Generalized core inverses of matrices

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1 Generalized core inverses of matrices Sanzhang Xu, Jianlong Chen, Julio Benítez and Dingguo Wang arxiv: v1 [math.ra 13 Sep 217 Abstract: In this paper, we introduce two new generalized inverses of matrices, namely, the i,m -core inverse and the (j,m)-core inverse. The i,m -core inverse of a complex matrix extends the notions of the core inverse defined by Baksalary and Trenkler [1 and the core-ep inverse defined by Manjunatha Prasad and Mohana [13. The (j,m)-core inverse of a complex matrix extends the notions of the core inverse and the DMP-inverse defined by Malik and Thome [12. Moreover, the formulae and properties of these two new concepts are investigated by using matrix decompositions and matrix powers. Key words: i, m -core inverse, (j, m)-core inverse, core inverse, DMP-inverse, core- EP inverse. AMS subject classifications: 15A9, 15A23. 1 Introduction LetC m n denote theset of all m n complex matrices. LetA, R(A) andrk(a) denotethe conjugate transpose, column space, and rank of A C m n, respectively. For A C m n, if X C n m satisfies AXA = A, XAX = X, (AX) = AX and (XA) = XA, then X is called a Moore-Penrose inverse of A. This matrix X is unique and denoted by A. A matrix X C n m is called an outer inverse of A if it satisfies XAX = X; is called a {2,3}-inverse of A if it satisfies XAX = X and (AX) = AX; is called a {1,3}-inverse of A if it satisfies AXA = A and (AX) = AX; is called a {1,2,3}-inverse of A if it satisfies AXA = A, XAX = X and (AX) = AX. The core inverse of a complex matrix was introduced by Baksalary and Trenkler [1. Let A C n n. A matrix X C n n is called a core inverse of A, if it satisfies AX = P A and R(X) R(A), here P A denotes the orthogonal projector onto R(A). If such a matrix exists, then it is unique and denoted by A #. For a square complex matrix A, one has that A is core invertible, A is group invertible, and rk(a) = rk(a 2 ) are three equivalent conditions (see [1). We denote C CM n = {A C n n rk(a) = rk(a 2 )}. LetA C n n. AmatrixX C n n suchthatxa k+1 = A k, XAX = X andax = XA is called the Drazin inverse of A and denoted by A D. The smallest integer k is called the Sanzhang Xu ( xusanzhang5222@126.com): School of Mathematics, Southeast University, Nanjing 2196, China. Jianlong Chen (Corresponding author: jlchen@seu.edu.cn): School of Mathematics, Southeast University, Nanjing 2196, China. Julio Benítez ( jbenitez@mat.upv.es): Universitat Politècnica de València, Instituto de Matemática Multidisciplinar, Valencia, 4622, Spain. Dingguo Wang ( dingguo95@126.com): School of Mathematical Sciences, Qufu Normal University, Qufu, , China. 1

2 Drazin index of A, denoted by ind(a). Any square matrix with finite index is Drazin invertible. If ind(a) 1, then the Drazin inverse of A is called the group inverse and denoted by A #. The DMP-inverse for a complex matrix was introduced by Malik and Thome [12. Let A C n n with ind(a) = k. A matrix X C n n is called a DMP-inverse of A, if it satisfies XAX = X, XA = A D A and A k X = A k A. If such a matrix X exists, then it is unique and denoted by A D,. Malik and Thome gave several characterizations of the DMP-inverse by using the decomposition of Hartwig and Spindelböck [1. The notion of the core-ep inverse for a complex matrix was introduced by Manjunatha Prasad and Mohana [13. A matrix X C n n is a core-ep inverse of A C n n if X is an outer inverse of A satisfying R(X) = R(X ) = R(A k ), where k is the index of A. If such a matrix X exists, then it is unique and denoted by A. In addition, 1 n and n will denote the n 1 column vectors all of whose components are 1 and, respectively. m n (abbr. ) denotes the zero matrix of size m n. If S is a subspace of C n, then P S stands for the orthogonal projector onto the subspace S. A matrix A C n n is called an EP matrix if R(A) = R(A ), A is called Hermitian if A = A and A is unitary if AA = I n, where I n denote the identity matrix of size n. Let N denotes the set of positive integers. 2 Preliminaries A related decomposition of the matrix decomposition of Hartwig and Spindelböck [1 was given in [2, Theorem 2.1 by Benítez, in [3 it can be found a simpler proof of this decomposition. Let us start this section with the concept of principal angles. Definition 2.1. [16 Let S 1 and S 2 be two nontrivial subspaces of C n. We define the principal angles θ 1,...,θ r [,π/2 between S 1 and S 2 by cosθ i = σ i (P S1 P S2 ), for i = 1,...,r, where r = min{dims 1,dimS 2 }. The real numbers σ i (P S1 P S2 ) are the singular values of P S1 P S2. The following theorem can be found in [2, Theorem 2.1. Theorem 2.2. Let A C n n, r = rk(a), and let θ 1,...,θ p be the principal angles between R(A) and R(A ) belonging to,π/2[. Denote by x and y the multiplicities of the angles and π/2 as a canonical angle between R(A) and R(A ), respectively. There exists a unitary matrix U C n n such that where M C r r is nonsingular, A = U [ MC MS U, (2.1) C = diag( y,cosθ 1,...,cosθ p,1 x ), [ diag(1y,sinθ S = 1,...,sinθ p ) p+y,n (r+p+y) x,p+y x,n (r+p+y), 2

3 and r = y + p + x. Furthermore, x and y + n r are the multiplicities of the singular values 1 and in P R(A) P R(A ), respectively. In this decomposition, one has C 2 +SS = I r. Recall that A always exists. We have that A # exists if and only if C is nonsingular in view of [2, Theorem 3.7. The following equalities hold A = U [ CM 1 S M 1 [ U, A # C = U 1 M 1 C 1 M 1 C 1 S U. By [3, Theorem 2, we have that A D = U [ (MC) D [(MC) D 2 MS U. (2.2) We also have [ AA Ir = U A # = A # AA = U U, (2.3) [ C 1 M 1 U. (2.4) Lemma 2.3. [17, Theorem 3.1 Let A C n n. Then A is core invertible if and only if there exists X C n n such that (AX) = AX, XA 2 = A and AX 2 = X. In this situation, we have A # = X. Lemma 2.4. Let A C n n. If there exists X C n n such that AX k+1 = X k and XA k+1 = A k for some k N, then for m N we have (1) A k = X m A k+m ; (2) X k = A m X k+m ; (3) A k X k = A k+m X k+m ; (4) X k A k = X k+m A k+m ; (5) A k = A m X m A k ; (6) X k = X m A m X k. Proof. (1). For m = 1, it is clear by the hypotheses. If the formula is true for m N, then X m+1 A k+m+1 = XX m A k+m A = XA k A = XA k+1 = A k. (3). It is easy to check that A k X k = A k+1 X k+1 by AX k+1 = X k. It is not difficult to check the equality A k X k = A k+m X k+m by induction. (5). From (1) we have A k = X k A 2k. Thus by AX k+1 = X k, we have A k = X k A 2k = AX k+1 A 2k = AX k XA 2k = A(AX k+1 )XA 2k = A 2 X k+2 A 2k = A 2 X 2 X k A 2k = = A m X m X k A 2k = A m X m A k. The proofs of (2), (4) and (6) are similar to the proofs of (1), (3) and (5), respectively. 3

4 Lemma 2.5. Let A C n n. If there exists X C n n such that AX k+1 = X k and XA k+1 = A k for some k N, then A D = X k+1 A k. Proof. Since A is Drazin invertible by XA k+1 = A k, we will check that A D = X k+1 A k. Have in mind, AX k+1 = X k and XA k+1 = A k, thus A(X k+1 A k ) = X k A k = X k (XA k+1 ) = X k+1 A k A. (2.5) That is, X k+1 A k and A are commute. Then by (1) and (4) in Lemma 2.4, we have that (X k+1 A k )A(X k+1 A k ) =X k+1 A k+1 X k+1 A k = X k A k (X k+1 A k ) =X k X k+1 A k A k = X k+1 X k A 2k = X k+1 A k. (2.6) From (1) in Lemma 2.4, we have that (X k+1 A k )A k+1 = X(X k A 2k )A = XA k+1 = A k. (2.7) Thus we have A D = X k+1 A k by the definition of the Drazin inverse and in view of (2.5), (2.6) and (2.7). Remark 2.6. From the proofs of Lemma 2.4 and Lemma 2.5, it is obvious that Lemma 2.4 and Lemma 2.5 are valid for rings. Moreover, we can get that for an element a R, a is Drazin invertible if and only if there exist x R and k N such that ax k+1 = x k and xa k+1 = a k, where R is a ring. The following lemma is similar to [12, Theorem 2.5. Lemma 2.7. Let A C n n be the form (2.1). Then A D, = U Proof. By (2.2) and (2.3), we can get A D = U [ Ir U [ (MC) D U. (2.8) [ (MC) D [(MC) D 2 MS U, respectively. Thus by the definition of DMP-inverse we have A D, = A D AA = U [ (MC) D [(MC) D 2 MS [ Ir U and AA = [ U (MC) = U D U. Lemma 2.8. [15, Corollary 3.3 Let A C n n be a matrix of index k. Then AA = A k (A k ). 4

5 3 i, m -core inverse Let us start this section by introducing the definition of the i, m -core inverse. Definition 3.1. Let A C n n and m,i N. A matrix X C n n is called an i,m -core inverse of A, if it satisfies X = A D AX and A m X = A i (A i ). (3.1) It will be proved that if X exists, then it is unique and denoted by A i,m. Theorem 3.2. Let A C n n. If exists X C n n such that (3.1) holds, then X is unique. Proof. Assume that X satisfies the system in (3.1), that is X = A D AX and A m X = A i (A i ). Thus X = A D AX = (A D ) m A m X = (A D ) m A i (A i ). Therefore, X is unique by the uniqueness of A D and A i (A i ). Theorem 3.3. The system in (3.1) is consistent if and only if i ind(a). In this case, the solution of (3.1) is X = (A D ) m A i (A i ). Proof. Assume that i ind(a). Let X = (A D ) m A i (A i ). We have A D AX = A D A(A D ) m A i (A i ) = (A D ) m A D AA i (A i ) = (A D ) m A i (A i ) = X; A m X = A m (A D ) m A i (A i ) = A D AA i (A i ) = A i (A i ). Thus, the system in (3.1) is consistent and the solution of (3.1) is X = (A D ) m A i (A i ). Ifthesystemin(3.1) isconsistent, thenexistsx suchthat X = A D AX anda m X = A i (A i ). Then X = A D AX = (A D ) m A m X = (A D ) m A i (A i ) and A i (A i ) = A m X = A m (A D ) m A i (A i ) = AA D A i (A i ). Hence A i = A i (A i ) A i = AA D A i (A i ) A i = AA D A i, that is i ind(a). Example 3.4. We will give [ an example that shows if i < ind(a), then the system in (3.1) 1 is not consistent. Let A =. It is easy to get ind(a) = 2 and A D =. Let i = 1 and suppose that X is the solution of system in (3.1), then X = A D AX =, which gives AA = A m X =, thus A = AA A =, this is a contradiction. Remark 3.5. If i ind(a), then A i,m+1 = AD A i,m. Remark 3.6. The i,m -core inverse is a generalization of the core inverse and the core- EP inverse. More precisely, we have the following statements: (1) If m = i = ind(a) = 1, then the 1,1 -core inverse coincides with the core inverse; (2) If m = 1 and i = ind(a), then the i,1 -core inverse coincides with the core-ep inverse. 5

6 For the convenience of the readers, in the following, we give some notes of (1) and (2) in Remark 3.6. (1). If m = i = ind(a) = 1, then A is group invertible and A D = A # and (3.1) is equivalent to X = A # AX and AX = AA. Thus X = A # AX = A # AA, (AX) = (AA ) = AA = AX, AX 2 = AA # AA A # AA = AA # AA AA # A = AA # A = X and XA 2 = A # AA A 2 = A # A 2 = A. Hence, 1,1 -core inverse coincides with the core inverse by Lemma 2.3. Note that if A is group invertible, then we have that X is the core inverse of A if and only if X = A # AX and AX = AA. (2). If m = 1 and i = ind(a), then by Theorem 3.3, A i,1 exists and A i,1 = AD A i (A i ). Let us denote X = A i,1 = AD A i (A i ). Observe that AX = A i (A i ) is Hermitian. Now, XAX = A D A i (A i ) A i (A i ) = A D A i (A i ) = X, that is X is an outer inverse of A. From A i = A D A i+1 = A D A i (A i ) A i A = XA i+1 we get R(A i ) R(X). Also, AX 2 = (AX)X = A i (A i ) A D A i (A i ) = A i (A i ) A i A D (A i ) = A D A i (A i ) = X, which implies X = (AX) X R(X ), therefore, R(X) R(X ). Finally, X = [A D A i (A i ) = A i (A i ) (A D ) implies R(X ) R(A i ). Hence R(X) = R(X ) = R(A i ). Therefore, the i,1 -core inverse coincides with the core-ep inverse by the definition of the core-ep inverse. From the above statement, we have the following theorem. Theorem 3.7. Let A C n n with i = ind(a). Then X is the core-ep inverse of A if and only if X = A D AX and AX = A i (A i ). Corollary 3.8. Let A C n n with 1 = ind(a). Then X is the core inverse of A if and only if X = A # AX and AX = AA. For any A C n n, either A l = for some l N, or A l for all positive integers. Moreover, if ind(a) = k, then G k B k is nonsingular (see [4, 6, 8), where A = B 1 G 1 is a full rank factorization of A and G l B l = B l+1 G l+1 is a full rank factorization of G l B l, l = 1,...,k 1. When A k, then it can be written as A k = k k B l G k+1 l. (3.2) We have the following results, (see [4, Theorem 4 or [7, Theorem 7.8.2): { k, when Gk B k is nonsingualr; ind(a) = k +1, when G k B k =. and k k A D B l (G k B k ) k 1 G k+1 l, when G k B k is nonsingualr; =, when G k B k =. (3.3) In the sequel, we always assume that A k. It is well-known that if A = EF is a full rank factorization of A, where r = rk(a), E C n r and F C r n, then (see [7, Theorem 1.3.2) A = F (FF ) 1 (E E) 1 E. (3.4) 6

7 Remark 3.9. The notations and results in above paragraph will be used many times in the sequel. We will investigate the i,m -core inverse of a matrix A C n n by using Remark 3.9. Theorem 3.1. Let A C n n with ind(a) = k. If i k, then A i,m = A k,m. Proof. Since ind(a) = k, we have R(A k ) = R(A i ) for any i k, and therefore, A k (A k ) = A i (A i ). Now, the conclusion follows from Theorem 3.3. Remark The proof of Theorem 3.1 also can be proved as follows. Since the proof in this remark will be used several times in the sequel, we write this proof here. Proof. If A is nilpotent, then A D =, hence by Theorem 3.3, one has A i,m = A k,m =. Therefore, we can assume that A k. By equality (3.2), we have k k A k = B l G k+1 l. (3.5) where A = B 1 G 1 is a full rank factorization of A and G l B l = B l+1 G l+1 is a full rank factorization of G l B l, l = 1,...,k 1. Let M = k B l, N = k G k+1 l and L = G k B k. Now, we will show that In fact, A i = A i = k k B l (G k B k ) i k G k+1 l = ML i k N. i i B l G k+1 l = B 1 B i G i G 1 = B 1 B i 1 (B i G i )G i 1 G 1 =B 1 B i 1 (G i 1 B i 1 )G i 1 G 1 = B 1 B i 2 (G i 2 B i 2 ) 2 G i 2 G 1 = =B 1 B k (G k B k ) i k G k G 1 = ML i k N. (3.6) If we let M 1 = ML i k, then A i = ML i k N = M 1 N is a full rank factorization of A i (see [8, p.183). Thus (A i ) = N (NN ) 1 (M 1 M 1) 1 M 1. (3.7) Note that NM = k k G k+1 l B l = L k. By Theorem 3.3, (3.3) and (3.7) we have A i,1 =AD A i (A i ) = ML k 1 NML i k N(A i ) =ML k 1 NML i k NN (NN ) 1 (M 1 M 1) 1 M 1 =ML i k 1 NN (NN ) 1 (M 1M 1 ) 1 M 1 =ML i k 1 (M 1M 1 ) 1 M 1 =ML i k 1 [(L i k ) M ML i k 1 (L i k ) M =ML i k 1 L k i (M M) 1 [(L i k ) 1 (L i k ) M =ML 1 (M M) 1 M. (3.8) 7

8 The last expression does not depend on i, then A i,1 = A k,1. Thus, by Remark 3.5, we have A i,m = AD A i,m 1 = AD (A D A i,m 2 ) = (AD ) 2 A i,m 2 = = (AD ) m 1 A i,1 = (A D ) m 1 A k,1 = A k,m. Remark By Theorem 3.1, it is enough to investigate the i = ind(a) = k case, when we discuss the i,m -core inverse of a matrix A C n n. That is, the Theorem 3.1 is a key theorem. Theorem Let A C n n with ind(a) = k and k,m N. If A = B 1 G 1 is a full rank factorization of A and G l B l = B l+1 G l+1 is a full rank factorization of G l B l, l = 1,...,k 1, then A k,m = ML m M, where M = k B l, N = k G k+1 l and L = G k B k. Proof. By the proof of Remark 3.11, we have A k,1 = ML 1 (M M) 1 M and NM = L k. Now, we will prove (A D ) s A k,1 = ML s 1 (M M) 1 M for any s N. By (3.3) we have A D = k B l (G k B k ) k 1 k G k+1 l = ML k 1 N. When s = 1, we have A D A k,1 =ML k 1 NML 1 (M M) 1 M = ML k 1 (NM)L 1 (M M) 1 M =ML k 1 L k L 1 (M M) 1 M = ML 2 (M M) 1 M. Assume that (A D ) s 1 A k,1 = ML s (M M) 1 M. Then (A D ) s A k,1 =AD (A D ) s 1 A k,1 = AD ML s (M M) 1 M Thus by Remark 3.5, we have =ML k 1 NML s (M M) 1 M = ML k 1 L k L s (M M) 1 M =ML s 1 (M M) 1 M. A k,m = (AD ) m 1 A k,1 = ML m (M M) 1 M = ML m M. In the following theorem, we will give a canonical form for the k,m -core inverse of a matrix A C n n by using the matrix decomposition in Theorem 2.2. We will also use the following simple fact: Let X C n m and b C n. If y C m satisfies X Xy = X b, then XX b = Xy. Theorem Let A C n n have the form (2.1) with ind(a) = k and m N. Then A k,m = U [ (MC) k 1,m Proof. Let r be the rank of A. By Theorem 3.3 we have A k,m = (AD ) m A k (A k ). U. (3.9) 8

9 Since A has the form given in Theorem 2.2 we have [ A k (MC) k (MC) = U k 1 MS Let b C n be arbitrary and let us decompose b = U [ b1 b 2 U. (3.1), where b 1 C r. Let x C n satisfy (A k ) A k x = (A k ) b [this x always exists because [ the normal equations always x1 have a solution. We can decompose x by writing x = U, where x 1 C r. Let us denote N = (MC) k 1 M. Using (3.1), [ [ CN NC NS U S N Therefore, [ x1 x 2 = U x 2 [ CN S N [ b1 CN N(Cx 1 +Sx 2 ) = CN b 1 and S N N(Cx 1 +Sx 2 ) = S N b 1. Premultiplying the first equality by C and the second equality by S and after, adding them, we get N N(Cx 1 +Sx 2 ) = N b 1, and hence, N(Cx 1 +Sx 2 ) = NN b 1. Now, [ [ A k (A k ) b = A k NC NS x1 x = U x 2 [ [ NCx1 +NSx = U 2 NN = U [ b 1 NN = U U b. Since b is arbitrary, A k (A k ) = U [ NN U. Now we will prove NN = (MC) k 1 [(MC) k 1. Recall that we have N = (MC) k 1 M, and so, R(N) R((MC) k 1 ). Since M is nonsingular, rk(n) = rk((mc) k 1 ), and therefore, R(N) = R((MC) k 1 ). Since NN and (MC) k 1 [(MC) k 1 are the orthogonal projectors onto R(N) andr((mc) k 1 ), respectively, weget NN = (MC) k 1 [(MC) k 1. By (2.2) we have A D = U [ (MC) D [(MC) D 2 MS b 2. U. (3.11) Thus, we have (A D ) m = U [ [(MC) D m [(MC) D m+1 MS U. Since ind(a) = k, we have A D A k+1 = A k. By using the above representations of A D and A k given in (3.1) and (3.11), respectively, [ (MC) D [(MC) D 2 [ MS (MC) k+1 (MC) k [ MS (MC) k (MC) = k 1 MS. 9

10 Therefore, (MC) D (MC) k M[C S = (MC) k 1 M[C S. (3.12) [ C Have in mind that we have C 2 + SS = I r. Thus, postmultiplying (3.12) by gives us (MC) D (MC) k M = (MC) k 1 M and from the nonsningularity of M we obtain (MC) D (MC) k = (MC) k 1, and so, ind(mc) k 1. Therefore we have A k,m =(AD ) m A k (A k ) [ [(MC) =U D m [(MC) D m+1 [ MS (MC) k 1 ((MC) k 1 ) [ [(MC) =U D m (MC) k 1 ((MC) k 1 ) U [ (MC) =U k 1,m U. U S Remark If we use the decomposition of Hartwig and Spindelböck [ in [1, Corollary (ΣK) 6, then an expression of the k,m -core inverse of A is A k,m = U k 1,m U, which is similar to the expression of A k,m in Theorem Since the proof of this result can be proved as the proof of Theorem 3.14, we omit this proof. Let A C n n with ind(a) = k. The Jordan Canonical form of A is P 1 AP = J, where P C n n is nonsingular and J C n n is a block diagonal matrix composed of Jordan blocks. In the following theorem, we will compute the k, m -core inverse by using the Jordan Canonical form of A. Theorem Let A C n n with ind(a) = k, then A k,m = P 1D m P [ 1, where A = D P P N 1 with D C r r is nonsingular, N is nilpotent and P = [P 1 P 2 with P 1 C n r. Proof. The Jordan Canonical form of A is P 1 AP = J, where P C n n is nonsingular and [ J C n n is a block diagonal matrix. Rearrange the elements of J such that A = D P P N 1, where D is nonsingular and N is nilpotent. It is well-known that A D = [ [ [ D 1 D P P 1 anda k k = P P 1. IfweletP = [P 1 P 2 andp 1 Q1 =, Q [ 2 (D then (A D ) m A k = [P 1 P 2 1 ) m [ [ D k Q1 = P Q 1 D k m Q 1. Observe that 2 A k = (P 1 D k )Q 1 is a full rank factorization of A k. Hence by (3.4) we have (A k ) =(P 1 D k Q 1 ) = Q 1(Q 1 Q 1) 1 [(P 1 D k ) P 1 D k 1 (P 1 D k ) =Q 1 (Q 1Q 1 ) 1 D k (P 1 P 1) 1 [(D k ) 1 (D k ) P 1 =Q 1 (Q 1Q 1 ) 1 D k (P 1 P 1) 1 P 1 =Q 1 D k P 1. 1

11 By Theorem 3.3, we have A k,m = (AD ) m A k (A k ). Thus we have A k,m =(AD ) m A k (A k ) = P 1 D k m Q 1 Q 1 D k P 1 = P 1D k m Q 1 Q 1(Q 1 Q 1) 1 D k P 1 =P 1 D m D k D k P 1 = P 1D m P 1. Proposition Let A C n n. If i ind(a), then A m A i,m R(A i ) along R(A i ). is the projector onto Proof. It is trivial. In the following proposition, we will investigate some properties of the i,m -core inverse. Proposition Let A C n n, m,i N. If i ind(a), then (1) A i,m is a {2,3}-inverse of Am ; (2) A i,m = (AD ) m P A i; (3) (A i,m )n = (A D ) m(n 1) P A i; (4) A i A i,m = A i,m Ai if and only if R(A i ) N((A D ) m ); (5) A i,m = A implies that A is EP. Proof. (1). By Theorem 3.3 we have A i,m = (AD ) m A i (A i ), thus A i,m Am A i,m =(AD ) m A i (A i ) A m (A D ) m A i (A i ) = (A D ) m A i (A i ) A i A m (A D ) m (A i ) =(A D ) m A i A m (A D ) m (A i ) = (A D ) m A m (A D ) m A i (A i ) =A D A(A D ) m A i (A i ) = (A D ) m A i (A i ) = A i,m. Thus A i,m is a {2,3}-inverse of Am in view of A m A i,m = Ai (A i ). (2) is trivial. (3). By (A i,m )2 = (A D ) m A i (A i ) (A D ) m A i (A i ) = (A D ) m (A D ) m A i (A i ) = (A D ) m A i,m and induction it is easy to check (3). (4). By R(I n A i (A i ) ) = N((A i ) ), we have A i A i,m = A i,m Ai A i (A D ) m A i (A i ) = (A D ) m A i (A i ) A i A i (A D ) m A i (A i ) = (A D ) m A i A i (A D ) m (I n A i (A i ) ) = R(I n A i (A i ) ) N(A i (A D ) m ) N((A i ) ) N((A D ) m ) N((A i ) ) N((A D ) m ) R(A i ) N((A D ) m ). 11

12 [ (MC) (5). Let A be written in the form (2.1). We have A i,m = U i 1,m U by Theorem Thus, A i,m = A implies MS =. From the nonsingularity of M, we have S =, which is equivalent to say that A is EP in view of [2, Theorem (j, m)-core inverse Let us start this section by introducing the definition of the (j, m)-core inverse. Definition 4.1. Let A C n n and m,j N. A matrix X C n n is called a (j,m)-core inverse of A, if it satisfies X = A D AX and A m X = A m (A j ). (4.1) Theorem 4.2. Let A C n n. If the system in (4.1) is consistent, then the solution is unique. Proof. Assume that X satisfies that (4.1), that is X = A D AX and A m X = A m (A j ). Then X = A D AX = (A D ) m A m X = (A D ) m A m (A j ) = A D A(A j ). Thus X is unique. By Theorem 4.2 if X exists, then it is unique and denoted by A j,m. Theorem 4.3. Let A C n n and m,j N. Then (1) If m ind(a), then the system in (4.1) is consistent and the solution is X = A D A(A j ) ; (2) If the system in (4.1) is consistent, then ind(a) max{j,m}. Proof. (1). Let X = A D A(A j ). We have A D AX = A D AA D A(A j ) = A D A(A j ) = X and A m X = A m A D A(A j ) = A D AA m (A j ) = A m (A j ). (2). If the system in (4.1) is consistent, then exits X C n n such that X = A D AX = (A D ) m A m X = (A D ) m A m (A j ) = A D A(A j ) and A m (A j ) = A m X = A m A D A(A j ) = A m (A D ) j A j (A j ). Thus A m (A j ) A j = A m (A D ) j A j (A j ) A j = A m (A D ) j A j = A m A D A. If m j, then A m A D A = A m (A j ) A j = A m j A j (A j ) A j = A m j A j = A m. That is ind(a) m. If j > m, then A j = A j (A j ) A j = A j m A m (A j ) A j = A j m A m A D A = A j A D A. That is ind(a) j. Therefore, ind(a) max{j,m}. Example 4.4. We will give an example that shows if m < ind(a), then the system in (4.1) is not consistent. Let A be the same matrix in Example 3.4. It is easy to get ind(a) = 2 and A D =. Let m = j = 1 and suppose that X is the solution of system in 4.1, then X = A D AX =, which gives AA = AX =, thus A = AA A =, this is a contradiction. 12

13 Example 4.5. The converse of Theorem 4.3 (1) is not true. Let m = 1 and j = 3. If we 1 let A = 1, then ind(a) = 3 and A 3 =. Hence X = is a solution of (4.1), but m < ind(a). Example 4.6. If ind(a) max{j, m}, then the system in (4.1) may be not consistent. If we let A = 1 1, then A 3 = A 2 = 1 1 1, A D = A and ind(a) = 2. Let m = 1 and j = 2, then ind(a) max{j,m}. It is easy to 2 1 check that (A 2 ) = If the system in (4.1) has a solution X, then 2 1 X = A D AX = A D A(A 2 ) and A(A 2 ) = AX = AA D A(A 2 ) = A 4 (A 2 ) = A 2 (A 2 ) would hold. But A(A 2 ) = 1 15 the system in (4.1) is not consistent = A 2 (A 2 ). Thus, Remark 4.7. If m ind(a) = k, it is not difficult to see that A j,m = A j,m+1. That is to say, the (j,m)-core inverse of A coincides with the (j,m+1)-core inverse of A. Thus, in the sequel, we only discuss the m = ind(a) case. Theorem 4.8. Let A,X C n n, k,j N. If ind(a) = k and X is the (j,k)-core inverse of A, then we have X j A j X j = (A D ) j(j 1) X j and XA j = A D A. Proof. By the definition of the (j,k)-core inverse, we have X = A D AX and A k X = A k (A j ). By X = A D A(A j ), it is easy to check that X n+1 = (A D ) j X n for arbitrary n N, which gives that X j = (A D ) j(j 1) X. XA j =A D A(A j ) A j = (A D ) j A j (A j ) A j = (A D ) j A j = A D A; X j A j X j =(A D ) j(j 1) XA j X j = (A D ) j(j 1) A D A(A j ) A j X j =(A D ) j(j 1) (A D ) j A j (A j ) A j X j = (A D ) j(j 1) (A D ) j A j X j =(A D ) j(j 1) A D AXX j 1 = (A D ) j(j 1) A D AA D A(A j ) X j 1 =(A D ) j(j 1) A D A(A j ) X j 1 = (A D ) j(j 1) XX j 1 =(A D ) j(j 1) X j. Corollary 4.9. Let A,X C n n and ind(a) = k. If X is the (1,k)-core inverse of A, then we have XAX = X and XA = A D A. The (j, m)-core inverse is a generalization of the core inverse and the DMP-inverse in view of Theorem

14 Remark 4.1. When j = m = 1 = ind(a), the equations in (4.1) are equivalent to XAX = X, XA = A # A and AX = AA. Thus AX = AA implies that (AX) = AX; XA = A # A gives that XA 2 = A and AXA = A; and X = XAX = A # AX = AA #, which means that R(X) R(A), then X = AY for some Y C n n, thus X = AY = AXAY = AX 2. Therefore, we have A # = X by Lemma 2.3. In a word, the (1,1)-core inverse coincides with the usual core inverse. Remark If we let j = 1 and m = ind(a), then the equations in (4.1) are equivalent to XAX = X, XA = A D A and A k X = A k A by Theorem 4.8. Thus (1,k)-core inverse coincides with the DMP-inverse. From Remark 4.11, Theorem 4.8 and the definition of the (j,k)-core inverse, we have the following theorem, which says that the conditions XAX = X and XA = A D A in the definition of the DMP-inverse can be replaced by X = A D AX. Theorem Let A C n n with k = ind(a). Then X C n n is the DMP-inverse of A if and only if X = A D AX and A k X = A k A. In the following theorem, we will give a canonical form for the (j,k)-core inverse of a matrix A C n n by using the matrix decomposition in Theorem 2.2. Theorem Let A C n n have the form (2.1) with ind(a) = k and j N. Then A j,k = U [ (MC) D (MC) j 1,k U. (4.2) Proof. By Theorem 4.3 and the idempotency of A D A we have A j,k = AD A(A j ) = (A D ) j A j (A j ). (4.3) From the proof of Theorem 3.14, we have A j (A j ) = U [ (MC) j 1 ((MC) j 1 ) U. (4.4) By (2.2) we have A D = U [ (MC) D [(MC) D 2 MS U, thus we have (A D ) j = U [ [(MC) D j [(MC) D j+1 MS U. (4.5) By the proof of Theorem 3.14, we have ind(mc) k 1 < k. From (4.3), (4.4) and (4.5), 14

15 we have A j,k =(AD ) j A j (A j ) [ [(MC) =U D j [(MC) D j+1 [ MS (MC) j 1 ((MC) j 1 ) [ [(MC) =U D j (MC) j 1 ((MC) j 1 ) U [ (MC) =U D [(MC) D j 1 (MC) j 1 ((MC) j 1 ) U [ (MC) =U D (MC) D MC((MC) j 1 ) U [ (MC) D (MC) =U j 1,k U. U Remark If we use the decomposition of Hartwig and Spindelböck [ in [1, Corollary (ΣK) 6, then an expression of the (j,k)-core inverse of A is A D j,k = U (ΣK) j 1,k U, which is similar to the expression of A j,k in Theorem Since the proof of this result can be proved like the proof of Theorem 4.13, we omit this proof. Theorem Let A C n n and ind(a) = k. If (A k X k ) = A k X k, AX k+1 = X k and XA k+1 = A k, then A is (k,k)-core invertible and A k,k = Xk. Proof. By Lemma 2.4 and Lemma 2.5, we have A k X k A k = A k, X k A k X k = X k, A k = X k A 2k and A D = X k+1 A k. Equalities (A k X k ) = A k X k and A k X k A k = A k imply that X k is a {1,3}-inverse of A k. From A D = X k+1 A k, we can obtain (A D ) k = X k 1 A D by induction. Thus A k,k =AD A(A k ) = (A D ) k A k (A k ) = (A D ) k A k (A k ) (1,3) =(A D ) k A k X k = (X k+1 A k ) k A k X k = X k 1 X k+1 A k A k X k =X 2k A 2k X k = X k (X k A 2k )X k = X k A k X k = X k. Proposition Let A C n n be a matrix with j ind(a) = k. If A is (j,k)-core invertible, then A j A j,k is the projector onto R(Aj ) along R(A j ). Proof. It is trivial. In the following proposition, we will investigate some properties of the (j, k)-core inverse. Proposition Let A C n n with j ind(a) = k. If A is (j,k)-core invertible, then (1) A j,k is a {1,2,3}-inverse of Aj ; 15

16 (2) A j,k = (AD ) j P A j; { (3) (A ((A D ) j (A j ) ) n 2 n is even j,k )n = A j ((A D ) j (A j ) ) n+1 2 n is odd ; (4) A j,k AD = (A D ) j+1 ; (5) A j A j,k = A j,k Aj if and only if R(A j ) N(A D ); (6) A j,k = A implies that A is EP. Proof. (1). By Theorem 4.3 we have A j,k = AD A(A j ) = (A D ) j A j (A j ), thus A j A j,k Aj =A j (A D ) j A j (A j ) A j = A j (A D ) j A j = A j A D A = A j ; A j,k Aj A j,k =(AD ) j A j (A j ) A j A j,k = AD AA j,k =A D AA D A(A j ) = A D A(A j ) = A j,k ; A j A j,k =Aj (A D ) j A j (A j ) = A j (A j ). (2) is trivial. (3). By (A j,k )2 = (A D ) j A j (A j ) (A D ) j A j (A j ) = (A D ) j (A j ) and induction it is easy to check (3). (4). A j,k AD = (A D ) j A j (A j ) A D = (A D ) j A j (A j ) (A D ) j A j A D = (A D ) j+1. (5). By R(I n A j (A j ) ) = N((A j ) ) and N(A D A) = N(A D ), we have A j A j,k = A j,k Aj A j (A D ) j A j (A j ) = (A D ) j A j (A j ) A j A j (A D ) j A j (A j ) = (A D ) j A j A j (A D ) j (I n A j (A j ) ) = R(I n A j (A j ) ) N(A D A) N((A j ) ) N(A D A) N((A j ) ) N(A D ) R(A j ) N(A D ). [ (MC) (6). LetAbewrittenintheform(2.1). WehaveA D j,k = U (MC) j 1,k U by Theorem Thus, A j,k = A implies MS =. From the nonsingularity of M, we have S =, which is equivalent to say that A is EP in view of [2, Theorem 3.7. In the following proposition, we shall give the the relationship between the (j, k)-core inverse and DMP-inverse and core-ep inverse. Proposition Let A C n n with ind(a) = k. Then A k,k = AD, (A D ) k 1 AA. 16

17 Proof. We have that A k (A k ) = AA by Lemma 2.8 and A D, = A D AA. Thus A k,k =AD A(A k ) = (A D ) k A k (A k ) = A D A k (A D ) k 1 (A k ) =A D AA A k (A D ) k 1 (A k ) = A D, (A D ) k 1 A k (A k ) =A D, (A D ) k 1 AA. In the following theorem, we will give a relationship between the i, m -core inverse and (j, m)-core inverse. Theorem Let A C n n with ind(a) = k. Then A k,m = A m,k for any m k. Proof. By Theorem 4.3, we have A m,k = AD A(A m ) = (A D ) k A k (A m ). By the proof of Remark 3.11, we have A k = MN and NM = L k, where M = k B l, N = k G k+1 l and L = G k B k. It is easy to see that (A D ) s = ML k s N for any s N by NM = L k. Thus (A D ) k = ML 2k N and (A D ) k A k = ML 2k NMN = ML 2k L k N = ML k N. By the proof of Remark 3.11, we have A m = ML m k N = M 1 N is a full rank factorization of A m, where M 1 = ML m k and (A m ) = N (NN ) 1 (M 1 M 1) 1 (M 1 ). By Theorem 3.13, we have A k,m = ML m M. In the following steps, we will show that A m,k = ML m M. From A m,k = (AD ) k A k (A m ), we have A k,m =(AD ) k A k (A m ) = ML k NN (NN ) 1 (M 1 M 1) 1 (M 1 ) =ML k (M 1 M 1) 1 (M 1 ) = ML k [(L m k ) M ML m k 1 (L m k ) M =ML k L k m (M M) 1 [(L m k ) 1 (L m k ) M =ML m (M M) 1 M = ML m M. Theorem 4.2. Let A C n n with i ind(a) = k, then A i,k = P 1D i P [ 1, where D A = P P N 1 with D C r r is nonsingular, N is nilpotent and P = [P 1 P 2 with P 1 C n r. Proof. It is easy to see that by Theorem 3.16 and Theorem ACKNOWLEDGMENTS This research is supported by the National Natural Science Foundation of China (No and No ), the Natural Science Foundation of Jiangsu Province (No. BK ). The first author is grateful to China Scholarship Council for giving him a purse for his further study in Universitat Politècnica de València, Spain. 17

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