Adjoint Mappings and Inverses of Matrices

Adjoint Mappings and Inverses of Matrices Predrag Stanimirović1∗, Stojan Bogdanović2 and Miroslav Ćirić3 1,3 University of Niš, Department of Mathematics, Faculty of Science, Ćirila i Metodija 2, 18000 Niš, Yugoslavia. 2 E-mail: 1 University of Niš, Faculty of Economics, Trg V.J. 11, P. O. Box 121, 18000 Niš, Yugoslavia 3 2 [email protected] [email protected] [email protected], Abstract In this paper we investigate a general and determinantal representation, and conditions for the existence of a nonzero {2}-inverse X of a given complex matrix A. We introduce a determinantal formula for X, representing its elements in terms of minors of the order s = rank(X), 1 ≤ s ≤ r = rank(A), taken from the matrix A and two adequately selected matrices. In accordance with these results we find restrictions of the adjoint mapping such that the set A{2} is equal to the union of their images. Minors of {2}-inverses are also investigated. Restrictions to the set of {1,2}-inverses produce the known results from [1], [2], [3] and [10]. Also, in a partial case we get known results from [11], relative to the Drazin inverse. AMS Subj. Class.: 15A09. Key words: {2}-inverses; determinantal representation; Adjoint mapping; Drazin inverse. 1 Introduction and preliminaries We consider the set of complex matrices. For any matrix A consider the following equations in X (1) AXA = A, (2) XAX = X, (3) (AX)∗ = AX, (4) (XA)∗ = XA. where ∗ denotes the conjugate and transpose. If A is a square matrix we also consider the following equations: (5) ∗ Corresponding AX = XA, (1k ) author 1 Ak+1 X = Ak . 2 P. Stanimirović, S. Bogdanović and M. Ćirić For a sequence S of elements from the set {1, 2, 3, 4, 5}, the set of matrices obeying the equations represented in S is denoted by A{S}. A matrix from A{S} is called an S-inverse of A and denoted by A(S) . The Moore-Penrose inverse A† of A is the unique {1, 2, 3, 4}-inverse of A. The group inverse, denoted by A# , is the unique {1, 2, 5}inverse of A, and it exists if and only if ind(A) = min{k : rank(Ak+1 ) = rank(Ak )} = 1. A matrix X = AD is said to be the Drazin inverse of A if (1k ) (for some positive integer k), (2) and (5) are satisfied. −1 By A−1 R and AL we denote a right and a left inverse of A, respectively. The set of m × n matrices of rank r with entries in the set of complex numbers C is denoted by Crm×n . Also, O (resp. I) denote an appropriate zero (resp. identity) matrix. By Tr(A) we denote the trace of a square matrix A, the determinantal rank of A is denoted by ρ(A), and |A| denotes the determinant of A. We use the following notations from [3] and [11]. Let A be an m × n matrix of rank r; let α = {α1 , . . . , αp } and β = {β1 , . . . , βp } be subsets of {1, . . . , m} and {1, . . . , n}, respectively, of the order 1 ≤ p ≤ min{m, n}. By Aα β = Aαβ we denote the p × p submatrix of A determined by the entries in rows indexed by α and in columns indexed α by β, and |Aα β | is the determinant of Aβ . Denote the collection of strictly increasing sequences of p integers chosen from {1, . . . , n} by Qp,n = {α : α = (α1 , . . . , αp ), 1 ≤ α1 < · · · < αp ≤ n} . For any integer q satisfying 1 ≤ q ≤ rank(A) we define the set Nq = Qq,m × Qq,n . For fixed α, β ∈ Qp,n , 1 ≤ p ≤ q, let Iq (α) = Nq (α, β) = {I : I ∈ Qq,m , I ⊇ α} , Iq (α) × Jq (β). Jq (β) = {J : J ∈ Qq,n , J ⊇ β} , If A is a square matrix, then the coefficient of |Aα β | in the Laplace expansion of |A| is ∂ |A|. In the special case α = {i}, β = {j}, we have the cofactor ∂a∂ij |A| denoted by ∂|A α| β of aij . Following the notations from [3], it is clear that Nr = N , Nr (α, β) = N (α, β) and Ir (α) = I(α), Jr (β) = J (β). By Cp (A) we denote the pth compound matrix of A with rows indexed by p-element subsets of {1, . . . , m}, columns indexed by p-element subsets of {1, . . . , n}, and the (α, β) entry defined by |Aα β |. Representations and characterizations of {1, 2}-inverses for matrices over an integral domain are investigated in [2], [3] and [10]. Using an arbitrary n × m matrix W of rank r, an arbitrary {1, 2}-inverse of A is expressed as follows: (1.1) −1 (A(1,2) )ij = (Tr(Cr (W A))) · X (α,β)∈N (j,i) |Wαβ | ∂ |Aα |. ∂aji β In the partial cases W = A∗ , W = A and W = (M AN −1 )∗ analogous representations and characterizations for the Moore-Penrose inverse, the group inverse and the weighted 3 Adjoint Mappings and Inverses of Matrices Moore-Penrose inverse can be derived. These results are introduced in [1], [4] and [5], respectively. In the papers [7], [8] and [9] D.W. Robinson introduced the adjoint mapping for matrices over a commutative ring ℜ with identity 1. The adjoint mapping of A of the n m order s is denoted by A•,s : ℜ( s )×( s ) 7→ ℜn×m and defined by  P ´ ³ P  s>1 S bβα Tβ Aad αβ α , (1.2) A•,s : B 7→ AB,s = AB = α∈Qs,m β∈Qs,n  B, s=1 where A is m × n matrix of rank r over ℜ, s ≤ min{m, n}, Sα is s × m matrix with 1 in positions (1, α(1)), · · · , (s, α(s)) and 0 elsewhere, Tβ is n × s matrix with 1 in positions n m (β(1), 1), · · · , (β(s), s) and 0 elsewhere, B = (bβα ) ∈ ℜ( s )×( s ) and Aad αβ is the classical adjoint matrix of Aα β = Aαβ . A determinantal formula for the Drazin inverse of a given square matrix over an integral domain is investigated in [11]. This representation of the Drazin inverse is based on the usage of minors of the order rk = rank(Ak ), k = ind(A), selected from the matrices A and Al , l ≥ k: X ¡ ¢−1 ∂ (1.3) (AD )ij = Tr(Crk (Al+1 )) |(Al )βα | |Aα | ∂aji β (α,β)∈Nrk (j,i) In the second section we investigate conditions for the existence of nonzero {2}inverses of a given matrix A ∈ Crm×n . These results are continuation of the papers [2], [3] and [10], where corresponding conditions for the existence of {1,2}-inverses are derived. Also, these results generalize known facts for the Drazin inverse, stated in [11]. In the third section we investigate several characterizations and representations of {2}-inverses. A nonzero {2}-inverse X of A obeying rank(X) = s, 1 ≤ s ≤ r = rank(A) is expressed in terms of minors of the order s, taken from the matrix A and two appropriate matrices. In the case s = r we derive analogous results relative to {1,2}-inverses, introduced in [2], [3] and [10]. Also, we examine minors of nonzero {2}-inverses. These results generalize known facts about minors of {1,2}-inverses from [1] as well as the analogous characterizations of minors of the Drazin inverse from [11]. In accordance with the results obtained in the third section, in the last section we define restrictions A⋄,s of the adjoint mappings A•,s , 1 ≤ s ≤ r, whose images form the (2) set of nonzero {2}-inverses of A. The Drazin inverse and the generalized inverse AT,S are exactly characterized as the elements of the image of A⋄,s . 2 The existence of {2}-inverses The following lemma gives an useful characterization of the adjoint mapping. Lemma 2.1 Let A ∈ C m×n of rank r, 1 ≤ s ≤ r is selected integer and F ∈ C n×s and G ∈ C s×m are two arbitrary matrices such that GAF is invertible. Then the matrix X = F (GAF )−1 G can be expressed by means of the adjoint mappings as follows: (2.1) −1 X = F (GAF )−1 G = (Tr(Cs (F GA))) ACs (F G) . 4 P. Stanimirović, S. Bogdanović and M. Ćirić Proof. Assume that A = P Q is a full-rank factorization of A. First we prove the following statement: |GAF | = |GP QF | = Tr(Cs (F GA)). (2.2) In the case 1 < s ≤ r, equality (2.2) can be derived after two successive applications of the Cauchy-Binet theorem: X X |GP QF | = |(GP )ǫ ||(QF )ǫ | = |GPǫ ||Qǫ F | ǫ∈Qs,r = X ǫ∈Qs,r = ǫ∈Qs,r   X X γ∈Qs,n = |Gγ ||Pǫγ |   |(F G)δγ | X |(F G)δγ ||Aγδ | X  δ∈Qs,m  |Pǫγ ||Qǫδ | = ǫ∈Qs,r (γ,δ)∈Ns X  |Qǫδ ||F δ | X |(F G)δγ ||P γ Qδ | (γ,δ)∈Ns = Tr(Cs (F GA)). (γ,δ)∈Ns In the case s = 1 we put F = uT , G = v, where (2.3) u = {u1 , . . . , un }, v = {v1 , . . . , vm }, and obtain (2.4) |GAF | = GAF = m X n X uj vi aij = Tr(F GA) = Tr(C1 (F GA)). i=1 j=1 In the rest of the proof we derive the determinantal representation (2.1) for X. Since −1 X = F (GAF )−1 G = (|GAF |) · F · adj(GP QF ) · G and we have just derived the representation for the determinant |GAF |, we now derive the determinantal formula for F · adj(GP QF ) · G. This representation can be derived in a similar way as in [11]. In the case s > 1 it is sufficient to replace the matrix PAl by the matrix F and the matrix QAl by the matrix G. Direct computation yields X ∂ |Aα |, 1 ≤ i ≤ n, 1 ≤ j ≤ m. (2.5) (F · adj(GAF ) · G)ij = |(F G)βα | ∂aji β (α,β)∈Ns (j,i) In the case s = 1 the matrices F and G are of the form (2.3). Also, using (2.4) we conclude that GAF = |GAF | = Tr(C1 (GAF )) is a nonzero complex number, and ¡ ¢ −1 (2.6) F (GAF )−1 G ij = (GAF )−1 (F G)ij = (Tr(F GA)) (F G)ij . Finally, we derive representation (2.1). Let 1 ≤ s ≤ r be an arbitrary integer. The adjoint mapping of A of the order s, defined in (1.2), can be rewritten in the form  P  bβα ∂a∂ji |Aα β |, 1 < s ≤ r  (α,β)∈Ns (j,i) (2.7) (A•,s (B))ij = (AB,s )ij = bij , s=1   Adjoint Mappings and Inverses of Matrices 5 ¡ ¢ ¡ ¢ n m where B = (bβα ) ∈ C ( s )×( s ) and i = 1, . . . , ns , j = 1, . . . , m s . Now, the representation (2.1) for X follows immediately from (2.2), (2.5), (2.6) and (2.7). In the following theorem we investigate conditions for the existence of nonzero {2}inverses. Theorem 2.1 Let A ∈ C m×n is of rank r and let 1 ≤ s ≤ r be an integer. The following conditions are equivalent: (i) There exists a {2}-inverse of A of rank s. (ii) There exist M∈ C n×s , N∈ C s×m such that N AM = Is . (iii) There exist F∈ C n×s , G∈ C s×m such that GAF is invertible matrix. (iv) There exists an X∈ C n×m of rank s, such that Tr(Cs (XA)) = 1 and ACs (X) = X. (v) There exist F ∈ C n×s and G ∈ C s×m , such that Tr(Cs (F GA)) 6= 0. Proof. (i) ⇒ (ii): If X is a nonzero {2}-inverse of A of rank s, then the matrices M and N obtained from the full-rank factorization X = M N satisfy the condition (ii). Indeed, from XAX = X we get M N AM N = M N . Since X = M N , M∈ C n×s , N∈ C s×m is the full-rank factorization of X, we have rank(M ) = rank(N ) = s. Hence, M is left invertible and N is right invertible [6, p. 19]. Any right inverses of N is of the form NR−1 = V N ∗ (N V N ∗ )−1 , where V is an arbitrary matrix such that rank(N V N ∗ ) = rank(N ) [6, p. 20]. Similarly, left inverses of M possess the form ML−1 = (M ∗ V M )−1 M ∗ V , where V is an arbitrary matrix satisfying rank(M ∗ V M ) = rank(M ) [6, p. 20]. Now, using left invertibility of M and right invertibility of N , we have N AM = Is . (ii) ⇒ (i): If M and N satisfy (ii), then one can verify that X = M N is {2}-inverse of A of rank s. Indeed, from (ii) we get XAX = M N AM N = M N = X. Also, using X = M N = M (N AM )−1 N , we get rank(X) ≥ rank(N AM ) = s, rank(X) ≤ min{rank(M ), rank(N )} = s which implies rank(X) = s. (i) ⇒ (iii): This statement follows from (i) ⇒ (ii) and the obvious implication (ii)⇒ (iii). (iii) ⇒ (i): When F and G satisfy conditions (iii), in a similar way as in the part (ii)⇒(i) of the proof, one can verify that X = F (GAF )−1 G is {2}-inverse of A of rank s. (ii) ⇒ (iv): Assume that M and N satisfy conditions (ii). We show that X = M (N AM )−1 N = M N satisfies conditions (iv). In accordance with (ii) we have (2.8) Tr(Cs (XA)) = Tr(Cs (M N A)) = Tr(Cs (N AM )) == |Is | = 1. 6 P. Stanimirović, S. Bogdanović and M. Ćirić Also, using (2.8) and Lemma 2.1 we obtain −1 X = (Tr(Cs (M N A))) −1 ACs (M N ) = (Tr(Cs (XA))) ACs (X) = ACs (X) . (iv) ⇒ (ii): Assume that X satisfies conditions in (iv). Select a full-rank factorization X = M N of X. As in the part (i) ⇒ (ii) we conclude that M is left invertible and N is right invertible. We have −1 X = ACs (X) = (Tr(Cs (XA))) −1 ACs (X) = (Tr(Cs (M N A))) ACs (M N ) . In accordance with Lemma 2.1 we obtain X = M (N AM )−1 N . Finally, using X = M (N AM )−1 N = M N, left invertibility of M and right invertibility of N , we get N AM = Is . (iii) ⇔ (v): This part of the proof can be verified using (2.2) and the following known fact: GAF is an invertible matrix if and only if |GAF | 6= 0. 3 Characterizations and representations of {2}-inverses In the following theorem we investigate several characterizations and representations of nonzero {2}-inverse. Theorem 3.1 Let A ∈ C m×n and let X ∈ C m×n be of determinantal rank s ≥ 1. Then the following conditions are equivalent: (i) XAX = X. (ii) N AM = Is for each full-rank factorization X = M N , M ∈ C n×s , N ∈ C s×m . (iii) Tr(Cs (XA)) = 1 and X = ACs (X) . −1 (iv) X = F (GAF )−1 G = (Tr(Cs (F GA))) ACs (F G) for an n × s matrix F and an s × m matrix G, such that GAF is an invertible matrix. Proof. (i) ⇒ (ii): Assume that X = M N is a full-rank factorization of X ∈ A{2}. As in Theorem 2.1 one can prove that M is left invertible and N is right invertible matrix. Now, this part of the proof follows from M N AM N = M N , left invertibility of M and right invertibility of N . (ii) ⇒ (i): Using N AM = Is we have XAX = M N AM N = M N = X. (i) ⇒ (iii): Assume that X is a nonzero {2}-inverse of A. Choose a rank factorization X = U V , U ∈ C n×s , V ∈ C s×m of X. Using (i) ⇔ (ii) we conclude V AU = Is , and get (3.1) Tr(Cs (XA)) = Tr(Cs (U V A)) = Tr(Cs (V AU )) = Tr(Cs (Is )) = 1. Also, using (3.1) and X = U V = U (V AU )−1 V , in view of Lemma 2.1 we obtain −1 X = (Tr(Cs (U V A))) −1 ACs (U V ) = (Tr(Cs (XA))) ACs (X) = ACs (X) . 7 Adjoint Mappings and Inverses of Matrices (iii) ⇒ (i): Assume that X satisfies (iii) and X = U V , U ∈ C n×s , V ∈ C s×m is a full-rank factorization of X. Then −1 X = ACs (X) = (Tr(Cs (XA))) −1 ACs (X) = (Tr(Cs (U V A))) ACs (U V ) . In accordance with Lemma 2.1 we obtain X = U (V AU )−1 V . Using X = U (V AU )−1 V = U V we get V AU = Is . Finally, we have XAX = U V AU V = U V = X. (i) ⇒ (iv): Assume that X is {2}-inverse of A of rank s. Using known result [6, Theorem 3.4.1], there exist two matrices C ∈ C n×p and D ∈ C q×m , such that X = C(DAC)(1,2) D and rank(X) = rank(DAC). In accordance with the general representation of {1, 2}-inverses from [1, Theorem 3], X can be represented in the form X = CVR−1 UL−1 D, where DAC = U V is a full-rank factorization of DAC. Using the general representations of left and right inverses from [6, Theorem 2.1.1], we conclude that there exist matrices T1 ∈ C p×p and T2 ∈ C q×q , such that X and = CT1 V ∗ (V T1 V ∗ )−1 (U ∗ T2 U )−1 U ∗ T2 D = CT1 V ∗ (U ∗ T2 DACT1 V ∗ )−1 U ∗ T2 D, rank(X) = rank(DAC) = rank(U ∗ T2 DACT1 V ∗ ) = s. Applying substitutions F = CT1 V ∗ ∈ C n×s and G = U ∗ T2 D ∈ C s×m we obtain rank(GAF ) = s and (3.2) X = F (GAF )−1 G, Using Lemma 2.1 we conclude that X satisfies all representations from (iv). (iv) ⇒ (i): If X is of the form X = F (GAF )−1 G for appropriate matrices F and G satisfying (iv), it is not difficult to verify rank(X) = s and XAX = F (GAF )−1 G = X. Remark 3.1 In the case X ∈ A{1, 2} we have rank(X) = rank(A) = r. Hence, in the case s = r the results of Theorem 2.1 and Theorem 3.1 give the known general representation, determinantal representation and conditions for the existence of the reflexive g-inverses from [10]. Moreover, in the case F = PAl , G = QAl , where Al = PAl AAl is a full-rank factorization of Al , l ≥ ind(A), we get analogous full-rank and determinantal representation of the Drazin inverse from [11]. (2) Using the general representation of the generalized inverse AT,S from [12], we conclude the following: if the matrices F and G satisfy R(F ) = T and R(G) = S, then Theorem 2.1 gives analogous representations and conditions for the existence of the (2) generalized inverse AT,S . Corollary 3.1 Given matrices A and X of the order m × n and n × m, respectively. Conditions (iii) and (v) from Theorem 2.1 as well as the representation (iv) from Theorem 3.1 are satisfied for the matrices F and G selected from the full-rank factorization X = F G. 8 P. Stanimirović, S. Bogdanović and M. Ćirić Proof. Follows from Theorem 2.1 and Theorem 3.1, using GAF = Is . Remark 3.2 We are in the position to state the following algorithm for the verification whether or not X ∈ Csn×m is a {2}-inverse of rank s for A ∈ Crm×n , in the case 1 ≤ s ≤ r. Find a full-rank factorization X = M N . Then X is {2}-inverse of A of rank s if and only if N AM = Is . Also, conditions (iii) and (v) from Theorem 2.1 as well as the representation (iv) from Theorem 3.1 are satisfied for the matrices F = M and G = N . Remark 3.3 In accordance with Theorem 2.1 and Theorem 3.1, each {2}-inverse X of A of rank s possesses the form (3.2). Of course, one can imagine a lot of various algorithms, less or more effective, in order to generate {2}-inverses of A of rank s. Several illustrative examples follow. Example 3.1 Consider the following 6 × 5 matrix of rank 4, which is generated by putting a = 1 in the test matrix M3 from [13, p. 143]:  1  1  2 A= 3  2 3 3 4 5 6 4 6 3 4 4 5 6 7 4 6 5 6 7 7 1 2 3 4 6 8    .  We choose appropriate matrices F and G in order to generate {2}-inverses F (GAF )−1 G of A. The matrices F and G must satisfy rank(F G) = s ≤ 4, and dimensions of the matrix F G must be 6 × 5. 1. For example, one can choose F and G using the following pattern: F = h Fs O i , G = [ Is O ], where Fs As = Is and As is the principal minor which contains first s rows and first s columns of A. For example, let us choose s = 2. In this case the block F2 is equal to F2 = h 3 −1 −2 1 i . Corresponding {2}-inverse X = F (GAF )−1 G = F G of rank 2 exists because of |GAF | = Tr(C2 (F GA)) = 1. Using (2.1) we have   X=  In view of (AC2 (X) )ij = ∂ ∂aji ¯ ¯ 1 ¯ 1 3 −1 0 0 0 2 3 −2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0   .  ¯ ¯ ¯ it is not difficult to verify X = AC2 (X) . 9 Adjoint Mappings and Inverses of Matrices 2. Also, we can use randomly generated matrices F ∈ C 5×2 and G ∈ C 2×6 , for example   0  2 F =  35 (3.3) 1 0 1 2 3 0  ,  G= h 0 1 1 0 0 1 1 0 0 1 1 0 i . Since rank(F G) = 2 and |GAF | = Tr(C2 (F GA)) = 174 we conclude that corresponding {2}-inverse X = F (GAF )−1 G of A of rank 2 exists. Simple calculation gives   0 −21 1   60 X= 174  39 −102 0 19 −46 −27 84 0 −21 60 39 −102 0 19 −46 −27 84 0 −21 60 39 −102 −1 One can verify the representation X = (Tr(C2 (F GA))) X satisfies rank(X) = 2 = rank(F G) = rank(GAF ). 0 19 −46 −27 84  .  AC2 (F G) . Let us mention that In the case F G = A∗ , using s = r = rank(A) = 4, from (2.1) we conclude that the Moore-Penrose inverse A† of A exists, because of Tr(C2 (F GA)) = Tr(C2 (A∗ A)) = 8. Also, we have   4 † ∗ −1 A = (Tr(Cr (A A))) 1  −8 10 ACr (A∗ ) =  8  −2 −4 −1 15 −13 3 −2 −8 −36 26 −2 12 7 23 −15 1 −10 −5 −5 1 1 6 3 3 −1 −1 −2  .  This example confirms known test matrix from [13, p. 143]. Minors of {1,2}-inverses for matrices over an integral domain are investigated in [1], [2] and [10]. We briefly restate these results (for the set of complex matrices). Proposition 3.1 [1], [2], [10]. Let A be a complex matrix of type m × n, whose rank is r. Then A admits a reflexive g-inverse X = (xij ) whose r × r minors are proportional to the corresponding minors of a given n × m matrix H if and only if Tr(Cr (AH)) is invertible. Then, for all (α, β) ∈ N the following is valid: −1 |Xβα | = (Tr(Cr (HA))) |Hαβ |. Also, in this case xij = X |Xαβ | (α,β)∈N (j,i) ∂ |Aα |. ∂aji β 10 P. Stanimirović, S. Bogdanović and M. Ćirić Moreover, in [11] we show that the Drazin inverse AD of a given square matrix A and Ak , k = ind(A) have proportional minors of the order rk = rank(Ak ). In the next statement we extend these results to the set of {2}-inverses. Corollary 3.2 Let A ∈ C m×n be of rank r and X be a nonzero {2}-inverse of A satisfying rank(X) = s, 1 ≤ s ≤ r and X = F (GAF )−1 G, for some matrices F ∈ C n×s , G∈ C s×m . Then we have (3.4) −1 |Xαβ | = (Tr(Cs (F GA))) |(F G)βα |, ∀(α, β) ∈ Ns . Proof. Using the equivalence (iii) ⇔ (iv) of Theorem 3.1 we have −1 X = ACs (X) = (Tr(Cs (F GA))) ACs (F G) . In the case 1 < s ≤ r we get xij X = |Xαβ | (α,β)∈Ns (j,i) ∂ |Aα | ∂aji β −1 = (Tr(Cs (F GA))) X |(F G)βα | (α,β)∈Ns (j,i) ∂ |Aα | ∂aji β for each 1 ≤ i ≤ n, 1 ≤ j ≤ m. In the case s = 1 we get in a similar way xij = (Tr(F GA)) Thus, we have −1 (F G)ij . −1 |Xαβ | = (Tr(Cs (F GA))) |(F G)βα |, for each (α, β) ∈ Ns (j, i) and each 1 ≤ i ≤ n, 1 ≤ j ≤ m. The proof can be completed using that i and j are arbitrary. Theorem 3.2 Let A be m × n complex matrix of rank r and let H be n × m complex matrix of rank s, for some integer s satisfying 1 ≤ s ≤ r. Then A admits {2}-inverse X = (xij ) of rank s ≥ 1 whose s × s minors are proportional to corresponding s × s minors of matrix H if and only if Tr(Cs (AH)) 6= 0. In this case, for all (α, β) ∈ Ns , 1 ≤ s ≤ r we have −1 |Xαβ | = (Tr(Cs (HA))) (3.5) |Hαβ |. Proof. Assume that X = (xij ) is a nonzero {2}-inverse of rank s whose s × s minors are proportional to corresponding minors of H. Consider firstly the case s = 1. This part of the proof can be proved in a similar way as the corresponding part of the proof in [2, Theorem 3]. Since XAX = X and xij xkl = xil xkj for each 1 ≤ i, k ≤ n, 1 ≤ l, j ≤ m, we have X X xil = xkj ajk = xil Tr(C1 (AX)). xij ajk xkl = xil j,k j,k 11 Adjoint Mappings and Inverses of Matrices Because of rank(X) ≥ 1 we conclude that xil is nonzero for some i, l, and we get Tr(C1 (AX)) = 1. Since X and H have proportional minors, there exists a nonzero complex number θ such that H = θX, and we have that Tr(C1 (AH)) = θ is invertible. In the case s > 1 we have Cs (H) = θCs (X), where θ 6= 0. Using part (iii) of Theorem 3.1, we have Tr(Cs (AH)) = θTr(Cs (AX)) = θ 6= 0. Now, assume that Tr(Cs (AH)) is invertible. If H = F G is a rank factorization for H, in accordance with Theorem 2.1 it follows that Tr(Cs (AH)) = Tr(Cs (F GA)) = |F GA| is invertible. Hence, the matrix GAF is invertible and X = F (GAF )−1 G is a nonzero {2}-inverse of A. Equality (3.5) can be verified applying Corollary 3.2. Example 3.2 Consider the matrix A from Example 3.1 and matrices F and G as in (3.3) from Example 3.1. In the case of the existence, minors of the order 2 of the matrix X = A(2) are arranged in the matrix 174−1 B, where  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0       B =     0 0 0 −1 −1 1 1 2 3 0 0 0 0 0 0 0 0 0 0 0 0 −1 −1 1 1 2 3 0 0 0 0 0 0 0 0 0 0 0 0 −1 −1 1 1 2 3 0 0 0 1 1 −1 −1 −2 −3 0 0 0 0 0 0 0 0 0 0 0 0 1 1 −1 −1 −2 −3 0 0 0 0 0 0 0 0 0 0 0 0 −1 −1 1 1 2 3 0 0 0 0 0 0 0 0 0 0 0 0 −1 −1 1 1 2 3 0 0 0 1 1 −1 −1 −2 −3 0 0 0 0 0 0 0 0 0 0 0 0 −1 −1 1 1 2 3      .     Also, we have C2 ((F G)T ) = B. Hence, as a verification of the results of Corollary 3.2, it is easy to see that elements of these matrices are proportional with the coefficient −1 174−1 = (Tr(C2 (F GA))) . Remark 3.4 In the case H = Ak , k = ind(A), s = rk = rank(Ak ) we get analogous result for the Drazin inverse: A admits the nonzero Drazin inverse whose rk × rk minors are proportional to corresponding rk × rk minors of matrix Ak if and only if Tr(Crk (Ak+1 )) is a nonzero number. In this case is ¡ ¢−1 Crk (AD ) = Tr(Crk (Ak+1 )) Crk (Ak ). This claim is an additional result with respect to known results concerning minors of the Drazin inverse from [11]. The following results are generalization of [11, Lemma 3.1] and can be proved in a similar way by the induction. Theorem 3.3 Let A ∈ Crn×n has a nonzero {2}-inverse X satisfying s = rank(X), 1 ≤ s ≤ r. Then for each p ≥ 1 the following two identities are valid: (i) Cs (X p+1 ) = Cs (X p )Tr(Cs (X)); p−1 (ii) Cs (X p ) = Cs (X) [Tr(Cs (X))] ; (iii) There exist constants ci , di , 1 ≤ i ≤ ¡n¢ s , such that (Cs (X p ))ij = ci dj · (Cs (X p ))11 , 1 ≤ i, j ≤ µ ¶ n . s 12 P. Stanimirović, S. Bogdanović and M. Ćirić Adjoint mapping and {2}-inverses 4 We introduce certain restrictions of the adjoint mappings of the order s, s = 1, . . . , r, such that the union of their images is equal to the set A{2}. Definition 4.1 For an arbitrary matrix A ∈ Crm×n and arbitrary integer s, 1 ≤ s ≤ r, the restricted adjoint mapping A⋄,s is the restriction of the adjoint mapping A•,s to the n m n m subset of the set C ( s )×( s ) , consisting of matrices B = (bβα ) ∈ C ( s )×( s ) of the form (4.1) bβα −1 = (Tr(Cs (W A))) |Wαβ |, W ∈ C n×m , rank(W ) = s, (α, β) ∈ Ns , 1 ≤ s ≤ r, i.e. (4.2) −1 B = (Tr(Cs (W A))) Cs (W ), W ∈ C n×m , rank(W ) = s. The following theorem is the main result of this section. Theorem 4.1 The set of nonzero {2}-inverses of A ∈ Crm×n is equal to (4.3) A{2}\{O} = Im(A⋄,r ) ∪ Im(A⋄,r−1 ) ∪ · · · ∪ Im(A⋄,1 ). Proof. Consider an arbitrary X ∈ A{2}\{O} of rank s, 1 ≤ s ≤ r. According to Theorem 3.1, part (iv), there exist matrices F ∈ C n×s , G ∈ C s×m such that X can be −1 expressed in the form X = (Tr(Cs (F GA))) ACs (F G) . Hence we can write ³ ´ −1 X = A⋄,s (Tr(Cs (F GA))) Cs (F G) , or X = A⋄,s (B) ∈ Im(A⋄,s ), where the matrix B is defined as in (4.2), in the case W = F G. On the other hand, assume X ∈ Im(A⋄,r )∪Im(A⋄,r−1 )∪· · ·∪Im(A⋄,1 ). Then, we can n m write X = A⋄,s (B), for some 1 ≤ s ≤ r, where the matrix B = (bβα ) ∈ C ( s )×( s ) is defined by means of (4.1) or (4.2). If W = F G is a rank factorization of W , we conclude that −1 the matrix X possesses the form X = (Tr(Cs (F GA))) ACs (F G) . According to Lemma 2.1, we get X = F (GAF )−1 G. Finally, using Theorem 3.1 we have X ∈ A{2}\{O}. Corollary 4.1 For A ∈ Crm×n the following statements are valid: (a) A{1, 2} = Im(A⋄,r ) ⊆ Im(A•,r ). (b) A{2}\ (A{1, 2} ∪ {O}) = Im(A⋄,r−1 ) ∪ · · · ∪ Im(A⋄,1 ). Proof. (a) The identity A{1, 2} = Im(A⋄,r ) follows from Theorem 4.1 and the known fact that each X ∈ A{2}\{O} satisfies X ∈ A{1, 2} if and only if rank(X) = rank(A) = r [6]. The rest of the proof follows from the part (a) and Theorem 4.1. (2) The Moore-Penrose inverse, the Drazin inverse and the generalized inverse AT,S are certain elements of images of restricted adjoint mappings. 13 Adjoint Mappings and Inverses of Matrices Corollary 4.2 Consider an arbitrary matrix A ∈ C m×n of rank r. The following statements are valid: (a) A has the Moore-Penrose inverse if and only if Tr(Cr (A∗ A)) 6= 0. In this case ³ ´ −1 A† = A⋄,r (Tr(Cr (A∗ A))) Cr (A∗ ) . (b) A has the weighted Moore-Penrose inverse if and only if Tr(Cr (M AN −1 )) 6= 0, and A†M,N = A⋄,r ³¡ ´ ¢−1 Tr(Cr ((M AN −1 )∗ A)) Cr ((M AN −1 )∗ ) . (c) In the case m = n the group inverse of A exists if and only if Tr(Cr (A)) 6= 0, and ³ ´ −2 A# = A⋄,r (Tr(Cr (A))) Cr (A) . (d) In the case m = n the Drazin inverse of A exists if and only if Tr(Crk (Al+1 )) 6= 0, and AD = A⋄,rk ³¡ ´ ¢−1 Tr(Crk (Al+1 )) Crk (Al ) , where k = ind(A), l ≥ k, rk = rank(Al ) and Al = PAl QAl is a full-rank factorization of the matrix Al . (2) (e) There exists a nonzero generalized inverse AT,S of rank s, 1 ≤ s ≤ r if and only if there exist matrices F ∈ Csn×s and G ∈ Css×m whose columns form bases for T = R(F ) and S = N (G), such that Tr(Cs (U V A)) 6= 0. In this case is ³ ´ (2) −1 AT,S = A⋄,s (Tr(Cs (U V A))) Cs (U V ) . Proof. The statements (a), (b), (c) and (d) follow from the known determinantal representations of the Moore-Penrose inverse, weighted Moore-Penrose inverse, the group inverse and the Drazin inverse from [1], [4], [5] and [11], respectively. The statement (2) (e) follows from the general representation of the generalized inverse AT,S , which is introduced in [12]. 14 P. Stanimirović, S. Bogdanović and M. Ćirić References [1] R.B. Bapat, K.P.S. Bhaskara and K. Manjunatha Prasad, Generalized inverses over integral domains, Linear Algebra Appl. 140 (1990), 181–196. [2] R.B. Bapat, Generalized inverses with proportional minors, Linear Algebra Appl. 211 (1994), 27–35. [3] J. 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