JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 408457 10.1155/2014/408457 408457 Research Article Some Results on Characterizations of Matrix Partial Orderings Wang Hongxing Xu Jin Zhang Yang Department of Mathematics, Huainan Normal University, Anhui 232001 China hnnu.edu.cn 2014 2852014 2014 19 02 2014 21 04 2014 28 5 2014 2014 Copyright © 2014 Hongxing Wang and Jin Xu. 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.

Some characterizations of the left-star, right-star, and star partial orderings between matrices of the same size are obtained. Based on those results, several characterizations of the star partial ordering between EP matrices are given. At last, one characterization of the sharp partial ordering between group matrices is obtained.

1. Introduction

In this paper we use the following notation. Let Cm×n be the set of complex m×n matrices. For any matrix ACm×n, A*, R(A), and r(A) denote the conjugate transpose, the range, and the rank of A, respectively. The symbol In denotes the n×n identity matrix, and 0 denotes a zero matrix of appropriate size. The Moore-Penrose inverse of a matrix ACm×n, denoted by A, is defined to be the unique matrix XCn×m satisfying the four matrix equations (1)(1)  AXA=A,(2)  XAX=X,(3)  (AX)*=AX,(4)  (XA)*=XA, and  A- denotes any solution to the matrix equation AXA=A with respect to X; A{1} denotes the set of A-; that is, A{1}={XAXA=A}. Moreover, A# denotes the group inverse of A with r(A2)=r(A), that is, the unique solution to (2)(1)  AXA=A,(2)  XAX=X,(5)  AX=XA. It is well known that A# exists if and only if r(A2)=r(A), where case A is also called a group matrix. A matrix A is EP if and only if A is a group matrix with A#=A. The symbols CGPn and CEPn stand for the subset of Cn×n consisting of group matrices and EP matrices, respectively (see, e.g., [1, 2] for details).

Five matrix partial orderings defined in Cm×n are considered in this paper. The first of them is the minus partial ordering defined by Hartwig  and Nambooripad  independently in 1980: (3)ABA-A=A-B,AA==BA=, where A-,A=A{1}. In  it was shown that (4)ABr(B-A)=r(B)-r(A). The rank equality indicates why the minus partial ordering is also called the rank-subtractivity partial ordering. In the same paper  it was also shown that (5)A*Br[AB]=r[AB]=r(B),AB-A=A, where B-B{1}.

The second partial ordering of interest is the star partial ordering introduced by Drazin , which is determined by (6)A*BAA=AB,AA=BA. It is well known that (7)A*BA*A=A*B,AA*=BA*.

In 1991, Baksalary and Mitra  defined the left-star and right-star partial orderings characterized as (8)A*BA*A=A*B,R(A)R(B),A*BAA*=BA*,R(A*)R(B*).

The last partial ordering we will deal with in this paper is the sharp partial ordering, introduced by Mitra  in 1987, and is defined in the set CGPn by (9)A#BA#A=A#B,AA#=BA#. A detailed discussion of partial orderings and their applications can be found in [1, 810].

It is well known that rank of matrix is an important tool in matrix theory and its applications, and many problems are closely related with the ranks of some matrix expressions under some restrictions (see  for details). Our aim in this paper is to characterize the left-star, right-star, star, and sharp partial orderings by applying rank equalities. In the following, when A is considered below B with respect to one partial ordering, then the partial ordering should entail the assumption r(A)>r(B)1.

2. The Star Partial Ordering

Let A and B be m×n complex matrices with ranks a and b, respectively. Let A*B. Then there exist unitary matrices UCm×m and VCn×n such that (10)U*AV=(Da000),U*BV=(Da000D0000), where both the a×a matrix Da and the (b-a)×(b-a) matrix D are real, diagonal, and positive definite (see [16, Theorem 2]). In [1, Theorem 5.2.8], it was also shown that (11)A*BAA=BA,AA=AB. In , Wang obtained the following characterizations of the left-star and right-star partial orderings for matrices: (12)A*Br[B*BA*ABA]=r(B),(13)A*Br[BB*AA*B*A*]=r(B),(14)A*Br[B*BA*ABA]=r(B),r[BB*AA*B*A*]=r(B).

Theorem 1.

Let A,BCm×n. Then

(15)A*Br[BBAAB*A*]=r(B);

(16)A*Br[BBAABA]=r(B);

(17)A*Br[BBAAB*BA*ABA]=r(B);

(18)A*Br[BBAABB*AA*B*A*]=r(B).

Proof.

From (19)r[BBAAB*A*]r([BBAAB*A*][B00A])=r[BAB*BA*A]r([BAB*BA*A][B00A])=r[BBAAB*A*], we have (20)r[B*BA*ABA]=r[BBAAB*A*]. Applying (12) gives (i).

In the same way, applying (21)r[BBAABA]=r[BB*AA*B*A*] and (13) gives (ii).

If (22)r[BBAAB*BA*ABA]=r(B), then (23)r[BB*AA*B*A*]=r(B),r[BBAAB*A*]=r(B). Applying (i), (ii), and (14), we obtain A*B. Conversely, if A*B, by using (11) and (14), we have AA-BA=0, and (24)r[B*BA*ABA]=r[0AA-BAB*BA*ABA]=r([In0B0In000Im][0AA-BAB*BA*ABA])=r[BBAAB*BA*ABA],r(B)=r[BBAAB*BA*ABA]. Hence, we have (iii).

Similarly, applying A*B, (11), and (14), we obtain AA-AB=0,   AB=(AB)*=(B*)A*, and (25)r[BB*AA*B*A*]=r[0AA-ABBB*AA*B*A*]=r([Im0(B*)0In000In][0AA-ABBB*AA*B*A*])=r(B). Then, we obtain (iv).

In [9, Theorem 2.1], Benítez et al. deduce the characterizations of the left-star, right-star, and star partial orderings for matrices, when at least one of the two involved matrices is EP. When both ACn×n and BCn×n are EP matrices, [1, Theorems 5.4.15 and 5.4.2] give the following results: (26)A*BAB,AB*  and  B*Aare  Hermitian.A*B(AB)=BA=AB=A2. In addition, it was also shown that A*B if and only if A and B have the form (27)A=U[T00000000]U*,B=U[T000K0000]U*, where TCr(A)×r(A) is nonsingular, KC(r(B)-r(A))×(r(B)-r(A)) is nonsingular, and UCn×n is unitary (see [1, Theorem 5.4.1]).

Based on these results, we consider the characterizations of the star partial ordering for matrices in the set of CEPn.

Theorem 2.

Let A,BCEPn, r(B)r(A). Then

(28)A*Br[BAB2A2]=r(B);

(29)A*Br[BB2AA2]=r(B).

Proof.

By A,BCEPn, it is obvious that AA=AA and BB=BB. Then (30)r[BAB2A2]=r(B)r[BBAABA]=r(B). Hence, we have (v).

The proof of (vi) is similar to that of (v).

Theorem 3.

Let A,BCEPn. Then

(31)A*Br[BBAABAB*A*]=r(B);

(32)A*Br[B*BA*ABAB*A*]=r(B);

(33)A*Br[BBABABAAB]=r(B);

(34)A*Br[BBABABAAB]=r(B);

(35)A*Br[BBA*BA*BAA*B]=r(B).

Proof.

By A,BCEPn, it is obvious that AA=AA and BB=BB. Applying (i), (ii), and the rank equality in (vii) we obtain (36)r[BBAABA]=r(B),r[BBAAB*A*]=r(B); that is, A*B. Conversely, suppose that A*B. Applying A-AAB=0 and B*BB=B*, we obtain (37)r(B)=r[BBAABA]=r[BBAABA0A*-B*AA]=r[BBAABAB*A*].

Applying (11), we obtain B*BBB=B*B and B*BAA=A*A and also (B*B)B*B=BB and (B*B)A*A=AA. Then (38)r[BBAABAB*A*]=r[BBAABAB*A*]r([B*B000In000In][BBAABAB*A*])=r[B*BA*ABAB*A*]r([(B*B)000In000In][B*BA*ABAB*A*])=r[BBAABAB*A*]; that is, (39)r[BBAABAB*A*]=r[B*BA*ABAB*A*]. Hence, we have (viii).

Suppose that A*B. Since A,BCEPn, applying (27), it is easy to check the rank equality in (ix). Conversely, under the rank equality in (ix), we have (40)r[BBABAB]=r[BBA0AB-BA]=r(B)AB=BA,r[BBAAAB]=r[B0AAB-AA2]=r(B)AB=A2. Since A is EP, there exists a unitary matrix U1Cn×n and a nonsingular matrix TCr(A)×r(A) such that (41)A=U1[T000]U1*. Correspondingly denote P-1BP by (42)B=U1[B1B2B3B4]U1*, where B1Cr(A)×r(A). It follows that (43)[TB1TB200]=[B1T0B3T0],[TB1TB200]=[T2000]. Since T is a unitary matrix, (44)B1=T,B2=0,B3=0. Thus (45)B=U[T00B4]U*. Since B is EP, B4 is EP, and there exists a unitary matrix U2C(n-r(A))×(n-r(A)) and a nonsingular matrix KC(r(B)-r(A))×(r(B)-r(A)) such that (46)B4=U2[K000]U2*. Write (47)U=U1[000U2]. Then A and B have the form (48)A=U[T00000000]U*,B=U[T000K0000]U*. Applying (27), we have A*B.

The proofs of (x) and (xi) are similar to that of (ix).

3. The Sharp Partial Ordering

Let A,BCGPn with ranks a and b, respectively. It is well known that (49)A#BA2=AB=BA. In addition, A#B if and only if A and B can be written as (50)A=P[E00000000]P-1,B=P[E000E0000]P-1, where ECa×a is nonsingular, EC(b-a)×(b-a) is nonsingular, and PCn×n is nonsingular (see ).

In Theorem 4, we give one characterization of the sharp partial ordering by using one rank equality.

Theorem 4.

Let A,BCGPn. Then (51)A#Br[ABAABABA]=r(ABA).

Proof.

Let A have the core-nilpotent decomposition (see [19, Exercise 5.10.12]) (52)A=P[Σ000]P-1, with nonsingular matrices ΣCr(A)×r(A) and PCn×n. Correspondingly denote P-1BP by (53)P-1BP=[B1B2B3B4], where B1Cr(A)×r(A). It follows that (54)r(ABA)=r(ΣB1Σ),r[ABAABABA]=r[Σ0B1Σ00B3ΣΣB1ΣB2ΣB1Σ]=r[Σ0000B3Σ0ΣB2ΣB1Σ-ΣB1Σ-1B1Σ]=r(Σ)+r[0B3ΣΣB2ΣB1Σ-ΣB1Σ-1B1Σ].

Applying (54) to the rank equality in (51), we obtain (55)r[0B3ΣΣB2ΣB1Σ-ΣB1Σ-1B1Σ]+r(Σ)=r(ΣB1Σ). Hence r(ΣB1Σ)=r(Σ), ΣB2=0, B3Σ=0, and ΣB1Σ=ΣB1Σ-1B1Σ. Since ΣCr(A)×r(A) is invertible and B1Cr(A)×r(A), it follows immediately that (56)r(B1)=r(Σ),B3=0,B2=0,B1=Σ. Therefore (57)B=P[Σ00B4]P-1. Applying (58)A2=P[Σ2000]P-1=P[Σ000]P-1P[Σ00B4]P-1=AB=P[Σ00B4]P-1P[Σ000]P-1=BA, and (49), we obtain that A#B.

Conversely, it is a simple matter.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the referees for their helpful comments and suggestions. The work of the first author was supported in part by the Foundation of Anhui Educational Committee (Grant no. KJ2012B175) and the National Natural Science Foundation of China (Grant no. 11301529). The work of the second author was supported in part by the Foundation of Anhui Educational Committee (Grant no. KJ2013B256).

Mitra S. K. Bhimasankaram P. Malik S. B. Matrix Partial Orders, Shorted Operators and Applications 2010 Singapore World Scientific MR2647903 Wang G. Wei Y. Qiao S. Generalized Inverses: Theory and Computations 2004 Beijing, China Science Press Hartwig R. E. How to partially order regular elements Mathematica Japonica 1980 25 1 1 13 MR571255 ZBL0442.06006 Nambooripad K. S. S. The natural partial order on a regular semigroup Proceedings of the Edinburgh Mathematical Society 1980 23 3 249 260 10.1017/S0013091500003801 MR620922 ZBL0459.20054 Drazin M. P. Natural structures on semigroups with involution Bulletin of the American Mathematical Society 1978 84 1 139 141 MR0486234 10.1090/S0002-9904-1978-14442-5 ZBL0395.20044 Baksalary J. K. Mitra S. K. Left-star and right-star partial orderings Linear Algebra and Its Applications 1991 149 73 89 10.1016/0024-3795(91)90326-R MR1092870 ZBL0717.15004 Mitra S. K. On group inverses and the sharp order Linear Algebra and Its Applications 1987 92 17 37 10.1016/0024-3795(87)90248-5 MR894635 ZBL0619.15006 Baksalary J. K. Baksalary O. M. Liu X. Further properties of the star, left-star, right-star, and minus partial orderings Linear Algebra and Its Applications 2003 375 83 94 10.1016/S0024-3795(03)00609-8 MR2013457 ZBL1048.15016 Benítez J. Liu X. Zhong J. Some results on matrix partial orderings and reverse order law Electronic Journal of Linear Algebra 2010 20 254 273 MR2653538 ZBL1207.15035 Groß J. Remarks on the sharp partial order and the ordering of squares of matrices Linear Algebra and Its Applications 2006 417 1 87 93 10.1016/j.laa.2005.10.036 MR2238597 ZBL1103.15013 Bai Z.-J. Bai Z.-Z. On nonsingularity of block two-by-two matrices Linear Algebra and Its Applications 2013 439 8 2388 2404 10.1016/j.laa.2013.06.004 MR3091311 ZBL1283.15009 Chu D. Hung Y. S. Woerdeman H. J. Inertia and rank characterizations of some matrix expressions SIAM Journal on Matrix Analysis and Applications 2009 31 3 1187 1226 10.1137/080712945 MR2558819 Liu Y. Tian Y. A simultaneous decomposition of a matrix triplet with applications Numerical Linear Algebra with Applications 2011 18 1 69 85 10.1002/nla.701 MR2769034 ZBL1249.15020 Wang H. The minimal rank of A-BX with respect to Hermitian matrix Applied Mathematics and Computation 2014 233 55 61 10.1016/j.amc.2014.01.116 Wang Q.-W. He Z.-H. Solvability conditions and general solution for mixed Sylvester equations Automatica 2013 49 9 2713 2719 10.1016/j.automatica.2013.06.009 MR3084457 Hartwig R. E. Styan G. P. H. On some characterizations of the “star” partial ordering for matrices and rank subtractivity Linear Algebra and Its Applications 1986 82 145 161 10.1016/0024-3795(86)90148-5 MR858968 ZBL0603.15001 Wang H. X. Rank characterizations of some matrix partial orderings Journal of East China Normal University 2011 5 5 11 MR2920261 Wang Z. J. Liu X. J. On three partial orderings of matrices Journal of Mathematical Study 2003 36 1 75 81 MR2032598 ZBL1065.15028 Meyer C. D. Matrix Analysis and Applied Linear Algebra 2000 Philadelphia, Pa, USA Society for Industrial and Applied Mathematics MR1777382