Some Results on Characterizations of Matrix Partial Orderings

Abstract

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 ℂ^𝑚×𝑛 be the set of complex 𝑚×𝑛 matrices. For any matrix 𝐴∈ℂ^𝑚×𝑛, 𝐴^∗, ℜ(𝐴), and 𝑟(𝐴) denote the conjugate transpose, the range, and the rank of 𝐴, 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^𝑚×𝑛, denoted by 𝐴^†, is defined to be the unique matrix 𝑋∈ℂ^𝑛×𝑚 satisfying the four matrix equations (1)(2)(3)𝐴𝑋𝐴=𝐴,𝑋𝐴𝑋=𝑋,)(𝐴𝑋^∗(4)=𝐴𝑋,)(𝑋𝐴^∗=𝑋𝐴, (1) 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}={𝑋∣𝐴𝑋𝐴=𝐴}. Moreover, 𝐴^# denotes the group inverse of 𝐴 with 𝑟(𝐴²)=𝑟(𝐴), that is, the unique solution to (1)(2)(5)𝐴𝑋𝐴=𝐴,𝑋𝐴𝑋=𝑋,𝐴𝑋=𝑋𝐴. (2) 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 ℂ^𝑛_GP and ℂ^𝑛_EP stand for the subset of ℂ^𝑛×𝑛 consisting of group matrices and EP matrices, respectively (see, e.g., [1, 2] for details).

Five matrix partial orderings defined in ℂ^𝑚×𝑛 are considered in this paper. The first of them is the minus partial ordering defined by Hartwig [3] and Nambooripad [4] independently in 1980: 𝐴≤𝐵⟺𝐴⁻𝐴=𝐴⁻𝐵,𝐴𝐴⁼=𝐵𝐴⁼, (3) where 𝐴⁻,𝐴⁼∈𝐴{1}. In [3] it was shown that 𝐴≤𝐵⟺𝑟(𝐵−𝐴)=𝑟(𝐵)−𝑟(𝐴). (4) 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 𝐴^∗𝐴𝐵≤𝐵⟺𝑟=𝑟𝐴𝐵=𝑟(𝐵),𝐴𝐵⁻𝐴=𝐴, (5) where 𝐵⁻∈𝐵{1}.

The second partial ordering of interest is the star partial ordering introduced by Drazin [5], which is determined by 𝐴≤^∗𝐵⟺𝐴^†𝐴=𝐴^†𝐵,𝐴𝐴^†=𝐵𝐴^†. (6) It is well known that 𝐴≤^∗𝐵⟺𝐴^∗𝐴=𝐴^∗𝐵,𝐴𝐴^∗=𝐵𝐴^∗. (7)

In 1991, Baksalary and Mitra [6] defined the left-star and right-star partial orderings characterized as 𝐴^∗≤𝐵⟺𝐴^∗𝐴=𝐴^∗𝐵,ℜ(𝐴)⊆ℜ(𝐵),𝐴≤^∗𝐵⟺𝐴𝐴^∗=𝐵𝐴^∗,ℜ𝐴^∗⊆ℜ𝐵^∗. (8)

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^𝑛_GP by 𝐴≤^#𝐵⟺𝐴^#𝐴=𝐴^#𝐵,𝐴𝐴^#=𝐵𝐴^#. (9) 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 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 𝑈∈ℂ^𝑚×𝑚 and 𝑉∈ℂ^𝑛×𝑛 such that 𝑈^∗𝐷𝐴𝑉=^𝑎000,𝑈^∗𝐷𝐵𝑉=^𝑎,000𝐷0000 (10) where both the 𝑎×𝑎 matrix 𝐷_𝑎 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 𝐴≤^∗𝐵⟺𝐴^†𝐴=𝐵^†𝐴,𝐴𝐴^†=𝐴𝐵^†. (11) In [17], Wang obtained the following characterizations of the left-star and right-star partial orderings for matrices: 𝐴^∗𝐵≤𝐵⟺𝑟^∗𝐵𝐴^∗𝐴(𝐵),𝐵𝐴=𝑟𝐴≤^∗𝐵⟺𝑟𝐵𝐵^∗𝐴𝐴^∗𝐵^∗𝐴^∗=𝑟(𝐵),𝐴^∗𝐵≤𝐵⟺𝑟^∗𝐵𝐴^∗𝐴(𝐵),𝑟𝐵𝐴=𝑟𝐵𝐵^∗𝐴𝐴^∗𝐵^∗𝐴^∗=𝑟(𝐵). (12)(13)(14)

Theorem 1. Let 𝐴,𝐵∈ℂ^𝑚×𝑛. Then (i)𝐴^∗≤𝐵⟺𝑟𝐵𝐵^†𝐴𝐴^†𝐵^∗𝐴^∗=𝑟(𝐵); (15)(ii)𝐴≤^∗𝐵𝐵⟺𝑟^†𝐵𝐴^†𝐴(𝐵);𝐵𝐴=𝑟 (16)(iii)𝐴^∗𝐵≤𝐵⟺𝑟^†𝐵𝐴^†𝐴𝐵^∗𝐵𝐴^∗𝐴(𝐵);𝐵𝐴=𝑟 (17)(iv)𝐴≤^∗𝐵⟺𝑟𝐵𝐵^†𝐴𝐴^†𝐵𝐵^∗𝐴𝐴^∗𝐵^∗𝐴^∗=𝑟(𝐵). (18)

Proof. From 𝑟𝐵𝐵^†𝐴𝐴^†𝐵^∗𝐴^∗≥𝑟𝐵𝐵^†𝐴𝐴^†𝐵^∗𝐴^∗𝐵𝐵00𝐴=𝑟𝐵𝐴_∗𝐵𝐴^∗𝐴𝐵≥𝑟𝐵𝐴_∗𝐵𝐴^∗𝐴𝐵^†00𝐴^†=𝑟𝐵𝐵^†𝐴𝐴^†𝐵^∗𝐴^∗, (19) we have 𝑟𝐵^∗𝐵𝐴^∗𝐴𝐵𝐴=𝑟𝐵𝐵^†𝐴𝐴^†𝐵^∗𝐴^∗. (20) Applying (12) gives (i).
In the same way, applying 𝑟𝐵^†𝐵𝐴^†𝐴𝐵𝐴=𝑟𝐵𝐵^∗𝐴𝐴^∗𝐵^∗𝐴^∗ (21) and (13) gives (ii).
If 𝑟𝐵^†𝐵𝐴^†𝐴𝐵^∗𝐵𝐴^∗𝐴(𝐵),𝐵𝐴=𝑟 (22) then 𝑟𝐵𝐵^∗𝐴𝐴^∗𝐵^∗𝐴^∗=𝑟(𝐵),𝑟𝐵𝐵^†𝐴𝐴^†𝐵^∗𝐴^∗=𝑟(𝐵). (23) Applying (i), (ii), and (14), we obtain 𝐴≤^∗𝐵. Conversely, if 𝐴≤^∗𝐵, by using (11) and (14), we have 𝐴^†𝐴−𝐵^†𝐴=0, and 𝑟𝐵^∗𝐵𝐴^∗𝐴𝐵𝐴=𝑟0𝐴^†𝐴−𝐵^†𝐴𝐵^∗𝐵𝐴^∗𝐴𝐼𝐵𝐴=𝑟^𝑛0𝐵^†0𝐼_𝑛000𝐼_𝑚0𝐴^†𝐴−𝐵^†𝐴𝐵^∗𝐵𝐴^∗𝐴𝐵𝐵𝐴=𝑟^†𝐵𝐴^†𝐴𝐵^∗𝐵𝐴^∗𝐴,𝑟(𝐵)𝐵𝐵𝐴=𝑟^†𝐵𝐴^†𝐴𝐵^∗𝐵𝐴^∗𝐴.𝐵𝐴 (24) Hence, we have (iii).
Similarly, applying 𝐴≤^∗𝐵, (11), and (14), we obtain 𝐴𝐴^†−𝐴𝐵^†=0,   𝐴𝐵^†=(𝐴𝐵^†)^∗=(𝐵^∗)^†𝐴^∗, and 𝑟𝐵𝐵^∗𝐴𝐴^∗𝐵^∗𝐴^∗=𝑟0𝐴𝐴^†−𝐴𝐵^†𝐵𝐵^∗𝐴𝐴^∗𝐵^∗𝐴^∗𝐼=𝑟^𝑚0𝐵^∗†0𝐼_𝑛000𝐼_𝑛0𝐴𝐴^†−𝐴𝐵^†𝐵𝐵^∗𝐴𝐴^∗𝐵^∗𝐴^∗=𝑟(𝐵). (25) 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 𝐴∈ℂ^𝑛×𝑛 and 𝐵∈ℂ^𝑛×𝑛 are EP matrices, [1, Theorems 5.4.15 and 5.4.2] give the following results: 𝐴≤^∗𝐵⟺𝐴≤𝐵,𝐴𝐵^∗and𝐵^∗𝐴𝐴areHermitian._∗≤𝐵⟺(𝐴𝐵)^†=𝐵^†𝐴^†=𝐴^†𝐵^†=𝐴^†2. (26) In addition, it was also shown that 𝐴≤^∗𝐵 if and only if 𝐴 and 𝐵 have the form 𝐴=𝑈𝑈𝑇00000000^∗,𝐵=𝑈𝑈𝑇000𝐾0000^∗, (27) where 𝑇∈ℂ^{𝑟(𝐴)×𝑟(𝐴)} is nonsingular, 𝐾∈ℂ^{(𝑟(𝐵)−𝑟(𝐴))×(𝑟(𝐵)−𝑟(𝐴))} is nonsingular, and 𝑈∈ℂ^𝑛×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 ℂ^𝑛_EP.

Theorem 2. Let 𝐴,𝐵∈ℂ^𝑛_EP, 𝑟(𝐵)≥𝑟(𝐴). Then(v)𝐴≤^∗𝐵𝐵⟺𝑟𝐵𝐴₂𝐴²=𝑟(𝐵); (28)(vi)𝐴^∗≤𝐵⟺𝑟𝐵𝐵²𝐴𝐴²=𝑟(𝐵). (29)

Proof. By 𝐴,𝐵∈ℂ^𝑛_EP, it is obvious that 𝐴𝐴^†=𝐴^†𝐴 and 𝐵𝐵^†=𝐵^†𝐵. Then 𝑟𝐵𝐵𝐴₂𝐴²(𝐵)𝐵=𝑟⟺𝑟^†𝐵𝐴^†𝐴(𝐵).𝐵𝐴=𝑟 (30) Hence, we have (v).
The proof of (vi) is similar to that of (v).

Theorem 3. Let 𝐴,𝐵∈ℂ^𝑛_EP. Then (vii)𝐴≤^∗𝐵⟺𝑟𝐵𝐵^†𝐴𝐴^†𝐵𝐵𝐴_∗𝐴^∗=𝑟(𝐵); (31)(viii)𝐴^∗𝐵≤𝐵⟺𝑟^∗𝐵𝐴^∗𝐴𝐵𝐵𝐴_∗𝐴^∗=𝑟(𝐵); (32)(ix)𝐴≤^∗𝐵⟺𝑟(𝐵);𝐵𝐵𝐴𝐵𝐴𝐵𝐴𝐴𝐵=𝑟 (33)(x)𝐴≤^∗𝐵⟺𝑟𝐵𝐵𝐴^†𝐵𝐴^†𝐵𝐴𝐴^†𝐵=𝑟(𝐵); (34)(xi)𝐴≤^∗𝐵⟺𝑟𝐵𝐵𝐴^∗𝐵𝐴^∗𝐵𝐴𝐴^∗𝐵=𝑟(𝐵). (35)

Proof. By 𝐴,𝐵∈ℂ^𝑛_EP, it is obvious that 𝐴𝐴^†=𝐴^†𝐴 and 𝐵𝐵^†=𝐵^†𝐵. Applying (i), (ii), and the rank equality in (vii) we obtain 𝑟𝐵^†𝐵𝐴^†𝐴(𝐵)𝐵𝐴=𝑟,𝑟𝐵𝐵^†𝐴𝐴^†𝐵^∗𝐴^∗=𝑟(𝐵); (36) that is, 𝐴≤^∗𝐵. Conversely, suppose that 𝐴≤^∗𝐵. Applying 𝐴−𝐴𝐴^†𝐵=0 and 𝐵^∗𝐵𝐵^†=𝐵^∗, we obtain 𝑟(𝐵)=𝑟𝐵𝐵^†𝐴𝐴^†𝐵𝐴=𝑟𝐵𝐵^†𝐴𝐴^†𝐵𝐴0𝐴^∗−𝐵^∗𝐴𝐴^†=𝑟𝐵𝐵^†𝐴𝐴^†𝐵𝐵𝐴_∗𝐴^∗. (37)
Applying (11), we obtain 𝐵^∗𝐵𝐵^†𝐵=𝐵^∗𝐵 and 𝐵^∗𝐵𝐴^†𝐴=𝐴^∗𝐴 and also (𝐵^∗𝐵)^†𝐵^∗𝐵=𝐵^†𝐵 and (𝐵^∗𝐵)^†𝐴^∗𝐴=𝐴^†𝐴. Then 𝑟𝐵𝐵^†𝐴𝐴^†𝐵𝐵𝐴_∗𝐴^∗𝐵=𝑟^†𝐵𝐴^†𝐴𝐵𝐵𝐴_∗𝐴^∗𝐵≥𝑟^∗𝐵000𝐼_𝑛000𝐼_𝑛𝐵^†𝐵𝐴^†𝐴𝐵𝐵𝐴_∗𝐴^∗𝐵=𝑟^∗𝐵𝐴^∗𝐴𝐵𝐵𝐴_∗𝐴^∗𝐵≥𝑟^∗𝐵^†000𝐼_𝑛000𝐼_𝑛𝐵^∗𝐵𝐴^∗𝐴𝐵𝐵𝐴_∗𝐴^∗𝐵=𝑟^†𝐵𝐴^†𝐴𝐵𝐵𝐴_∗𝐴^∗; (38) that is, 𝑟𝐵𝐵^†𝐴𝐴^†𝐵𝐵𝐴_∗𝐴^∗𝐵=𝑟^∗𝐵𝐴^∗𝐴𝐵𝐵𝐴_∗𝐴^∗. (39) Hence, we have (viii).
Suppose that 𝐴≤^∗𝐵. Since 𝐴,𝐵∈ℂ^𝑛_EP, applying (27), it is easy to check the rank equality in (ix). Conversely, under the rank equality in (ix), we have 𝑟(𝐵)𝑟𝐵𝐵𝐴𝐵𝐴𝐵=𝑟𝐵𝐵𝐴0𝐴𝐵−𝐵𝐴=𝑟⟹𝐴𝐵=𝐵𝐴,𝐵𝐵𝐴𝐴𝐴𝐵=𝑟𝐵0𝐴𝐴𝐵−𝐴𝐴²=𝑟(𝐵)⟹𝐴𝐵=𝐴². (40) Since 𝐴 is EP, there exists a unitary matrix 𝑈₁∈ℂ^𝑛×𝑛 and a nonsingular matrix 𝑇∈ℂ^{𝑟(𝐴)×𝑟(𝐴)} such that 𝐴=𝑈₁𝑈𝑇000^∗1. (41) Correspondingly denote 𝑃⁻¹𝐵𝑃 by 𝐵=𝑈₁𝐵₁𝐵₂𝐵₃𝐵₄𝑈^∗₁, (42) where 𝐵₁∈ℂ^{𝑟(𝐴)×𝑟(𝐴)}. It follows that 𝑇𝐵₁𝑇𝐵₂=𝐵00𝐵¹𝑇0₃,𝑇0𝑇𝐵₁𝑇𝐵₂=𝑇00²0.00 (43) Since 𝑇 is a unitary matrix, 𝐵₁=𝑇,𝐵₂=0,𝐵₃=0. (44) Thus 𝐵=𝑈𝑇00𝐵₄𝑈^∗. (45) Since 𝐵 is EP, 𝐵₄ is EP, and there exists a unitary matrix 𝑈₂∈ℂ^{(𝑛−𝑟(𝐴))×(𝑛−𝑟(𝐴))} and a nonsingular matrix 𝐾∈ℂ^{(𝑟(𝐵)−𝑟(𝐴))×(𝑟(𝐵)−𝑟(𝐴))} such that 𝐵₄=𝑈₂𝑈𝐾000^∗2. (46) Write 𝑈=𝑈₁000𝑈₂. (47) Then 𝐴 and 𝐵 have the form 𝐴=𝑈𝑈𝑇00000000^∗,𝐵=𝑈𝑈𝑇000𝐾0000^∗. (48) 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,𝐵∈ℂ^𝑛_GP with ranks 𝑎 and 𝑏, respectively. It is well known that 𝐴≤^#𝐵⟺𝐴²=𝐴𝐵=𝐵𝐴. (49) In addition, 𝐴≤^#𝐵 if and only if 𝐴 and 𝐵 can be written as 𝐴=𝑃𝑃𝐸00000000⁻¹,𝐵=𝑃𝐸000𝐸^′0𝑃000⁻¹, (50) where 𝐸∈ℂ^𝑎×𝑎 is nonsingular, 𝐸^′∈ℂ^{(𝑏−𝑎)×(𝑏−𝑎)} is nonsingular, and 𝑃∈ℂ^𝑛×𝑛 is nonsingular (see [18]).

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

Theorem 4. Let 𝐴,𝐵∈ℂ^𝑛_GP. Then 𝐴≤^#𝐵⟺𝑟().𝐴𝐵𝐴𝐴𝐵𝐴𝐵𝐴=𝑟𝐴𝐵𝐴 (51)

Proof. Let 𝐴 have the core-nilpotent decomposition (see [19, Exercise 5.10.12]) 𝐴=𝑃𝑃Σ000⁻¹, (52) with nonsingular matrices Σ∈ℂ^{𝑟(𝐴)×𝑟(𝐴)} and 𝑃∈ℂ^𝑛×𝑛. Correspondingly denote 𝑃⁻¹𝐵𝑃 by 𝑃⁻¹𝐵𝐵𝑃=¹𝐵₂𝐵₃𝐵₄, (53) where 𝐵₁∈ℂ^{𝑟(𝐴)×𝑟(𝐴)}. It follows that 𝑟(𝐴𝐵𝐴)=𝑟Σ𝐵₁Σ,𝑟𝐴𝐵𝐴𝐴𝐵𝐴𝐵𝐴=𝑟Σ0𝐵₁Σ00𝐵₃ΣΣ𝐵₁Σ𝐵₂Σ𝐵₁Σ=𝑟Σ0000𝐵₃Σ0Σ𝐵₂Σ𝐵₁Σ−Σ𝐵₁Σ⁻¹𝐵₁Σ=𝑟(Σ)+𝑟0𝐵₃ΣΣ𝐵₂Σ𝐵₁Σ−Σ𝐵₁Σ⁻¹𝐵₁Σ. (54)
Applying (54) to the rank equality in (51), we obtain 𝑟0𝐵₃ΣΣ𝐵₂Σ𝐵₁Σ−Σ𝐵₁Σ⁻¹𝐵₁Σ+𝑟(Σ)=𝑟Σ𝐵₁Σ. (55) Hence 𝑟(Σ𝐵₁Σ)=𝑟(Σ), Σ𝐵₂=0, 𝐵₃Σ=0, and Σ𝐵₁Σ=Σ𝐵₁Σ⁻¹𝐵₁Σ. Since Σ∈ℂ^{𝑟(𝐴)×𝑟(𝐴)} is invertible and 𝐵₁∈ℂ^{𝑟(𝐴)×𝑟(𝐴)}, it follows immediately that 𝑟𝐵₁=𝑟(Σ),𝐵₃=0,𝐵₂=0,𝐵₁=Σ. (56) Therefore 𝐵=𝑃Σ00𝐵₄𝑃⁻¹. (57) Applying 𝐴²Σ=𝑃²0𝑃00⁻¹=𝑃𝑃Σ000⁻¹𝑃Σ00𝐵₄𝑃⁻¹=𝐴𝐵=𝑃Σ00𝐵₄𝑃⁻¹𝑃𝑃Σ000⁻¹=𝐵𝐴, (58) 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).