s be non-negative integers such that 0 ď s ă s ă n and0 ă s ă n . Let a, b be non-zero real numbers (actually all the results holdsfor non-zero complex numbers). We are going to work with the followingtype of n ˆ n circulant matrices: aP s n ` bP s n . (1)Here an example of how looks like one of these matrices: aP ` bP “ »————————————– a b a b a b
00 0 0 0 0 a bb a b a
00 0 b aa b a b fiffiffiffiffiffiffiffiffiffiffiffiffifl . ` n ˘ forms of circulant matrices with two non-zero entries.Since aP s n ` bP s n “ P s n p aI n ` bP s ´ s n q and aI n ` bP n ´ s n “ p aI n ` bP s n q T ,it looks like we only need to understand p n ´ q{ aI n ` bP n .As it is usual in number theory, the greatest common divisor of twointegers n and s is denoted by p n, s q . We find useful the following notation n z s : “ n p n,s q . We read it “n without s”. Notice that s ¨ n z s “ n ¨ s z n “ r n, s s , where r n, s s is the lowest common multiple of n and s . The rest of the integer divisionof x by n is denoted by x mod n .Let A “ p a ij q be a matrix of order n . It is usual to associate a digraphof order n to A , denoted by D p A q , see [5]. The vertices of D p A q are labeledby the integers t , , ..., n ´ u . If a i,j ‰
0, there is an arc from the vertex i to the vertex j of weight a ij , for each i, j P t , , . . . , n ´ u . Whether σ is a permutation, we just write D p σ q instead of D p P σ q , and we talk aboutdigraphs and permutations associated.In this section, we show that aP s n ` bP s n is associated with a digraphwho has p n, s ´ s q main cycles of length n zp s ´ s q each, and we give twopermutations that allow to untangle this digraphs, in a sense that will beclear later.The first untangle is given by the permutation associated with the matrix P s n , because aP s n ` bP s n “ P s n p aI n ` bP s ´ s n q . Therefore, we just need tostudy only digraphs associated to matrices of the form aI n ` bP sn . In Figures1 and 2 it can be seen how this permutation untangle these digraphs.Notice that D p aI n ` P sn q has p n, s q connected components of order n z s ,each of them has a main spanning cycle. Notice that in D p aI n ` P sn q , v „ u if and only if v ´ u “ s mod n .Given n, s positive integers such that 0 ă s ă n , the permutation p n ´ s q -canonical permutation is ν n,s : “ p n,s q ź i “ ν n,s,i , (2)where ν n,s,i : “ p i p n z s q ´ i p n z s q ´ ¨ ¨ ¨ i p n z s q ´ pp n z s q ´ q i p n z s q ´ p n z s qq . The permutation ν n,s has p n, s q cycles of length n z s in the natural-cyclic-order. For instance, if we consider n “ s “
6, we have that ν , “p qp q . Notice that P ν n,s “ I p n,s q b P n z s , (3)where b denote the usual Kronecker product between matrices, see [22].4 bb b b b bbba a aaaaa a a a a a a a a a b bbb bbb b Figure 1: On the left D ` aP ` bP ˘ , and on the right D ` aI ` bP ˘ . a a a a a a a a a b bbbbbb b b Figure 2: The digraph at left is D ` aP ` bP ˘ ( b -arcs in blue and a -arcs in black-dashed),and the digraph at right is D ` aI ` bP ˘ . Permutations τ sn and ν n,s have the same cycle type. Therefore, for eachpair n, s with 0 ă s ă n , there exists a permutation σ such that στ sn σ ´ “ ν n,s . The digraph D p ν , q can be seen in Figure 3. Notice that D p ν , q isan untangling version of D p aI ` bP q , up to loops. Untangling version ofdigraphs of form D p aP s n ` bP s n q are digraphs of form D p aI n ` bP ν n,s ´ s p q q ,see Figure 4 and 5. Notice that, let n, s and s be integers such that0 ă s ă s ă n . If n z s “ n z s , then ν n,s “ ν n,s . Therefore, D p ν n,s q “ D p ν n,s q . 5he permutations τ s n and σ give us a block diagonal form of aP s n ` bP s n . P σ P ` aP ` bP ˘ P Tσ “ »————————————– a b a b b a a b a b b a a b
00 0 0 0 0 0 0 a b b a fiffiffiffiffiffiffiffiffiffiffiffiffifl “ aI ` bP ν , .i ̺ , p i q mod p , q ℓ , p i q Figure 3: D p ν , q “ D p ν , q . Note that p , q “ p , q “ z “ z “ bb b b b bbba a aaaaa a a a a a a a a b b bbbbb b Figure 4: On the left D ` aP ` bP ˘ , and on the right its untangled version D ` aI ` bP ν , ˘ . a a a a a a a a a bbbb b bbb b Figure 5: The Digraph at left is D ` aP ` bP ˘ (in blue b -arcs and in black-dashed a -arcs),and the digraph at right is its untangled version D ` aI ` bP ν , ˘ .
3. Finding σ n,s In order to compute the Drazin (group) inverse of a matrix of the form aP s n ` bP s n (among others computations) we need to find one σ explicitly.This can be done whether we find which vertices are together is the sameconnected component of D p aI n ` bP sn q . Owing to this digraph appears manytimes in our work, we just write D n,s p a, b q instead of D p aI n ` bP sn q .Let n and s be non-negative integers such that 0 ă s ă n . For each i P Z , we define R p n, s, i q : “ t i ` ks mod n : k P Z u . (4)This is the set of all reachable vertices of from i in D n,s p a, b q . Notice thatthe following statements are all equivalent.1. R p n, s, i q “ R p n, s, i q ,2. i mod n P R p n, s, i q ,3. R p n, s, i q X R p n, s, i q ‰ H , and4. i ´ i “ p n, s q .In Figure 6, we can see R p , , q and R p , , q .In order to obtain a σ explicitly we introduce the following functions:1. ̺ n,s : Z ÝÑ rp n, s qs , ̺ n,s p i q : “ max t k P Z : i ě k p n z s q ě u .2. ℓ n,s : Z ÝÑ Z n z s , ℓ n,s p i q : “ rp ̺ n,s p i q ` q p n z s q ´ i s mod n z s .3. cycle n,s : Z ÝÑ rp n, s qs ,cycle n,s p i q : “ i mod p n, s q ,i.e., cycle n,s p i q is the rest of dividing i by p n, s q .4. pos n,s : Z ÝÑ Z n z s ,pos n,s p i q “ x ,where x satisfies: 0 ď x ă n z s and i ´ cycle n,s p i q “ sx mod n .7 p , , q “ t , , , u R p , , q “ t , , , u a a a a a a a a b bbb bbb b Figure 6: D , p a, b q . Note that p , q “ z “ i cycle , p i q pos , p i q a a a a a a a a a bbb bb b bb b Figure 7: D , p a, b q . Note that p , q “ z “ Notice that the equation i ´ cycle n,s p i q “ sx mod n has always a uniquesolution in r n z s s , because of i ´ cycle n,s p i q “ p n, s q .The functions ̺ n,s , and ℓ n,s give us, essentially, the cycle and the positionon the cycle of a vertex of D p ν n,s q , respectively. While the functions cycle n,s and pos n,s give us the main cycle and the position on the main cycle in D p aI n ` bP sn q , respectively.We embed these digraphs in the cylinder rp n, s qs ˆ Z n z s . The followingfunctions and its pullbacks are these embedding.s5. J n,s : Z ÝÑ rp n, s qs ˆ Z n z s J n,s p i q “ p ̺ n,s p i q mod p n, s q , ℓ n,s p i qq .6. p J n,s : rp n, s qs ˆ Z n z s ÝÑ Z n p J n,s p c, p q “ p c ` ´ δ ,p q p n z s q ´ p ,where δ ,p is the Kronecker delta.8. F n,s : Z ÝÑ rp n, s qs ˆ Z n z s F n,s p i q “ p cycle n,s p i q , pos n,s p i qq .8. p F n,s : rp n, s qs ˆ Z n z s ÝÑ Z n ˆ F n,s p c, p q “ c ` ps mod n .The function J n,s embeds D p ν n,s q in rp n, s qs ˆ Z n z s “in the same way” that F n,s embeds D n,s p a, b q in rp n, s qs ˆ Z n z s .We denote the identification map by ι n : r n s Ñ Z n . Let A be a set, wedenote the identity map by id A : A Ñ A . If n and s are two integers suchthat 0 ă s ă n , then the following identities are direct ´ ι ´ n ˝ p J n,s ¯ ˝ p J n,s ˝ ι n q “ id r n s , p J n,s ˝ ι n q ˝ ´ ι ´ n ˝ p J n,s ¯ “ id rp n,s qsˆ Z n z s , ´ ι ´ n ˝ p F n,s ¯ ˝ p F n,s ˝ ι n q “ id r n s , p F n,s ˝ ι n q ˝ ´ ι ´ n ˝ p F n,s ¯ “ id rp n,s qsˆ Z n z s . These expressions give us the next lemma.
Lemma 3.1.
Let n and s be integers such that ă s ă n . Then, thefunction σ n,s : r n s ÝÑ r n s , defined by σ n,s : “ ι ´ n ˝ p F n,s ˝ J n,s ˝ ι n , is apermutation of r n s and σ ´ n,s “ ι ´ n ˝ p J n,s ˝ F n,s ˝ ι n . The function Shift n,s : rp n, s qs ˆ Z n z s ÝÑ rp n, s qs ˆ Z n z s , defined byShift n,s p c, p q : “ p c, p ´ n z s q , allows us to express τ sn in terms of F n,s and p F n,s , and ν n,s in terms of J n,s and p J n,s . Lemma 3.2.
Let n and s be integers such that ă s ă n . Then τ sn “ ι ´ n ˝ p F n,s ˝ Shift n,s ˝ F n,s ˝ ι n ,ν n,s “ ι ´ n ˝ p J n,s ˝ Shift n,s ˝ J n,s ˝ ι n . Corollary 3.3.
Let n and s be integers such that ă s ă n . Then ν n,s “ σ ´ n,s ˝ τ n,s ˝ σ n,s ,I p n,s q b P n z s “ P Tσ n,s P sn P σ n,s . Theorem 3.4.
Let n, s , s be integers such that ď s ă s ă n , and let a, b non-zero complex numbers. Then, aP s n ` bP s n “ P s n P σ n,s ´ s “ I p n,s ´ s q b ` aI n zp s ´ s q ` bP n zp s ´ s q ˘‰ P Tσ n,s ´ s . roof. Let s “ s ´ s . Then, by Corollary 3.3, P Tσ n,s P n ´ s n p aP s n ` bP s n q P σ n,s “ aI n ` bP Tσ n,s P sn P σ n,s “ aI n ` b ` I p n,s q b P n z s ˘ “ I p n,s q b ` aI n z s ` bP n z s ˘ . This conclude the proof.The following theorem showed that we can untangle ř n z s ´ k “ a k P ksn , inthe same way as in the case of P sn . Theorem 3.5.
Let s, n positive integers such that s ă n and let a k be non-zero real numbers for k P r n z s s . Then there exists a permutation σ n,s P S n of r n s such that P ´ σ n,s ´ n z s ´ ÿ k “ a k P ksn ¯ P σ n,s “ I p n,s q b ´ n z s ´ ÿ k “ a k P kn z s ¯ . Proof.
Clearly ν n,s has the same cycle type that τ sn . Then, there exists apermutation σ n,s such that σ ´ n,s τ sn σ n,s “ ν n,s . By taking into account that the application P : S n Ñ GL p n q , where S n is the symmetric group and GL p n q is the general lineal group, defined by P p σ q “ P σ consider in the above section is a group homomorphism, we havethat P ´ σ n,s P sn P σ n,s “ P σ ´ n,s τ sn σ n,s “ P ν n,s “ I p n,s q b P n z s . In the same way, if we consider the powers P skn with k P r n z s s , we obtainthat P ´ σ n,s P ksn P σ n,s “ p P ´ σ n,s P sn P σ n,s q k “ P kσ ´ n,s ˝ τ sn ˝ σ n,s “ P kν n,s . By the property p A b B qp C b D q “ AC b BD of the Kronecker product, wehave that P ´ σ n,s P ksn P σ n,s “ P kν n,s “ p I p n,s q b P n z s q k “ I p n,s q b P kn z s , for all k P r n z s s . By taking into account that A b p B ` C q “ A b B ` A b C ,we obtain P ´ σ n,s ´ n z s ´ ÿ k “ a k P ksn ¯ P σ n,s “ n z s ´ ÿ k “ a k ´ I p n,s q b P kn z s ¯ “ I p n,s q b ´ n z s ´ ÿ k “ a k P kn z s ¯ , as asserted. 10 . Determinant and permanent of aP s n ` bP s n A linear subdigraph L of a digraph D is a spanning subdigraph of D inwhich each vertex has indegree 1 and outdegree 1, see [5]. Theorem 4.1 ([5]) . Let A “ p a ij q be a square matrix of order n . Then det p A q “ ÿ L P L p D p A qq p´ q n ´ c p L q w p L q , and perm A “ ÿ L P L p D p A qq w p L q . Where L p D p A qq is the set of all linear subdigraphs of the digraph D p A q , c p L q is the number of cycles contained in L , and w p L q is the product of theweights of the edges of L . Theorem 4.2.
Let n be a non-negative integer and let a, b be non-zero realnumbers. Then det p aI n ` bP n q “ a n ´ p´ b q n . Proof.
Notice that the digraph D n, p a, b q has only two linear sub-digraphs,the whole cycle and n loops. The result follows from Theorem 4.1. Corollary 4.3.
Let n , s and s be integers such that ď s ă s ă n . Let a and b be non-zero real numbers. Then det p aP s n ` bP s n q “ p´ q p n ´ q s ´ a n zp s ´ s q ´ p´ b q n zp s ´ s q ¯ p n,s ´ s q . Hence, aP s n ` bP s n is singular if and only if a n zp s ´ s q ´ p´ b q n zp s ´ s q “ . Notice that a n zp s ´ s q ´ p´ b q n zp s ´ s q “ a “ ˘ b when n z p s ´ s q is even or a “ ´ b when n z p s ´ s q is odd. Proof.
Let s “ s ´ s . By Theorem 3.4det p aP s n ` bP s n q “ det p P s n q det ` I p n,s q b ` aI n z s ` bP n z s ˘˘ . Owing to det ` P in ˘ “ p´ q p n ´ q¨ i , the result follows from Theorem 4.2. Example 4.4.
If we consider the matrix aI ` bP , whose associated digraphis in Figure 6, is singular if and only if | a | “ | b | , the same occur with thematrix aP ` bP . The matrix aI ` bP , whose associated digraph is inFigure 7, is singular if and only if a “ ´ b . The matrix aP ` bP is alsosingular if and only if a “ ´ b . orollary 4.5. Let n , s and s be integers such that ď s ă s ă n . Let a and b be non-zero real numbers. Then perm p aP s n ` bP s n q “ ´ a n zp s ´ s q ` b n zp s ´ s q ¯ p n,s ´ s q . Proof.
The result follows from Theorem 1.1. of [14]: if A is an square matrixand Q is a permutation matrix of the same order, then perm QA “ perm A ,Theorem 4.1 and Corollary 4.3.
5. Inverse of aP s n ` bP s n Theorem 5.1.
Let n be a non-negative integer such and let a and b benon-zero real numbers. If aI n ` bP n is non-singular (i.e. a n ´ p´ b q n ‰ ),then p aI n ` bP n q ´ “ a n ´ p´ b q n n ´ ÿ i “ p´ q i b i a n ´ ´ i P in . (5) Proof.
It is just check that p aI n ` bP n q « a n ´ p´ b q n n ´ ÿ i “ p´ q i b i a n ´ ´ i P in ff “ I n . Example 5.2.
Let a, b non-zero real numbers. If a ´ p´ b q ‰ , then p aI ` bP q ´ “ a ´ b Circ ` a , ´ ba , b a, ´ b ˘ . If a ´ p´ b q ‰ , then p aI ` bP q ´ “ a ` b Circ ` a , ´ ba , b a , ´ b a, b ˘ . In order to obtain an explicit formula for the inverse of a non-singularcirculant matrices of the form aP s n ` bP s n , we define ρ n,s p i q “ p´ q pos n,s p i q δ , cycle n,s p i q b pos n,s p i q a n z s ´ ´ pos n,s p i q , (6)where δ is the usual Kronecker delta. Notice that ρ satisfies the followingproperties:1. For n ą ρ n, p i q “ p´ q i b i a n ´ i ´ for all i “ , . . . , n ´ ă s ă n , ρ n z s, p i q “ ρ n,s p i ¨ s q for all i “ , . . . , n z s ´ Corollary 5.3.
Let n , s and s be integers such that ď s ă s ă n . Let a and b be non-zero real numbers such that a n zp s ´ s q ´ p´ b q n zp s ´ s q ‰ .Then p aP s n ` bP s n q ´ “ a n zp s ´ s q ´ p´ b q n zp s ´ s q n ´ ÿ i “ ρ n,s ´ s p i ` s q P in . (7)12 roof. Let s “ s ´ s . By Theorem 3.4 we have that p aP s n ` bP s n q ´ isequal to P σ n,s ” I p n,s q b ` aI n z s ` bP n z s ˘ ´ ı P Tσ n,s P n ´ s n . Thus, by Theorem 5.1 and (6) p aP s n ` bP s n q ´ “ P σ n,s ¨˝ p n z s q´ ÿ i “ I p n,s q b ρ n z s, p i q a n z s ´ p´ b q n z s P in z s ˛‚ P Tσ n,s P n ´ s n . By Corollary 3.3 we have that P sn “ P σ n,s ` I p n,s q b P n z s ˘ P Tσ n,s , (8)and P σ n,s ´ I p n,s q b P in z s ¯ P Tσ n,s “ P i ¨ sn . (9)Then p aP s n ` bP s n q ´ “ p n z s q´ ÿ i “ ρ n z s, p i q a n z s ´ p´ b q n z s P p i ¨ s q´ s n “ p n z s q´ ÿ i “ ρ n,s p i ¨ s q a n z s ´ p´ b q n z s P p i ¨ s q´ s n “ n ´ ÿ j “ ρ n,s p j ` s q a n z s ´ p´ b q n z s P jn . Example 5.4.
Let a and b be two real numbers such a ´ p´ b q ‰ . Wewill compute ` aP ` bP ˘ ´ . Since ´ “ and z “ , by Theorems3.4 and 5.1, we know that the inverse of aI ` bP is composed essentiallyby 3 blocks of Circ ` a , ´ ba , b a, ´ b ˘ . They are merged in a ˆ matrix via P and P σ , . Since s “ , we havethat p , q “ , so there are 3 major cycles of length 4 in D ` aI ` bP ˘ :0 Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ Ñ cycle , p i ` q δ , cycle , p i ` q pos , p i ` q ρ , p i ` q ´ ba b a ´ b a Therefore, ` aP ` bP ˘ ´ “ a ´ b Circ ` , , ´ ba , , , b a, , , ´ b , , , a ˘ .
6. Drazin inverse of aP s n ` bP s n Given a matrix A , the column space of A is denoted by Rank p A q andits dimension by rank A . The null space of A is denoted by Null p A q andits dimension, called nullity, by null p A q . The index of a square matrix A , denoted by ind p A q , is the smallest non-negative integer k for whichRank p A k q “ Rank p A k ` q . It is well-known that a circulant matrix hasindex at most 1, see [8]. Let A be a matrix of index k , the Drazin inverse of A , denoted by A D , is the unique matrix such that1. AA D “ A D A ,2. A k ` A D “ A k ,3. A D AA D “ A D .The following Bjerhammar-type condition for the Drazin inverse wasproved in [9]. We find it useful in order to check the Drazin conditions incombinatorial settings. Theorem 6.1 ([9]) . Let A and D be square matrices of order n , with ind p A q “ k , such that Null ` A k ˘ “ Null p D q , and AD “ DA . Then A k ` D “ A k if and only if D A “ D . Theorem 6.2.
Let n , s and s be integers such that ď s ă s ă n . Let a and b be non-zero real numbers such that a n zp s ´ s q ´ p´ b q n zp s ´ s q “ .Then p aP s n ` bP s n q D equals P n ´ s n P σ n,s ´ s ” I p n,s ´ s q b ` aI n zp s ´ s q ` bP n zp s ´ s q ˘ D ı P Tσ n,s ´ s . (10)14 roof. The following two fact are well-known. If A “ XBX ´ , then A D “ XB D X ´ . If AB “ BA , then p AB q D “ B D A D “ A D B D . See [4] or [6].In 1997, Wang, proved that p A b B q D “ A D b B D and ind p A b B q “ max t ind p A q , ind p B qu , see Theorem 2.2. of [24]. Let s “ s ´ s . Then, byTheorem 3.4 p aP s n ` bP s n q D “ P n ´ s n ” P σ n,s “ I p n,s q b ` aI n z s ` bP n z s ˘‰ P Tσ n,s ı D . “ P n ´ s n P σ n,s “ I p n,s q b ` aI n z s ` bP n z s ˘‰ D P Tσ n,s “ P n ´ s n P σ n,s ” I p n,s q b ` aI n z s ` bP n z s ˘ D ı P Tσ n,s . Therefore, we just need to study the Drazin inverse of the matrices ofthe form aI n ` bP n .By Corollary 4.3, we know that aP s n ` bP s n is singular if and only if a n z s ´ p´ b q n z s “
0, where s “ s ´ s . If n z s is even, then a n z s ´ p´ b q n z s “ | a | “ | b | . If n z s is odd, then a n z s ´ p´ b q n z s “ a “ ´ b . Hence, the matrices of the form a p I n ´ P sn q are always singular,but the matrices of form a p I n ` P sn q are singular if and only if n z s is even.All of these facts reduce the calculus of Drazin inverse of singular circulantmatrices of form aP s n ` bP s n to the calculus of Drazin inverse of just twomatrices: I n ´ P n and I n ` P n . Drazin inverse of I n ´ P n Theorem 6.3.
The Drazin inverse of aI n ´ aP n is p aI n ´ aP n q D : “ an n ´ ÿ i “ p n ´ i ´ q P in . (11) Example 6.4.
Let a be a non-zero real number. Thus, we have that p aI ´ aP q D “ a Circ p , , ´ , ´ q , and p aI ´ aP q D “ a Circ p , , , ´ , ´ q . In order to prove this theorem, we will use some polynomial tools. Theidea is to prove that both matrices in (11) have the same null space, andthen use the Theorem 6.1.By 11 n we denoted the vector of all ones. The direct proof of the nextlemma is left to the reader. Lemma 6.5.
Under the above notation we have that n P Null p aI n ´ aP n q X Null ˜ an n ´ ÿ i “ p n ´ i ´ q P in ¸ . p c , . . . , c n ´ q has its associated polinomial P C p x q “ ř n ´ i “ c i x i . Ingleton, in 1955, proved the following proposition. Proposition 6.6 (Proposition 1.1 in [12]) . The rank of a circulant matrix C of order n is n ´ d , where d is the degree of the greatest common divisorof x n ´ and the associated polynomial of C . This means that in order to know the rank of some circulant matrix, it isenough to know how many n -th roots of the unit are roots of its associatedpolynomial.Let us consider the polynomials H nj p x q “ n ´ ÿ i “ p n ´ p i ` j q ´ q x i , (12)for j “ , . . . , n ´
1. Notice that H n p x q is the associated polynomial of thecirculant matrix ř n ´ i “ p n ´ i ´ q P in .We denote by Ω ℓ “ t x P C : x ℓ “ u , the set of ℓ -th root of the unit. Notice that,Ω n “ t ω kn : k “ , . . . , n ´ u , where ω “ exp p πin q , i.e. ω kn are exactly the roots of the polynomial p p x q “ x n ´
1. Moreover, they are all simple roots of p p x q “ x n ´
1, because theyare all different and p has degree n . Moreover, if we denoteΦ n p x q : “ x n ´ x ´ “ n ´ ÿ i “ x i , then ω kn are all simple roots of Φ n for any k ‰
0. In terms of derivatives,this means that Φ n p ω q “ n p ω q ‰ , for all ω P Ω n r t u . Proposition 6.7.
Under the above notation we have that H nj p ω q ‰ , for all j “ , . . . , n ´ and ω P Ω n r t u .Proof. We have that H nj p x q “ n ´ ÿ i “ p n ´ p i ` j q ´ q x i “ p n ´ j ´ q n ´ ÿ i “ x i ´ n ´ ÿ i “ ix i . ř n ´ i “ ix i “ x Φ n p x q , and so we obtain that H nj p x q “ p n ´ j ´ q Φ n p x q ´ x Φ n p x q . (13)Now, assume that H nj p ω q “ ω P Ω n r t u , thus (13) implies that0 “ p n ´ j ´ q Φ n p ω q ´ ω Φ n p ω q . Hence, we have that ω Φ n p ω q “
0, since Φ n p ω q “
0. But this cannot occursince ω ‰ ω are all simple roots of Φ n . Therefore H nj p ω q ‰ j “ , . . . , n ´ ω P Ω n r t u , as asserted.We are ready to prove the main result of the subsection. Proof. of Theorem 6.3:
Since index of all singular circulant matrices is 1and the rank of I n ´ P n is clearly n ´
1, by Propositions 6.6 and 6.7, wehave that Null p aI n ´ aP n q “ Null ˜ an n ´ ÿ i “ p n ´ i ´ q P in ¸ . By Theorem 6.1, we just need to check that p aI n ´ aP n q ˜ an n ´ ÿ i “ p n ´ i ´ q P in ¸ “ aI n ´ aP n . which is left to the reader.In order to obtain an explicit formula for the Drazin inverse of the cir-culant matrices of form aP s n ´ aP s n we define ρ ˚ n,s p i q : “ δ , cycle n,s p i q ` n z s ´ n,s p i q ´ ˘ , (14)where δ is the usual Kronecker delta. In this case, ρ ˚ satisfies the following:1. For n ą ρ ˚ n, p i q “ p n ´ i ´ q for all i “ , . . . , n ´ ă s ă n , ρ ˚ n z s, p i q “ ρ ˚ n,s p i ¨ s q for all i “ , . . . , n z s ´ Corollary 6.8.
Let n , s and s be integers such that ď s ă s ă n . Let a be a non-zero real number. Then p aP s n ´ aP s n q D “ a r n z p s ´ s qs n ´ ÿ i “ ρ ˚ n,s ´ s p i ` s q P in . (15) Proof.
Let s “ s ´ s . By Theorem 6.2 we have that p aP s n ´ aP s n q D isequal to P n ´ s n P σ n,s ” I p n,s q b ` aI n z s ´ aP n z s ˘ D ı P Tσ n,s . p aP s n ´ aP s n q D “ P n ´ s n P σ n,s ¨˝ p n z s q´ ÿ i “ I p n,s q b ρ ˚ n z s, p i q a p n z s q P in z s ˛‚ P Tσ n,s . Hence, by Corollary 3.3, and expresions (8) and (9) p aP s n ´ aP s n q D “ p n z s q´ ÿ i “ ρ ˚ n z s, p i q a p n z s q P p i ¨ s q´ s n “ p n z s q´ ÿ i “ ρ ˚ n,s p i ¨ s q a p n z s q P p i ¨ s q´ s n “ n ´ ÿ j “ ρ ˚ n,s p j ` s q a p n z s q P jn . Example 6.9.
For example, we compute ` aP ´ aP ˘ D . Since z “ .By Theorems 6.2 and 6.3, we know that the Drazin inverse is essentially 3blocks of Circ p , , ´ , ´ q merge in a ˆ matrix via P and P σ , . i cycle , p i ` q δ , cycle , p i ` q pos , p i ` q ρ ˚ , p i ` q ` aP ´ aP ˘ D “ a Circ p , , , , , ´ , , , ´ , , , q .6.2. Drazin inverse of I n ` P n Theorem 6.10.
The Drazin inverse of aI n ` aP n is p aI n ` aP n q D “ an n ´ ÿ i “ p´ q i p n ´ i ´ q P i n . (16)The idea of the proof is the same as before, i.e. we want to show thatboth matrices in (16) have the same null space, and then use Theorem 6.1.We define ˘ n “ pp´ q i q n ´ i “ P R n , the vector of ones and minus ones.The following lemma, can be proved easily and it is left to the reader.18 emma 6.11. Under the above notation we have that ˘ n P Null p aI n ` aP n q X Null ˜ an n ´ ÿ i “ p´ q i p n ´ i ´ q P i n ¸ . Let us consider the polynomials p H nj p x q “ n ´ ÿ i “ p´ q i ` j p n ´ p i ` j q ´ q x i , (17)for j “ , . . . , n ´
1. As before, notice that p H n p x q is the associated poly-nomial of the circulant matrix ř n ´ i “ p´ q i p n ´ i ´ q P i n , the followingresult assures that its rank is 2 n ´ Proposition 6.12.
Let n be a positive integer. Then p H nj p ω q ‰ for all j “ , . . . , n ´ and ω P Ω n r t´ u .Proof. On the first hand, the linear transformation T p x q “ ´ x for x P C isa permutation on Ω n , i.e. we have that ω is a p n q -th root of 1 ðñ ´ ω is a p n q -th root of 1 . (18)On the other hand, if C i ` j “ n ´ p i ` j q ´ p H nj p x q “ n ´ ÿ i “ p´ q i ` j C i ` j x i “ p´ q j n ´ ÿ i “ C i ` j p´ x q i “ p´ q j H nj p´ x q , where H nj as in (12). By (18) and Proposition 6.7 we obtain that p H nj p ω q “ ðñ H nj p´ ω q “ ðñ ω “ ´ . So, if ω ‰ ´
1, then p H nj p ω q ‰
0, as desired.We are ready to prove the main result of the subsection.
Proof. of Theorem 6.10:
The rank of aI n ` aP n is clearly 2 n ´
1. Hence,by Lemma 6.11 and Proposition 6.12Null p aI n ` aP n q “ Null ˜ an n ´ ÿ i “ p´ q i p n ´ i ´ q P i n ¸ . By Theorem 6.1, it suffices to check that p aI n ` aP n q ˜ an n ´ ÿ i “ p´ q i p n ´ i ´ q P i n ¸ “ aI n ` aP n . which is left to the reader. 19 orollary 6.13. Let n , s and s be integers such that ď s ă s ă n .Let a be a non-zero real number. Then p aP s n ` aP s n q D “ a r n z p s ´ s qs n ´ ÿ i “ p´ q i ` s ρ ˚ n,s ´ s p i ` s q P i n . (19) Example 6.14.
Let us compute ` aP ` aP ˘ D . Since z “ . ByTheorems 6.2 and 6.3, we know that the Drazin inverse is essentially 3blocks of Circ p , ´ , ´ , q merge in a ˆ matrix via P and P σ , .Since ´ “ , then z “ , and p , q “ . Therefore, we have i cycle , p i ` q δ , cycle , p i ` q pos , p i ` q p´ q i ` ρ ˚ , p i ` q ` aP ` aP ˘ D “ a Circ p , , ´ , , , ´ , , , , , , q . Acknowledgment
Even though the text does not reflect it, we carry on many numericalexperiments on [23]. They gives us the insight for this paper.Funding: This work was partially supported by the Universidad Nacionalde San Luis, grant PROICO 03-0918, and MATH AmSud, grant 18-MATH-01. Denis Videla was partially supported by CONICET and SECyT-UNC.Andr´es M. Encinas has been partially supported by Comisi´on Interministe-rial de Ciencia y Tecnolog´ıa under project MTM2017-85996-R.
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