Some Results on Characterizations of Matrix Partial Orderings

In this paper we use the following notation. Let Cm×n be the set of complexm× nmatrices. For any matrix A ∈ C, A, R(A), and r(A) denote the conjugate transpose, the range, and the rank ofA, respectively.The symbol I n denotes the n × n identity matrix, and 0 denotes a zero matrix of appropriate size. The Moore-Penrose inverse of a matrix A ∈ C, denoted by A, is defined to be the unique matrix X ∈ Cn×m satisfying the four matrix equations


Introduction
In this paper we use the following notation.Let C × be the set of complex  ×  matrices.For any matrix  ∈ C × ,  * , R(), and () denote the conjugate transpose, the range, and the rank of , respectively.The symbol   denotes the  ×  identity matrix, and 0 denotes a zero matrix of appropriate size.The Moore-Penrose inverse of a matrix  ∈ C × , denoted by  † , is defined to be the unique matrix  ∈ C × satisfying the four matrix equations and  − denotes any solution to the matrix equation  =  with respect to ; {1} denotes the set of  − ; that is, {1} = { |  = }.Moreover,  # denotes the group inverse of  with ( 2 ) = (), that is, the unique solution to (2) It is well known that  # exists if and only if ( 2 ) = (), where case  is also called a group matrix.A matrix  is EP if and only if  is a group matrix with  # =  † .The symbols C  GP and C  EP stand for the subset of C × consisting of group matrices and EP matrices, respectively (see, e.g., [1,2] for details).
Five matrix partial orderings defined in C × are considered in this paper.The first of them is the minus partial ordering defined by Hartwig [3] and Nambooripad [4] independently in 1980: where  − ,  = ∈ {1}.In [3] 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 [3] it was also shown that where  − ∈ {1}.
The second partial ordering of interest is the star partial ordering introduced by Drazin [5], which is determined by It is well known that In 1991, Baksalary and Mitra [6] 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 [7] in 1987, and is defined in the set C  GP by A detailed discussion of partial orderings and their applications can be found in [1,[8][9][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][12][13][14][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 () > () ≥ 1.

The Star Partial Ordering
Let  and  be  ×  complex matrices with ranks  and , respectively.Let  * ≤ .Then there exist unitary matrices  ∈ C × and  ∈ C × 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 [17], Wang obtained the following characterizations of the left-star and right-star partial orderings for matrices: Theorem 1.Let ,  ∈ C × .Then (i) (ii) (iii) (iv) Proof. From we have Applying ( 12) gives (i).
In the same way, applying and ( 13) gives (ii).If

The Sharp Partial Ordering
Let ,  ∈ C  GP with ranks  and , respectively.It is well known that In addition,  ≤ #  if and only if  and  can be written as where  ∈ C × is nonsingular,   ∈ C (−)×(−) is nonsingular, and  ∈ C × is nonsingular (see [18]).
In Theorem 4, we give one characterization of the sharp partial ordering by using one rank equality.
Conversely, it is a simple matter.

Then 𝐴 and 𝐵 have the
), we have  *
Proof.By ,  ∈ C  EP , it is obvious that  † =  †  and 1, Theorem 5.4.1]).Based on these results, we consider the characterizations of the star partial ordering for matrices in the set of C  EP .