TSWJ The Scientific World Journal 1537-744X Hindawi Publishing Corporation 765732 10.1155/2013/765732 765732 Research Article Some New Algebraic and Topological Properties of the Minkowski Inverse in the Minkowski Space Zekraoui Hanifa 1 http://orcid.org/0000-0002-8543-5497 Al-Zhour Zeyad 2 Özel Cenap 3 Hoff da Silva J. Székelyhidi L. 1 Department of Mathematics and Informatic, Faculty of Exact and Natural Sciences Oum-El-Bouaghi University Oum-El-Bouaghi 04000 Algeria univ-oeb.academia.edu 2 Department of Basic Sciences and Humanities College of Engineering University of Dammam P.O. Box 1982 Dammam 34151 Saudi Arabia ud.edu.sa 3 Department of Mathematics, Faculty of Sciences University of Abant Izzet Baysal Bolu Turkey, 14280 Bolu Turkey ibu.edu.tr 2013 29 10 2013 2013 09 08 2013 18 09 2013 2013 Copyright © 2013 Hanifa Zekraoui et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We introduce some new algebraic and topological properties of the Minkowski inverse A of an arbitrary matrix AMm,n (including singular and rectangular) in a Minkowski space μ. Furthermore, we show that the Minkowski inverse A in a Minkowski space and the Moore-Penrose inverse A+ in a Hilbert space are different in many properties such as the existence, continuity, norm, and SVD. New conditions of the Minkowski inverse are also given. These conditions are related to the existence, continuity, and reverse order law. Finally, a new representation of the Minkowski inverse A is also derived.

1. Introduction and Preliminaries

In this work, we consider matrices over the field of complex numbers as and real numbers as . The set of m-by-n complex matrices is denoted by Mm,n()=m×n. For simplicity, we write Mm,n instead of Mm,n() or Mm,n(), and when m=n, we write Mn instead of Mn,n. The notations At, A*, A~,  r(A), R(A), N(A), tr(A), det(A), A2, A+, A, and σ(A) stand for the transpose, conjugate transpose, μ-symmetric, rank, range, null space, trace, determinant, Frobenius norm, Moore-Penrose inverse, Minkowski inverse, and set of all eigenvalues of a matrix A, respectively.

The Moore-Penrose inverse is widely used in perturbation theory, singular systems, neural network problems, least-squares problems, optimization problems, and many other subjects . The Moore-Penrose inverse of an arbitrary matrix AMm,n is defined to be the unique solution of the following four matrix equations [3, 4, 810]: (1)AXA=A,XAX=X,(AX)*=AX,(XA)*=XA, and it is often denoted by X=A+. Note that if we designate any matrix X that satisfying the ith matrix equation (i{1,2,3,4}) in (1) is called the i-inverse and denoted by A(i).

The Moore-Penrose inverse can be explicitly expressed by the singular value decomposition (SVD) due to van Loan . For any matrix AMm,n with r(A)=r, there exist unitary matrices UMm and VMn satisfying U*U=Im and V*V=In such that (2)A=U[D000]V*, where D=diag(δ1,δ2,,δr)Mr, δ1δ2δr>0, and δi2 (i=1,2,,r) are the nonzero eigenvalues of A*A. Then, the Moore-Penrose inverse can be represented as (3)A+=V[D-1000]U*.

Some algebraic properties concerning the null space, range, rank, continuity, and some representations of some types of the generalized inverses of a given matrix over complex and real fields are widely studied by many researchers . The Minkowski inverse A of an arbitrary matrix AMm,n is one of the important generalized inverses for solving matrix equations in the Minkowski space μ with respect to the generalized reflection antisymmetric matrix A~ . Some methods such as iterative, Borel summable, Euler-Knopp summable, Newton-Raphson, and Tikhonov’s methods are used for representation and computation of the Minkowski inverse A in the Minkowski space μ [18, 19].

By letting n be the space of complex n-tuples, we will index the components of a complex vector in n from 0 to n-1; that is, u=(u0,u1,u2,,un-1). In addition to that, let G be the Minkowski metric tensor defined by Gu=(u0,-u1,-u2,,-un-1). Clearly, the Minkowski metric matrix is defined by [18, 20] (4)G=[100-In-1]Mn, and G*=G and G2=In.

In [21, 22], the Minkowski inner product on n is defined by (u,v)=[u,Gv], where [·,·] denotes the conventional Hilbert (unitary) space inner product. The space with the Minkowski inner product is called a Minkowski space and is denoted by μ. For any square matrix AMn and vectors x and yn, we have (5)  (Ax,y)=[Ax,Gy]=[x,A*Gy]=[x,G(GA*G)y]=[x,GA~y]=(x,A~y), where A~=GA*G is called the Minkowski conjugate transpose of A in the Minkowski space μ. Naturally, the matrix AMn is called μ-symmetric in the Minkowski space μ if A=A~. Now, it is easy to show that A is μ-symmetric if and only if AG is Hermitian if and only if GA is Hermitian. Also, it is easy to verify that G-1=G and σ(A~)=  σ(A). More generally, if AMm,n, then the Minkowski conjugate transpose of A is defined by A~=G1A*G2 (where  G1 and G2 are the Minkowski metric matrices of orders n×n and m×m, resp.), and it satisfies the following algebraic properties as in the following result.

Lemma 1.

Let AMm,n. Then, the following one given:

A~ is unique,

(A~)~=A,

(AB)~=B~A~,

~-cancellation rule A~AX=A~AYAX=AY,

r(A)=r(A~),

R(A*)=R(A~),

N(A*)=N(A~).

Finally, a matrix AMm,n is said to be a range symmetric in unitary space (or equivalently A is said to be EP) if N(A*)=N(A). For further properties of EP matrices, one may refer to [3, 4, 10, 11].

In this paper, some algebraic properties concerning the rank, range, existence, uniqueness, continuity, and reverse order law of the Minkowski inverse A are introduced. The relationships between A and A~ are also discussed. Furthermore, a new representation of A related to the full-rank factorization of the matrix A is derived, and new conditions for the existence and continuity of A are also given.

2. Some Algebraic Properties of the Minkowski Inverse

In this section, we derive some attractive algebraic properties and the reverse order law property of the Minkowski inverse in a Minkowski space.

The Minkowski inverse of an arbitrary matrix AMm,n (including singular and rectangular), analogous to the Moore-Penrose inverse, is defined as follows.

Definition 2.

Let AMm,n be any matrix in the Minkowski space μ. Then, the Minkowski inverse of A is the matrix AMn,m which satisfies the following four matrix equations: (6)AAA=A,AAA=A,(AA)~=AA,(AA)~=AA.

Theorem 3.

Let AMm,n be any matrix in the Minkowski space μ. Then, the Minkowski inverse A satisfies the following properties:

(A)~=(A~),

A is a unique matrix,

(A)=A,

AA and AA are idempotents (i.e., AA and AA are projectors on R(A) and R(A), resp.),

AA+αIn and AA+αIm are invertible matrices, where α>0,

(A~A)A~=A=A~(AA~),

R(A)=R(A~).

Proof.

(i)Since the following four matrix equations are satisfied: (7)A~(A)~A~=(AAA)~=A~,(A)~A~(A)~=(AAA)~=(A)~,(A~(A)~)~=((A)~)~(A~)~=AA=(AA)~=A~(A)~,((A)~A~)~=((A~)~)((A)~)~=AA=(AA)~=(A)~A~, then, by (6), we get the result.

(ii)Let G1 and G2 be two Minkowski metric tensors such that A1 and A2 are two Minkowski inverses of a matrix A; then, by using Lemma 1 and Theorem 3(i), we have (8)A1=A1AA1=A1(AA1)~=A1(A1)~A~=A1(A1)~A~(A2)~A~=A1(AA1)~(AA2)~=A1(AA1)(AA2)=A1AA2=A1AA2AA2=(A1A)~(A2A)~A2=A~(A1)~A~(A2)~A2=A~(A2)~A2=(A2A)~A2=(A2A)A2=A2.

This means that A is a unique matrix.

(iii) It follows by applying the four matrix equations in (6).

(iv) By using the matrix equations in (6), we have (AA)2=A(AAA)=AA and (AA)2=A(AAA)=AA.

(v) Since AA is an idempotent matrix, then eigenvalues of AA are 0 or 1. That is, det(AA+αIm)=0α=0 or α=-1. So, for all α>0, we have det(AA+αIm)0 (i.e., AA+αIn  is an invertible matrix). Similarly, we can prove that AA+αIm is also an invertible matrix.

(vi) Since A~A(A~A)A~A=A~A, then, from ~-cancellation, we have (9)A(A~A)A~A=A. Now, by using the four matrix equations in (6), Theorem 3, and Lemma 1, we have (10)((A~A)A~)A((A~A)A~)=((A~A)(A~A)(A~A))A~=(A~A)A~,(11)(A(A~A)A~)~=A((A~A))~A~=A((A~A)~)A~=A(A~A)A~,(12)((A~A)A~A)~=A~A(A~A)=(A~A(A~A))~=(A~A)A~A. Consequently, (9), (10), and (11) show that (A~A)A~=A.

(vii) Equations (9) and (10) show that r(A)=r(A)=r(A~). Now, by applying Theorem 3(vi), we have R(A)R(A~); then, the equality holds.

The reverse order law property for the Moore-Penrose inverse of the product of two matrices is investigated by many researchers; one may refer to . Analogous to Greville’s conditions that were stated in , we reached the following result.

Theorem 4.

Let AMm,n and BMn.p be two matrices in the Minkowski space μ such that the Minkowski inverses A, B, and (AB) exist. Then, (AB)=BA if and only if R(A~AB)R(B) and R(BB~A~)R(A~).

Proof.

Since BB is a projector on R(B) as in Theorem 3(iv), then (13)BBA~AB=A~AB. Now, by Definition 2 and Theorem 3, we have (14)AABB~A~=BB~A~. Taking the Minkowski conjugate transpose of the two sides of (13), we have (15)B~A~ABB=B~A~A. Multiplying the right side and the left side of (15) by A and ((AB)~), respectively, we have (16)((AB)~)(AB)~(AB)BA=((AB)~)(AB)~AA. Since R(AB)R(A)=R(AA), then we have (17)ABBA=((AB)(AB))~(AB)BA=((AB)(AB))~AA=(AB)(AB). Also, multiplying the right side and the left side of (14) by ((AB)~) and B, respectively, and applying Theorem 3 and Definition 2 for (AB), we have (18)BAAB(AB)(AB)=BBB~A~((AB)~). Since BB is a projector on R(B~), we have (19)BAAB=B~A~((AB)~)=((AB))(AB). Equations (17) and (19) imply that BA satisfies the first, third, and fourth equations in (6). Finally, by taking the Minkowski conjugate transpose of the two sides of the first and the second equations in (6) for matrices A and B and by using Theorem 3(vi), we have (20)B~A~=B~BBAAA~,BA=B(B)~B~A~(AA~). This equation shows that (21)r(BA)=r(B~A~)=r(AB)~=r(AB). Consequently, BA satisfies the second equation in (6).

3. Existence of the Minkowski Inverse

The Minkowski inverse of a matrix A exists if and only if r(AA~)=r(A~A)=r(A) . In this section, we give some equivalent conditions for the existence and derive a new representation of the Minkowski inverse. If AMm,n is a matrix of full row rank (column rank), then AA* and A*A are invertible matrices of orders m×m and n×n, respectively, in a Hilbert (Euclidian) space. Here, in a Minkowski space, if we define A02=(tr(AA~))1/2, then the following example shows that A~A and AA~ are, in general, not invertible matrices and also A02A2.

Example 5.

Let A=[1-11111]. Then, r(A)=2, and (22)A~=G1A*G2=[1000-1000-1][11-1111][100-1]=[1-111-11],AA~=[1-11111][1-111-11]=[-1-111]. Note that r(AA~)=1 (i.e., det(AA~)=0), and hence AA~ is not invertible. Also, A2=tr(AA*)=6, and A02=0, which are not equal.

Lemma 6.

Let AMm,n and BMn.p be two matrices. Then, the following are considered.

If r(A)=n, then r(AB)=r(B).

If r(B)=n, then r(AB)=r(A).

Proof.

(i) Since r(A)=n, then A is a left invertible; thus, there exists a matrix X  Mn,m such that XA=In. Hence, r(B)=r(XAB)r(AB)r(B), which implies that r(AB)=r(B).  Similarly, we can prove (ii).

Theorem 7.

Let A=BCMm,n be a rank factorization of rank r. Then, (CC~)-1 and (B~B)-1 exist if and only if r(AA~)=r(A~A)=r(A).

Proof.

Since A=BC, then B and C are of orders m×r and r×n, respectively, and r(A)=r(B)=r(C)=r. Hence, (23)AA~=BC(G1C*G)(GB*G2)=BCC~B~, where G, G1, and G2 are the Minkowski metric matrices of orders r  ×  r, n  ×  n, and m×m, respectively. Since B and B~ are matrices of full ranks (i.e., r(B)=r(B~)=r), then, by Lemma 6, we have r(AA~)=r(CC~). Similarly, we can prove that r(A~A)=r(B~B).

Now, since B~B and CC~ are square matrices of order r  ×  r, then they are invertible if and only if they are of rank r.

By applying the four matrix equations in (6), we can get a new representation of the Minkowski inverse as shown in the following result.

Theorem 8.

Let A=BCMm,n be a rank factorization of rank r. Then, (24)A=C~(CC~)-1(B~B)-1B~.

4. Some Topological Properties of the Minkowski Inverse

In this section, we establish some attractive topological properties and new conditions for the continuity of the Minkowski inverse in a Minkowski space.

It is known that, in normed algebra of bounded linear operators, the map of linear invertible operators associated with its usual inverse is continuous. The following example shows that this property is not valid in the Minkowski space.

Example 9.

Let An=[1001/n] be a sequence of matrices for n; then, An=[100n], limnAn=, and (limnAn)=. Note that limnAn does not exist. That is, for the map T, we have (25)T(limnAn)limnT(An).

For A020, the following results are very important for finding the new conditions for the continuity of the Minkowski inverse of rectangular matrices in a Minkowski space.

Lemma 10.

Let AMm,n. Then, for any xR(A~), (26)Ax02x02A02.

Proof.

By using Theorem 3((iv) and (vii)), then, for any xR(A~), we have x=AAx. Thus, x02=AAx02A02Ax02, and then we get the result.

Lemma 11.

Let A and EMm,n such that  A020 and E02<1/A02. Then, (27)r(A+E)r(A).

Proof.

Suppose that r(A)=r and {v1,,vr} are the basis of R(A~). Then, the set {(A+E)(v1),,(A+E)(vr)} is a subset of R(A+E). Now, suppose that i=1rαi(A+E)(vi)=0, for αi; then, x=i=1rαivi0, and we have (28)0=ri=1αi(A+E)(vi)02=(A+E)ri=1αivi02=Ax+Ex02Ax02-E02x02>Ax02-x02A02. Now, by using Lemma 10, we have 0>(x02/A02)-(x02/A02)=0, which is impossible, and thus x=0. As {v1,,vr} is linearly independent, it follows that αi=0 (for all i=1,,r). Consequently, r(A+E)r=r(A).

Corollary 12.

Let A and BMm,n such that A020,  B020, and A-B02<1/max{A02,B02}. Then, (29)r(A)=r(B).

Proof.

Set E=A-B, and since B020, then, by using Lemma 11, we have E02<1/B02 which implies that (30)r(A)=r(B+E)r(B). Since A020, then we also have E2<1/A02 which implies that (31)r(B)=r(A-E)r(A). Now, the result follows by using (30) and (31).

If AMm,n and λ0 is the largest eigenvalue of A~A, then A02=λ0 in the Minkowski space μ. Here, if P is the μ-symmetric projector, then it is easy to show that P=P. Since the eigenvalues of a projector are only 0 and 1, then we have P02=P02=1, and by applying Corollary 12, we get the following result.

Corollary 13.

Let P and Q be two μ-symmetric projectors such that P-Q02<1. Then, (32)r(P)=r(Q).

Theorem 14.

The matrix AMm,n can be written by using SVD as in the form A=USV* with U*U=Im, V*V=In, and S is a diagonal if and only if the following conditions hold:

σ(A~A) are nonnegative real numbers,

A~A is diagonalizable,

N(A~A)=N(A).

If only assumption (i) is violated but (ii) and (iii) hold of Theorem 14, then we can still get singular value decomposition (SVD). But in the Minkowski space, each of the assumptions can fail even if the other two hold. This is illustrated by the following three counterexamples .

Example 15.

Let A=[-1111]. Then, A~=[-1-1-1-1], and AA~=[0-220]. Hence, σ(A~A)={±2i} which are not real numbers.

Example 16.

Let A=[1.510.51]. Then, A~=[1.5-0.5-11], and AA~=[21-10]. Hence, σ(A~A)={1} and cannot be diagonalized.

Example 17.

Let A=[1-1-11]. Then, A~=, and AA~=. Hence, N(A~A)N(A).

The following result gives the equivalent conditions for the continuity to be held for the Minkowski inverse of any rectangular matrix.

Theorem 18.

Let (An)n  Mm,p be a sequence of matrices such that limnAn=A. Then, limnAn=A=(limnAn) if and only if r(An)=r(A).

Proof.

Suppose that limnAn=A=(limnAn), and set An=A+En  such that limnEn=0 and limn(A+En)=A. Then, (33)limn(A+En)(A+En)=AA, which means that there exists n0 such that, for any nn0, (34)(A+En)(A+En)-AA02<1. Since (A+En)(A+En) and AA are μ-symmetric projectors, then, by Corollary 13, we have (35)r((A+En))=r((A+En)(A+En))=r(AA)=r(A). Conversely, suppose that A satisfies the SVD conditions as in Theorem 14; then, B=U*AV=[Dr000], where Dr is a diagonal matrix and U and V are unitary matrices. Suppose also that limnEn=0 and r(A+En)=r(A)=r for any nn0. Now, set (36)Fn=U*EnV=[F11(n)F12(n)F21(n)F22(n)]. Then, (37)r(B+Fn)=r(A+En)=r,B=V~A(U*)~,(B+Fn)=V~(A+En)(U*)~. Since limnFn=0, then, for Dr-1020 and A020, we have (38)Fn02<1Dr-102=1A02. But Fn02  supi,j{Fij(n)02}, and then, by Lemma 11, we have (39)r(Dr+F11(n))r. On the other hand, (40)r=r(B+Fn)r(Dr+F11(n))r. Now, by using (39) and (40), we get that r(Dr+F11(n))=r, which means that (Dr+F11(n))-1 exists. Also by using the Schur complement, we have (41)(B+Fn)=[Dr+F11(n)F12(n)F21(n)F21(n)(Dr+F11(n))-1F12(n)]=[IrM](Dr+F11(n))[IrN], where M=F21(n)(Dr+F11(n))-1 and N=(Dr+F11(n))-1F12(n). Now, if M~M-Ir and NN~-Ir, then, by Definition 2, we can see that (42)(B+Fn)=[IrN~]((Ir+M~M)(Dr+F11(n))(Ir+NN~))-1×[IrM~] is the Minkowski inverse of B+Fn. Since limnFn=0, then limnF11(n)=limnF21(n)=limnF12(n)=0. Therefore, limn(Dr+F11(n))-1=Dr, limnM=0=limnM~, and limnN=0=limnN~. Now, by using the fact that the map which transforms an invertible matrix to its inverse is continuous, consequently, we find that (43)limn(B+Fn)=[Ir0]Dr-1[Ir0]=[Dr-1000]=B, which completes the proof of Theorem 18.

5. Conclusion

Several attractive properties and conditions of the Minkowski inverse A in the Minkowski space μ are presented. In our opinion, it is worth extending these properties and establishing some necessary and sufficient conditions for the reverse order rule of the weighted Minkowski inverse AM,N in the Minkowski space μ of two and multiple matrix products.

Acknowledgment

The authors express their sincere thanks to the referees for the careful reading of the paper and several helpful suggestions.

Gulliksson M. Jin X. O. Wei Y. Perturbation bounds for constrained and weighted least squares problems Linear Algebra and Its Applications 2002 349 1–3 221 232 10.1016/S0024-3795(02)00262-8 Wei Y. Perturbation bound of singular linear systems Applied Mathematics and Computation 1999 105 2-3 211 220 2-s2.0-0001778647 Wei Y. Recurrent neural networks for computing weighted Moore-Penrose inverse Applied Mathematics and Computation 2000 116 3 279 287 2-s2.0-0034547867 10.1016/S0096-3003(99)00147-2 Wang D. Some topics on weighted Moore-Penrose inverse, weighted least squares and weighted regularized Tikhonov problems Applied Mathematics and Computation 2004 157 1 243 267 2-s2.0-4344612912 10.1016/j.amc.2003.08.035 Bapat R. B. Linear Algebra and Linear Models 2000 Springer Ben-Israel A. Greville T. N. E. Generalized Inverses: Theory and Applications 2003 New York, NY, USA Springer Groetsch C. W. Generalized Inverse of Linear Operators 1977 New York, NY, USA Marcel Dekker Wei Y. Wu H. The representation and approximation for the weighted Moore-Penrose inverse Applied Mathematics and Computation 2001 121 1 17 28 2-s2.0-0035947468 10.1016/S0096-3003(99)00275-1 Wei Y. The representation and approximation for the weighted Moore-Penrose inverse in Hilbert space Applied Mathematics and Computation 2003 136 2-3 475 486 2-s2.0-0037443324 10.1016/S0096-3003(02)00071-1 Wei Y. A characterization and representation of the generalized inverse A(2)T,S and its applications Linear Algebra and Its Applications 1998 280 2-3 87 96 2-s2.0-0041713889 van Loan C. F. Generalizing the singular value decomposition SIAM Journal on Numerical Analysis 1976 13 1 76 83 2-s2.0-0016926423 Zekraoui H. Propriétés algébriques des Gk—inverses des matrices [Thèse de Doctorat] 2011 Batna, Algeria Université de Batna Chen W. On EP elements, normal elements and partial isometries in rings with involution Electronic Journal of Linear Algebra 2012 23 553 561 Mosić D. Djordjević D. S. Moore-Penrose-invertible normal and Hermitian elements in rings Linear Algebra and Its Applications 2009 431 5–7 732 745 10.1016/j.laa.2009.03.023 Campbell S. L. Meyer C. D. Generalized Inverses of Linear Transformations 2009 56 SIAM Classics in Applied Mathematics Rakocevi V. On continuity of the Moore-Penrose and Drazin inverses Matematiqki Vesnik 1997 49 163 172 Krishnaswamy D. Punithavalli G. The anti-reflexive solutions of the matrix equation A×B=C in Minkowski space International Journal of Research and Reviews in Applied Sciences 2013 15 2 2 9 Kiliçman A. Al-Zhour Z. The representation and approximation for the weighted Minkowski inverse in Minkowski space Mathematical and Computer Modelling 2008 47 3-4 363 371 2-s2.0-38149024826 10.1016/j.mcm.2007.03.031 Liu X. Qin Y. Iterative methods for computing the weighted Minkowski inverses of matrices in Minkowski space World Academy of Science, Engineering and Technology 2011 51 1082 1084 Meenakshi A. R. Range symmetric matrices in Minkowski space Bulletin of the Malaysian Mathematical Sciences Society 2000 23 45 52 Meenakshi A. R. Krishnaswamy D. Product of range symmetric block matrices in Minkowski space Bulletin of the Malaysian Mathematical Sciences Society 2006 29 1 59 68 Renardy M. Singular value decomposition in Minkowski space Linear Algebra and Its Applications 1996 236 53 58 2-s2.0-0030553668 10.1016/0024-3795(94)00124-3 Djordjević D. S. Dinčić N. Č. Reverse order law for the Moore-Penrose inverse Journal of Mathematical Analysis and Applications 2010 361 1 252 261 10.1016/j.jmaa.2009.08.056