The DMP Inverse for Rectangular Matrices

Transcription

1 Filomat 31:19 (2017, Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: The DMP Inverse for Rectangular Matrices Lingsheng Meng a a College of Mathematics Statistics, Northest Normal University, Lanzhou , PR China Abstract. The definition of the DMP inverse of a square matrix ith complex elements is extended to rectangular matrices by shoing that for any A W, m by n n by m, respectively, there exists a unique matrix X, such that XAX = X, XA = WA d, WA (WA k+1 X = (WA k+1 A, here A d, denotes the W-eighted Drazin inverse of A k = Ind(AW, the index of AW. 1. Introduction Let C m n denotes the set of complex m n matrices C m n r, the subset of all rank r matrices in C m n. The symbols A, R(A N(A respectively st for the conjugate transpose, the column space null space of a matrix A C m n. As usual, I m denotes the identity matrix of order m. Moreover, P L,M denotes the projector onto L along M, here L M are to complementary subspaces of C n. For a given matrix A C n n, this notation ill be reduced to P A hen L = R(A M is the subspace orthogonal to L. The Moore-Penrose inverse of a matrix A C m n is the unique matrix A satisfying the four equations: AA A = A, A AA = A, (AA = AA, (A A = A A, as described by Penrose [9]. A matrix X that satisfies the equality AXA = A is called a g-inverse of A if X satisfies XAX = X, it is called an outer inverse of A. For each square matrix A C n n, the index of A, ritten as Ind(A, is the smallest non-negative integer k for hich rank(a k = rank(a k+1. The Drazin inverse of a square matrix A, denoted by A D, is the unique matrix satisfying the folloing equations: A k XA = A, XAX = X AX = XA, here k = Ind(A. In particular, if the Ind(A 1, the Drazin inverse is called the group inverse A #. In [5], Cline Greville extended the Drazin inverse of square matrix to rectangular matrix. If A C m n W C n m, then X = ( (AW D 2 A C m n is the unique solution to the equations: (AW k+1 XW = (AW k, XWAWX = X, AWX = XWA, k = Ind(AW. ( Mathematics Subject Classification. 15A09 Keyords. DMP inverse, W-eighted Drazin inverse, Moore-Penrose inverse Received: 22 March 2017; Accepted: 27 April 2017 Communicated by Dijana Mosić Research supported by the National Natural Science Foundation of China ( No the Science Technology Project of Northest Normal University (No. NWNU-LKQN address: menglsh@nnu.edu.cn (Lingsheng Meng

2 L. Meng / Filomat 31:19 (2017, The matrix X is called the W-eighted Drazin inverse of A is ritten as A d,. Recently, Malik Thome [6] defined a ne generalized inverse, namely the DMP inverse, of a square matrix A C n n of an arbitrary index. The DMP inverse of A C n n, denoted by A D, is defined to be the matrix A D, = A D AA, hich is the unique solution of the folloing equations: XAX = X, XA = A D A, A k X = A k A, k = Ind(A. Specially, if Ind(A = 1, the DMP inverse is reduced to the core inverse A Θ (see [1], hich is the unique matrix satisfying AX = P A R(X R(A. The Moore-Penrose inverse, the Drazin inverse the DMP inverse of a matrix alays exist, hile the group inverse as ell as the core inverse of a square matrix A exist if only if Ind(A = 1. We refer the readers to [1, 2, 11] for basic results on these generalized inverses. All of these generalized inverses are knon to be used in important applications. For example, the Moore-Penrose inverse is used to solve the least-squares problems, the group inverse has applications in Markov chain theory, the Drazin inverse has applications in singular differential equations iterative methods, the core inverse has applications in partial order theory (see for example [2 4, 7, 8, 10, 11]. Motivated by the extension of Drazin inverse to W-eighted Drazin inverse, e extend the DMP inverse of square matrix to rectangular matrix in this paper. First, a canonical form for the ne generalized inverse is established. Second, the equivalence of the algebraic definition (Definition 2.1 the geometrical approach (Theorem 3.2 has been stated. We finally give some of the properties that this ne generalized inverse possesses. 2. The W-eighted DMP Inverse of A In this section, e give the definition of W-eighted DMP inverse of rectangular matrix A C m n present the canonical form of it by using the singular value decompositions of A W C n m. Theorem 2.1. Let A C m n W C n m, then the matrix X = WA d, WAA is the unique solution to the equations XAX = X, XA = WA d, WA (WA k+1 X = (WA k+1 A, (2.1 here k = Ind(AW. Proof. From the definition of A d,, e have WA d, WAA AWA d, WAA = WA d, WAA, WA d, WAA A = WA d, WA (WA k+1 WA d, WAA = W(AW k+1 A d, WAA = (WA k+1 A, i.e., WA d, WAA satisfies the three equations in (2.1. To sho uniqueness, suppose both X 1 X 2 are to solutions to (2.1. Then using repeated applications of the three equations in (2.1 (1.1, e have X 1 = X 1 AX 1 = WA d, WAX 1 = (WA d, 2 (WA 2 X 1 = = (WA d, k+1 (WA k+1 X 1 = (WA d, k+1 (WA k+1 A = (WA d, k+1 (WA k+1 X 2 = WA d, WAX 2 = X 2 AX 2 = X 2. We have just finished the proof.

3 L. Meng / Filomat 31:19 (2017, It should be noted in Theorem 2.1 that A D, = A D AA = WA d, WAA hen A is square W = I n. In vie of the correspondence beteen the defining equations for A D,, those in Theorem 2.1, e define the DMP inverse of a rectangular matrix in the folloing manner: Definition 2.2. For any matrices A W, m by n n by m, respectively, the matrix X = WA d, WAA is called the W-eighted DMP inverse of A, is ritten as X = A D,. Remark 2.3. Obviously, hen A C m m W = I m, then A D, reduces to A D,. When A C m m, W = I m Ind(A = 1, then A D, reduces to the core inverse of A. Moreover, hen A is a nonsingular square matrix W = I m, then A D, = A 1. We no give the canonical form for the W-eighted DMP inverse of A by using the singular value decompositions of A W. Let A C m n r, W C n m s respectively have the folloing singular value decompositions: Σ1 0 A = U V W = 1 0, (2.2 here U = (U 1, U 2, = ( 1, 2 C m m V = (V 1, V 2, = ( 1, 2 C n n are unitary matrices, U 1 C m r, 1 C m s, V 1 C n r, 1 C n s, Σ 1 = dia (σ 1,, σ r, 1 = dia ( σ 1,, σ s, σ 1 σ r > 0 σ 1 σ s > 0. After a series of complicated calculation, e can get the folloing theorem. Theorem 2.4. Let A C m n W C n m have the singular value decompositions (2.2. Then A D, = 1 S 11 Λ 1 S 11 0 U, here S 11 = 1 U 1 Λ = (Σ 1 T 11 d, 1 S 11 ith T 11 = V 11. Here (Σ 1 T 11 d, 1 S 11 denotes the 1 S 11 -eighted Drazin inverse of matrix Σ 1 T 11. Proof. Denote S = U, T = V, assume that S T have the folloing block forms S11 S S = 12 T11 T T = 12, S 21 S 22 T 21 T 22 here S 11 C s r, T 11 C r s, then S S = I m T T = I n, i.e., S T are unitary matrices. Then, e have Σ1 0 A = U V Σ1 0 = U V Σ1 T = U 11 Σ 1 T 12 W = 1 0 = 1 0 UU = 1 S 11 1 S 12 U. X1 X Let X = U 2 X 3 X be the W-eighted Drazin inverse of A, here X 1 C r s. From AWX = 4 XWA, XWAWX = X (AW k+1 XW = (AW k, e can get the folloing equalities after some tedious manipulation X 1 = X 1 1 S 11 Σ 1 T 11 1 S 11 X 1, X 1 1 S 11 Σ 1 T 11 = Σ 1 T 11 1 S 11 X 1, X 2 = X 1 1 S 11 X 1 1 S 11 Σ 1 T 12, X 3 = 0, X 4 = 0, (Σ 1 T 11 1 S 11 k+1 X 1 1 S 11 = (Σ 1 T 11 1 S 11 k (Σ 1 T 11 1 S 11 k+1 X 1 1 S 12 = (Σ 1 T 11 1 S 11 k 1 Σ 1 T 11 1 S 12.

4 L. Meng / Filomat 31:19 (2017, These equations sho that the W-eighted Drazin inverse of A is the matrix Λ Λ A d, = U 1 S 11 Λ 1 S 11 Σ 1 T 12, here Λ = (Σ 1 T 11 d, 1 S 11. Furthermore, it is easy to check that T A = 11 Σ U. Thus, e have T 12 Σ A D, The proof of this theorem is no complete. = WA d, WAA = ( 1 S 11 Λ 1 S 11 0 U. 3. Properties of the W-eighted DMP Inverse In this section, e study the properties of the W-eighted DMP inverse. Theorem 3.1. Let A C m n W C n m. Then the folloing statements hold: (i. AA D, is a projector onto R(AWA d, along N(A d, A. (ii. A D, A is a projector onto R((WA l along N((WA l, here l = Ind(WA. Proof. (1. Since A D, is an outer inverse of A, it follos that AA D, is a projector. It follos from AA D, = AWA d, WAA AWA d, = (AWA d, WAA AWA d, that R(AA D, R(AWA d, rank(aa D, =rank(awa d,, hich implies that R(AA D, = R(AWA d,. Similarly, e can get N(AA D, = N(A d, A. (2. We can immediately prove the result (ii in this theorem by using the fact that The proof is complete. WA d, WA = P R((WA l,n((wa l (see [5] A D, A = WA d, WA. In Definition 2.1 the W-eighted DMP inverse has been introduced from an algebraic approach. Next result presents a characterization of the W-eighted DMP inverse from a geometrical point of vie. Theorem 3.2. Let A C m n W C n m. Then A D, is the unique matrix X that satisfies AX = P R(AWAd,,N(A d, A, R(X R((WA l, (3.1 here l = Ind(WA. Proof. According to Theorem 3.1, it follos that A D, is the solution to (3.1. It remains to prove that the system (3.1 has only one solution. Suppose that both X 1 X 2 are solutions to (3.1. Then Consequently, A(X 1 X 2 = 0, R(X 1 R((WA l R(X 2 R((WA l. R(X 1 X 2 N(WA R(X 1 X 2 R((WA l. Therefore, e have R(X 1 X 2 R((WA l N((WA l = {0} since WA has index l. Thus, X 1 = X 2.

5 L. Meng / Filomat 31:19 (2017, Obviously, A D, is the only matrix that satisfies (2.1 (3.1. Hence, both algebraic geometrical approaches are equivalent. The W-eighted DMP inverse also has the folloing properties. Theorem 3.3. Let A C m n W C n m. Then (a. A D, = WA d, WP A ; (b. A D, is an outer inverse of A. Proof. The proof follos from the definitions properties of the Moore-Penrose inverse W-eighted Drazin inverse. The folloing example shos that in general the W-eighted DMP inverse, the Moore-Penrose inverse the W-eighted Drazin inverse are different Example. If A = W =, then simple computations give 0 A = ( , A d, = 1 0 AD, = ( 1 0 Hence, the W-eighted DMP inverses provide a ne class of generalized inverses for rectangular matrices because in general the W-eighted DMP inverse of a matrix is different from each of its Moore- Penrose inverse W-eighted Drazin inverse.. Acknoledgement I m grateful to the hing editor referees for their helpful comments suggestions. References [1] O.M. Baksalary, G. Trenkler, Core inverse of matrices, Linear Multilinear Algebra 58 ( [2] A. Ben-Israel, T.N.E. Greville, Generalized Inverses: Theory Applications, (2nd edition, Springer, Ne York, [3] S.L. Campbell, C.D. Meyer Jr, Generalized Inverse of Linear Transformation, Dover, Ne York, [4] S.L. Campbell, C.D. Meyer Jr, N.J. Rose, Applications of the Drazin inverse to linear systems of differential equations ith singular constant coefficients, SIAM J. Appl. Math. 31 ( [5] R.E. Cline, T.N.E. Greville, A Drazln inverse for rectangular matrices, Linear Algebra Appl. 29 ( [6] S.B. Malik, N. Thome, On a ne generalized inverse for matrices of an arbitrary index, Appl. Math. Comput. 226 ( [7] C.D. Meyer Jr, The role of the group generalized inverse in the theory of finite Markov chains, SIAM Rev. 17 ( [8] S.K. Mitra, P. Bhimasankaram, S.B. Malik, Matrxi Partial Orders, Shorted Operators Applications, World Scientific, Singapore, [9] R. Penrose, A generalized inverse for matrices, Proc. Cambridge Philos. Soc. 52 ( [10] C.R. Rao, S.K. Mitra, S.B. Malik, Generalized Inverse of Matrices its Applications, John Wiley Sons, Ne York, [11] G. Wang, Y. Wei, S. Qiao, Generalized inverses: Theory Computations, Science Press, Beijing, 2004.