Advances in Matrices, Finite and Infinite, with Applications 2014View this Special Issue
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Hongxing Wang, Jin Xu, "Some Results on Characterizations of Matrix Partial Orderings", Journal of Applied Mathematics, vol. 2014, Article ID 408457, 6 pages, 2014. https://doi.org/10.1155/2014/408457
Some Results on Characterizations of Matrix Partial Orderings
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.
In this paper we use the following notation. Let be the set of complex matrices. For any matrix , , , and denote the conjugate transpose, the range, and the rank of , respectively. The symbol denotes the identity matrix, and denotes a zero matrix of appropriate size. The Moore-Penrose inverse of a matrix , denoted by , is defined to be the unique matrix satisfying the four matrix equations and denotes any solution to the matrix equation with respect to ; denotes the set of ; that is, . Moreover, denotes the group inverse of with , that is, the unique solution to It is well known that exists if and only if , where case is also called a group matrix. A matrix is EP if and only if is a group matrix with . The symbols and stand for the subset of consisting of group matrices and EP matrices, respectively (see, e.g., [1, 2] for details).
Five matrix partial orderings defined in are considered in this paper. The first of them is the minus partial ordering defined by Hartwig  and Nambooripad  independently in 1980: where . In  it was shown that 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 where .
The second partial ordering of interest is the star partial ordering introduced by Drazin , which is determined by It is well known that
In 1991, Baksalary and Mitra  defined the left-star and right-star partial orderings characterized as
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 by A detailed discussion of partial orderings and their applications can be found in [1, 8–10].
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 [11–15] 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 is considered below with respect to one partial ordering, then the partial ordering should entail the assumption .
2. The Star Partial Ordering
Let and be complex matrices with ranks and , respectively. Let . Then there exist unitary matrices and such that where both the matrix and the matrix are real, diagonal, and positive definite (see [16, Theorem 2]). In [1, Theorem 5.2.8], it was also shown that In , Wang obtained the following characterizations of the left-star and right-star partial orderings for matrices:
Theorem 1. Let . Then (i)(ii)(iii)(iv)
Applying (12) gives (i).
In the same way, applying and (13) gives (ii).
If then Applying (i), (ii), and (14), we obtain . Conversely, if , by using (11) and (14), we have , and Hence, we have (iii).
Similarly, applying , (11), and (14), we obtain , , and 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 and are EP matrices, [1, Theorems 5.4.15 and 5.4.2] give the following results: In addition, it was also shown that if and only if and have the form where is nonsingular, is nonsingular, and 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 .
Theorem 2. Let , . Then(v)(vi)
Proof. By , it is obvious that and . Then
Hence, we have (v).
The proof of (vi) is similar to that of (v).
Theorem 3. Let . Then (vii)(viii)(ix)(x)(xi)
Proof. By , it is obvious that and . Applying (i), (ii), and the rank equality in (vii) we obtain
that is, . Conversely, suppose that . Applying and , we obtain
Applying (11), we obtain and and also and . Then that is, Hence, we have (viii).
Suppose that . Since , applying (27), it is easy to check the rank equality in (ix). Conversely, under the rank equality in (ix), we have Since is EP, there exists a unitary matrix and a nonsingular matrix such that Correspondingly denote by where . It follows that Since is a unitary matrix, Thus Since is EP, is EP, and there exists a unitary matrix and a nonsingular matrix such that Write Then and have the form Applying (27), we have .
The proofs of (x) and (xi) are similar to that of (ix).
3. The Sharp Partial Ordering
Let with ranks and , respectively. It is well known that In addition, if and only if and can be written as where is nonsingular, is nonsingular, and is nonsingular (see ).
In Theorem 4, we give one characterization of the sharp partial ordering by using one rank equality.
Theorem 4. Let . Then
Proof. Let have the core-nilpotent decomposition (see [19, Exercise 5.10.12])
with nonsingular matrices and . Correspondingly denote by
where . It follows that
Applying (54) to the rank equality in (51), we obtain Hence , , , and . Since is invertible and , it follows immediately that Therefore Applying and (49), we obtain that .
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.
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).
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