Solving Optimization Problems on Hermitian Matrix Functions with Applications

Abstract

We consider the extremal inertias and ranks of the matrix expressions 𝑓(𝑋,𝑌)=𝐴₃−𝐵₃𝑋−(𝐵₃𝑋)^∗−𝐶₃𝑌𝐷₃−(𝐶₃𝑌𝐷₃)^∗, where 𝐴₃=𝐴^∗₃,𝐵₃,𝐶₃, and 𝐷₃ are known matrices and 𝑌 and 𝑋 are the solutions to the matrix equations 𝐴₁𝑌=𝐶₁, 𝑌𝐵₁=𝐷₁, and 𝐴₂𝑋=𝐶₂, respectively. As applications, we present necessary and sufficient condition for the previous matrix function 𝑓(𝑋,𝑌) to be positive (negative), non-negative (positive) definite or nonsingular. We also characterize the relations between the Hermitian part of the solutions of the above-mentioned matrix equations. Furthermore, we establish necessary and sufficient conditions for the solvability of the system of matrix equations 𝐴₁𝑌=𝐶₁, 𝑌𝐵₁=𝐷₁, 𝐴₂𝑋=𝐶₂, and 𝐵₃𝑋+(𝐵₃𝑋)^∗+𝐶₃𝑌𝐷₃+(𝐶₃𝑌𝐷₃)^∗=𝐴₃, and give an expression of the general solution to the above-mentioned system when it is solvable.

1. Introduction

Throughout, we denote the field of complex numbers by ℂ, the set of all 𝑚×𝑛 matrices over ℂ by ℂ^𝑚×𝑛, and the set of all 𝑚×𝑚 Hermitian matrices by ℂ_ℎ^𝑚×𝑚. The symbols 𝐴^∗ and ℛ(𝐴) stand for the conjugate transpose, the column space of a complex matrix 𝐴 respectively. 𝐼_𝑛 denotes the 𝑛×𝑛 identity matrix. The Moore-Penrose inverse [1] 𝐴^† of 𝐴, is the unique solution 𝑋 to the four matrix equations: (i)()()𝐴𝑋𝐴=𝐴,ii𝑋𝐴𝑋=𝑋,iii)(𝐴𝑋^∗()=𝐴𝑋,iv)(𝑋𝐴^∗=𝑋𝐴. (1) Moreover, 𝐿_𝐴 and 𝑅_𝐴 stand for the projectors 𝐿_𝐴=𝐼−𝐴^†𝐴,𝑅_𝐴=𝐼−𝐴𝐴^† induced by 𝐴. It is well known that the eigenvalues of a Hermitian matrix 𝐴∈ℂ^𝑛×𝑛 are real, and the inertia of 𝐴 is defined to be the triplet 𝕀_𝑛(𝐴)=𝑖₊(𝐴),𝑖₋(𝐴),𝑖₀(𝐴), (2) where 𝑖₊(𝐴), 𝑖₋(𝐴), and 𝑖₀(𝐴) stand for the numbers of positive, negative, and zero eigenvalues of 𝐴, respectively. The symbols 𝑖₊(𝐴) and 𝑖₋(𝐴) are called the positive index and the negative index of inertia, respectively. For two Hermitian matrices 𝐴 and 𝐵 of the same sizes, we say 𝐴≥𝐵 (𝐴≤𝐵) in the Löwner partial ordering if 𝐴−𝐵 is positive (negative) semidefinite. The Hermitian part of 𝑋 is defined as 𝐻(𝑋)=𝑋+𝑋^∗. We will say that 𝑋 is Re-nnd (Re-nonnegative semidefinite) if 𝐻(𝑋)≥0,𝑋 is Re-pd (Re-positive definite) if 𝐻(𝑋)>0, and 𝑋 is Re-ns if 𝐻(𝑋) is nonsingular.

It is well known that investigation on the solvability conditions and the general solution to linear matrix equations is very active (e.g., [2–9]). In 1999, Braden [10] gave the general solution to 𝐵𝑋+(𝐵𝑋)^∗=𝐴. (3) In 2007, Djordjević [11] considered the explicit solution to (3) for linear bounded operators on Hilbert spaces. Moreover, Cao [12] investigated the general explicit solution to 𝐵𝑋𝐶+(𝐵𝑋𝐶)^∗=𝐴. (4) Xu et al. [13] obtained the general expression of the solution of operator equation (4). In 2012, Wang and He [14] studied some necessary and sufficient conditions for the consistence of the matrix equation 𝐴₁𝑋+𝐴₁𝑋^∗+𝐵₁𝑌𝐶₁+𝐵₁𝑌𝐶₁^∗=𝐸₁ (5) and presented an expression of the general solution to (5).

Note that (5) is a special case of the following system: 𝐴₁𝑌=𝐶₁,𝑌𝐵₁=𝐷₁,𝐴₂𝑋=𝐶₂,𝐵₃𝑋+𝐵₃𝑋^∗+𝐶₃𝑌𝐷₃+𝐶₃𝑌𝐷₃^∗=𝐴₃. (6) To our knowledge, there has been little information about (6). One goal of this paper is to give some necessary and sufficient conditions for the solvability of the system of matrix (6) and present an expression of the general solution to system (6) when it is solvable.

In order to find necessary and sufficient conditions for the solvability of the system of matrix equations (6), we need to consider the extremal ranks and inertias of (10) subject to (13) and (11).

There have been many papers to discuss the extremal ranks and inertias of the following Hermitian expressions: 𝑝(𝑋)=𝐴₃−𝐵₃𝑋−𝐵₃𝑋^∗,𝑔(𝑌)=𝐴−𝐵𝑌𝐶−(𝐵𝑌𝐶)^∗,ℎ(𝑋,𝑌)=𝐴₁−𝐵₁𝑋𝐵^∗₁−𝐶₁𝑌𝐶^∗₁,𝑓(𝑋,𝑌)=𝐴₃−𝐵₃𝑋−𝐵₃𝑋^∗−𝐶₃𝑌𝐷₃−𝐶₃𝑌𝐷₃^∗. (7)(8)(9)(10) Tian has contributed much in this field. One of his works [15] considered the extremal ranks and inertias of (7). He and Wang [16] derived the extremal ranks and inertias of (7) subject to 𝐴₁𝑋=𝐶₁, 𝐴₂𝑋𝐵₂=𝐶₂. Liu and Tian [17] studied the extremal ranks and inertias of (8). Chu et al. [18] and Liu and Tian [19] derived the extremal ranks and inertias of (9). Zhang et al. [20] presented the extremal ranks and inertias of (9), where 𝑋 and 𝑌 are Hermitian solutions of 𝐴₂𝑋=𝐶₂,𝑌𝐵₂=𝐷₂, (11)(12) respectively. He and Wang [16] derived the extremal ranks and inertias of (10). We consider the extremal ranks and inertias of (10) subject to (11) and 𝐴₁𝑌=𝐶₁,𝑌𝐵₁=𝐷₁, (13) which is not only the generalization of the above matrix functions, but also can be used to investigate the solvability conditions for the existence of the general solution to the system (6). Moreover, it can be applied to characterize the relations between Hermitian part of the solutions of (11) and (13).

The remainder of this paper is organized as follows. In Section 2, we consider the extremal ranks and inertias of (10) subject to (11) and (13). In Section 3, we characterize the relations between the Hermitian part of the solution to (11) and (13). In Section 4, we establish the solvability conditions for the existence of a solution to (6) and obtain an expression of the general solution to (6).

2. Extremal Ranks and Inertias of Hermitian Matrix Function (10) with Some Restrictions

In this section, we consider formulas for the extremal ranks and inertias of (10) subject to (11) and (13). We begin with the following Lemmas.

Lemma 1 (see [21]). (a) Let 𝐴₁, 𝐶₁, 𝐵₁, and 𝐷₁ be given. Then the following statements are equivalent:(1)system (13) is consistent,(2)𝑅_𝐴1𝐶₁=0,𝐷₁𝐿_𝐵1=0,𝐴₁𝐷₁=𝐶₁𝐵₁. (14)(3)𝑟𝐴₁𝐶₁=𝑟𝐴₁,𝐷₁𝐵₁=𝑟𝐵₁,𝐴₁𝐷₁=𝐶₁𝐵₁. (15)In this case, the general solution can be written as 𝑌=𝐴^†₁𝐶₁+𝐿_𝐴1𝐷₁𝐵^†₁+𝐿_𝐴1𝑉𝑅_𝐵1, (16) where 𝑉 is arbitrary.
(b) Let 𝐴₂ and 𝐶₂ be given. Then the following statements are equivalent:(1)equation (11) is consistent,(2)𝑅_𝐴2𝐶₂=0, (17)(3)𝑟𝐴₂𝐶₂=𝑟𝐴₂. (18)In this case, the general solution can be written as 𝑋=𝐴^†𝐶+𝐿_𝐴𝑊, (19) where 𝑊 is arbitrary.

Lemma 2 ([22, Lemma 1.5, Theorem 2.3]). Let 𝐴∈ℂ_ℎ^𝑚×𝑚, 𝐵∈ℂ^𝑚×𝑛, and 𝐷∈ℂ_ℎ^𝑛×𝑛, and denote that 𝐵𝑀=𝐴𝐵_∗0,𝑄𝑁=0𝑄_∗0,𝐵𝐿=𝐴𝐵_∗𝐷,𝐺=𝑃𝑀𝐿_𝑁𝐿_𝑁𝑀^∗0. (20) Then one has the following(a)the following equalities hold 𝑖_±(𝑀)=𝑟(𝐵)+𝑖_±𝑅_𝐵𝐴𝑅_𝐵,𝑖_±(𝑁)=𝑟(𝑄), (21)(22)(b)if ℛ(𝐵)⊆ℛ(𝐴), then 𝑖_±(𝐿)=𝑖_±(𝐴)+𝑖_±(𝐷−𝐵^∗𝐴^†𝐵)). Thus 𝑖_±(𝐿)=𝑖_±(𝐴) if and only if ℛ(𝐵)⊆ℛ(𝐴) and 𝑖_±(𝐷−𝐵^∗𝐴^†𝐵)=0,(c)𝑖_±(𝐺)=𝑀𝑃𝑀0_∗0𝑁^∗(𝑁)0𝑁0−𝑟. (23)

Lemma 3 (see [23]). Let 𝐴∈ℂ^𝑚×𝑛, 𝐵∈ℂ^𝑚×𝑘, and 𝐶∈ℂ^𝑙×𝑛. Then they satisfy the following rank equalities:(a)𝑟[𝐴𝐵]=𝑟(𝐴)+𝑟(𝐸_𝐴𝐵)=𝑟(𝐵)+𝑟(𝐸_𝐵𝐴), (b)𝑟^𝐴_𝐶=𝑟(𝐴)+𝑟(𝐶𝐹_𝐴)=𝑟(𝐶)+𝑟(𝐴𝐹_𝐶), (c)𝑟^𝐴𝐵_𝐶0=𝑟(𝐵)+𝑟(𝐶)+𝑟(𝐸_𝐵𝐴𝐹_𝐶), (d)𝑟[𝐵𝐴𝐹_𝐶]=𝑟^𝐵𝐴_0𝐶−𝑟(𝐶), (e)𝑟^𝐶_𝐸𝐵𝐴=𝑟^𝐶0_𝐴𝐵−𝑟(𝐵), (f)𝑟^{𝐴𝐵𝐹𝐷}_{𝐸𝐸𝐶0}=𝑟^{𝐴𝐵0𝐶0𝐸}_0𝐷0−𝑟(𝐷)−𝑟(𝐸),

Lemma 4 (see [15]). Let 𝐴∈ℂ_ℎ^𝑚×𝑚, 𝐵∈ℂ^𝑚×𝑛, 𝐶∈ℂ_ℎ^𝑛×𝑛, 𝑄∈ℂ^𝑚×𝑛, and 𝑃∈ℂ^𝑝×𝑛 be given, and 𝑇∈ℂ^𝑚×𝑚 be nonsingular. Then one has the following (1)𝑖_±(𝑇𝐴𝑇^∗)=𝑖_±(𝐴), (2)𝑖_±^𝐴0_0𝐶=𝑖_±(𝐴)+𝑖_±(𝐶), (3)𝑖_±𝑄^0𝑄∗₀=𝑟(𝑄), (4)𝑖_±^{𝐴𝐵𝐿𝑃}_𝐿𝑃𝐵^∗₀+𝑟(𝑃)=𝑖_±𝐵^𝐴𝐵0∗_0𝑃^∗_0𝑃0.

Lemma 5 (see [22, Lemma  1.4]). Let 𝑆 be a set consisting of (square) matrices over ℂ^𝑚×𝑚, and let 𝐻 be a set consisting of (square) matrices over ℂ_ℎ^𝑚×𝑚. Then Then one has the following (a)𝑆 has a nonsingular matrix if and only if max_𝑋∈𝑆𝑟(𝑋)=𝑚; (b)any 𝑋∈𝑆 is nonsingular if and only if min_𝑋∈𝑆𝑟(𝑋)=𝑚; (c){0}∈𝑆 if and only if min_𝑋∈𝑆𝑟(𝑋)=0; (d)𝑆={0} if and only if max_𝑋∈𝑆𝑟(𝑋)=0;(e)𝐻 has a matrix 𝑋>0(𝑋<0) if and only if max_𝑋∈𝐻𝑖₊(𝑋)=𝑚(max_𝑋∈𝐻𝑖₋(𝑋)=𝑚);(f)any 𝑋∈𝐻 satisfies 𝑋>0(𝑋<0) if and only if min_𝑋∈𝐻𝑖₊(𝑋)=𝑚(min_𝑋∈𝐻𝑖₋(𝑋)=𝑚); (g)𝐻 has a matrix 𝑋≥0(𝑋≤0) if and only if min_𝑋∈𝐻𝑖₋(𝑋)=0(min_𝑋∈𝐻𝑖₊(𝑋)=0); (h)any 𝑋∈𝐻 satisfies 𝑋≥0(𝑋≤0) if and only if max_𝑋∈𝐻𝑖₋(𝑋)=0(max_𝑋∈𝐻𝑖₊(𝑋)=0).

Lemma 6 (see [16]). Let 𝑝(𝑋,𝑌)=𝐴−𝐵𝑋−(𝐵𝑋)^∗−𝐶𝑌𝐷−(𝐶𝑌𝐷)^∗, where 𝐴, 𝐵, 𝐶, and 𝐷 are given with appropriate sizes, and denote that 𝑀₁=𝐵𝐴𝐵𝐶_∗𝐶00_∗,𝑀00₂=𝐴𝐵𝐷^∗𝐵^∗,𝑀00𝐷00₃=𝐴𝐵𝐶𝐷^∗𝐵^∗,𝑀000₄=𝐴𝐵𝐶𝐷^∗𝐵^∗𝐶000_∗,𝑀000₅=𝐴𝐵𝐶𝐷^∗𝐵^∗000.𝐷000 (24) Then one has the following:(1)the maximal rank of 𝑝(𝑋,𝑌) is max_{𝑋∈ℂ𝑛×𝑚,𝑌}𝑟𝑝(𝑋,𝑌)=min𝑚,𝑟𝑀₁,𝑟𝑀₂,𝑟𝑀₃, (25)(2)the minimal rank of 𝑝(𝑋,𝑌) is min_{𝑋∈ℂ𝑛×𝑚,𝑌}𝑟𝑝(𝑀𝑋,𝑌)=2𝑟₃(𝐵)𝑢−2𝑟+max₊+𝑢₋,𝑣₊+𝑣₋,𝑢₊+𝑣₋,𝑢₋+𝑣₊, (26)(3)the maximal inertia of 𝑝(𝑋,𝑌) is max_{𝑋∈ℂ𝑛×𝑚,𝑌}𝑖_±𝑝(𝑋,𝑌)=min𝑖_±𝑀₁,𝑖_±𝑀₂, (27)(4)the minimal inertias of 𝑝(𝑋,𝑌) is min_{𝑋∈ℂ𝑛×𝑚,𝑌}𝑖_±𝑝(𝑋,𝑌)=𝑟𝑀₃(𝐵)𝑖−𝑟+max_±𝑀₁−𝑟𝑀₄,𝑖_±𝑀₂−𝑟𝑀₅, (28)where 𝑢_±=𝑖_±𝑀₁−𝑟𝑀₄,𝑣_±=𝑖_±𝑀₂−𝑟𝑀₅. (29)

Now we present the main theorem of this section.

Theorem 7. Let 𝐴₁∈ℂ^𝑚×𝑛, 𝐶₁∈ℂ^𝑚×𝑘, 𝐵₁∈ℂ^𝑘×𝑙, 𝐷₁∈ℂ^𝑛×𝑙, 𝐴₂∈ℂ^𝑡×𝑞, 𝐶₂∈ℂ^𝑡×𝑝, 𝐴₃∈ℂ_ℎ^𝑝×𝑝, 𝐵₃∈ℂ^𝑝×𝑞,   𝐶₃∈ℂ^𝑝×𝑛, and 𝐷₃∈ℂ^𝑝×𝑛 be given, and suppose that the system of matrix equations (13) and (11) is consistent, respectively. Denote the set of all solutions to (13) by 𝑆 and (11) by 𝐺. Put 𝐸₁=𝐴₃𝐶₃𝐷^∗₃𝐶^∗₁𝐵₃𝐶^∗₂𝐶^∗₃0𝐴^∗₁𝐶00₁𝐷₃𝐴₁𝐵000_∗3000𝐴^∗₂𝐶₂00𝐴₂0,𝐸₂𝐴=𝑟³𝐷^∗₃𝐶₃𝐷₁𝐵₃𝐶^∗₂𝐷₃0𝐵₁𝐷00_∗1𝐶^∗₃𝐵^∗₁𝐵000_∗3000𝐴^∗₂𝐶₂00𝐴₂0,𝐸₃=𝐴₃𝐵₃𝐶^∗₃𝐷^∗₃𝐶^∗₂𝐵^∗₃000𝐴^∗₂𝐷^∗₁𝐶^∗₃00𝐵^∗₁0𝐶₁𝐷₃0𝐴₁𝐶00₂𝐴₂,𝐸000₄=𝐴₃𝐵₃𝐶₃𝐷^∗₃𝐶^∗₂𝐷^∗₃𝐶^∗₁𝐵^∗₃000𝐴^∗₂0𝐶^∗₃0000𝐴^∗₁000𝐵^∗₁𝐶00₁𝐷₃0𝐴₁𝐶000₂𝐴₂,𝐸0000₅=𝐴₃𝐵₃𝐶₃𝐷^∗₃𝐶^∗₂𝐶₃𝐷₁𝐵^∗₃000𝐴^∗₂0𝐷₃0000𝐵₁𝐷^∗₁𝐶^∗₃00𝐴^∗₁0000𝐴₁𝐶000₂𝐴₂.0000 (30) Then one has the following:(a)the maximal rank of (10) subject to (13) and (11) is max_{𝑋∈𝐺,𝑌∈𝑆}𝑟𝑓(𝐸𝑋,𝑌)=min𝑝,𝑟₁−2𝑟𝐴₁−2𝑟𝐴₂,𝑟𝐸₂−2𝑟𝐵₁−2𝑟𝐴₂,𝑟𝐸₃−2𝑟𝐴₂−𝑟𝐴₁−𝑟𝐵₁, (31)(b)the minimal rank of (10) subject to (13) and (11) is min_{𝑋∈𝐺,𝑌∈𝑆}𝑟𝑓(𝐸𝑋,𝑌)=2𝑟₃𝐵−2𝑟³𝐴₂+max𝑟𝐸₁−2𝑟𝐸₄,𝑟𝐸₂−2𝑟𝐸₅,𝑖₊𝐸₁+𝑖₋𝐸₂−𝑟𝐸₄−𝑟𝐸₅,𝑖₋𝐸₁+𝑖₊𝐸₂−𝑟𝐸₄−𝑟𝐸₅, (32)(c)the maximal inertia of (10) subject to (13) and (11) is max_{𝑋∈𝐺,𝑌∈𝑆𝑥}𝑖_±𝑓(𝑋,𝑌)=min𝑖_±𝐸₁−𝑟𝐴₁−𝑟𝐴₂,𝑖_±𝐸₂−𝑟𝐵₁−𝑟𝐴₂, (33)(d)the minimal inertia of (10) subject to (13) and (11) is min_{𝑋∈𝐺,𝑌∈𝑆}𝑖_±𝑓(𝑋,𝑌)=𝑟𝐸₃𝐵−𝑟³𝐴₂+max𝑖_±𝐸₁−𝑟𝐸₄,𝑖_±𝐸₂−𝑟𝐸₅. (34)

Proof. By Lemma 1, the general solutions to (13) and (11) can be written as 𝑋=𝐴^†₂𝐶₂+𝐿_𝐴2𝑊,𝑌=𝐴^†₁𝐶₁+𝐿_𝐴1𝐷₁𝐵^†₁+𝐿_𝐴1𝑍𝑅_𝐵1, (35) where 𝑊 and 𝑍 are arbitrary matrices with appropriate sizes. Put 𝑄=𝐵₃𝐿_𝐴2,𝑇=𝐶₃𝐿_𝐴1,𝐽=𝑅_𝐵1𝐷₃,𝑃=𝐴₃−𝐵₃𝐴^†₂𝐶₂−𝐵₃𝐴^†₂𝐶₂^∗−𝐶₃𝐴^†₁𝐶₁+𝐿_𝐴1𝐷₁𝐵^†₁𝐷₃−𝐶₃𝐴^†₁𝐶₁+𝐿_𝐴1𝐷₁𝐵^†₁𝐷₃^∗. (36) Substituting (36) into (10) yields 𝑓(𝑋,𝑌)=𝑃−𝑄𝑊−(𝑄𝑊)^∗−𝑇𝑍𝐽−(𝑇𝑍𝐽)^∗. (37)
Clearly 𝑃 is Hermitian. It follows from Lemma 6 that max_{𝑋∈𝐺,𝑌∈𝑆}𝑟𝑓(𝑋,𝑌)=max_𝑊,𝑍𝑟𝑃−𝑄𝑊−(𝑄𝑊)^∗−𝑇𝑍𝐽−(𝑇𝑍𝐽)^∗=min𝑚,𝑟𝑁₁,𝑟𝑁₂,𝑟𝑁₃,min_{𝑋∈𝐺,𝑌∈𝑆}𝑟𝑓(𝑋,𝑌)=max_𝑊,𝑍𝑟𝑃−𝑄𝑊−(𝑄𝑊)^∗−𝑇𝑍𝐽−(𝑇𝑍𝐽)^∗=2𝑟𝑁₃(𝑄)𝑠−2𝑟+max₊+𝑠₋,𝑡₊+𝑡₋,𝑠₊+𝑡₋,𝑠₋+𝑡₊,max_{𝑋∈𝐺,𝑌∈𝑆}𝑖_±𝑓(𝑋,𝑌)=max_𝑊,𝑍𝑟𝑃−𝑄𝑊−(𝑄𝑊)^∗−𝑇𝑍𝐽−(𝑇𝑍𝐽)^∗=min𝑖_±𝑁₁,𝑖_±𝑁₂,min_{𝑋∈𝐺,𝑌∈𝑆}𝑖_±𝑓(𝑋,𝑌)=max_𝑊,𝑍𝑟𝑃−𝑄𝑊−(𝑄𝑊)^∗−𝑇𝑍𝐽−(𝑇𝑍𝐽)^∗=𝑟𝑁₃−𝑟(𝑄)+max𝑠_±,𝑡_±, (38)(39)(40)(41) where 𝑁₁=𝑄𝑃𝑄𝑇_∗𝑇00_∗,𝑁00₂=𝑃𝑄𝐽^∗𝑄^∗,𝑁00𝐽00₃=𝑃𝑄𝑇𝐽^∗𝑄^∗,𝑁000₄=𝑃𝑄𝑇𝐽^∗𝑄^∗𝑇000_∗,𝑁000₅=𝑃𝑄𝑇𝐽^∗𝑄^∗,𝑠000𝐽000_±=𝑖_±𝑁₁−𝑟𝑁₄,𝑡_±=𝑖_±𝑁₂−𝑟𝑁₅. (42) Now, we simplify the ranks and inertias of block matrices in (38)–(41).
By Lemma 4, block Gaussian elimination, and noting that 𝐿^∗_𝑆=𝐼−𝑆^†𝑆^∗=𝐼−𝑆^∗𝑆^∗†=𝑅_𝑆^∗, (43) we have the following: 𝑟𝑁₁=𝑟𝑄𝑃𝑄𝑇_∗𝑇00_∗𝐴00=𝑟³𝐶₃𝐷^∗₃𝐶^∗₁𝐵₃𝐶^∗₂𝐶^∗₃0𝐴^∗₁𝐶00₁𝐷₃𝐴₁𝐵000_∗3000𝐴^∗₂𝐶₂00𝐴₂0−2𝑟𝐴₁−2𝑟𝐴₂. (44) By 𝐶₁𝐵₁=𝐴₁𝐷₁, we obtain 𝑟𝑁₂=𝑟𝑃𝑄𝐽^∗𝑄^∗𝐴00𝐽00=𝑟³𝐷^∗₃𝐶₃𝐷₁𝐵₃𝐶^∗₂𝐷₃0𝐵₁𝐷00_∗1𝐶^∗₃𝐵^∗₁𝐵000_∗3000𝐴^∗₂𝐶₂00𝐴₂0−2𝑟𝐵₁−2𝑟𝐴₂,𝑟𝑁₃=𝑟𝑃𝑄𝑇𝐽^∗𝑄^∗𝐴000=𝑟³𝐵₃𝐶^∗₃𝐷^∗₃𝐶^∗₂𝐵^∗₃000𝐴^∗₂𝐷^∗₁𝐶^∗₃00𝐵^∗₁0𝐶₁𝐷₃0𝐴₁𝐶00₂𝐴₂𝐵000−𝑟₁−2𝑟𝐴₂−𝑟𝐴₁,𝑟𝑁₄=𝑟𝑃𝑄𝑇𝐽^∗𝑄^∗𝑇000_∗𝐴000=𝑟³𝐵₃𝐶₃𝐷^∗₃𝐶^∗₂𝐷^∗₃𝐶^∗₁𝐵^∗₃000𝐴^∗₂0𝐶^∗₃0000𝐴^∗₁000𝐵^∗₁𝐶00₁𝐷₃0𝐴₁𝐶000₂𝐴₂𝐵0000−𝑟₁−2𝑟𝐴₂−2𝑟𝐴₁,𝑟𝑁₅=𝑟𝑃𝑄𝑇𝐽^∗𝑄^∗𝑇000_∗𝐴000=𝑟³𝐵₃𝐶₃𝐷^∗₃𝐶^∗₂𝐶₃𝐷₁𝐵^∗₃000𝐴^∗₂0𝐷₃0000𝐵₁𝐷^∗₁𝐶^∗₃00𝐴^∗₁0000𝐴₁𝐶000₂𝐴₂𝐵0000−2𝑟₁−2𝑟𝐴₂−𝑟𝐴₁. (45) By Lemma 2, we can get the following: 𝑖_±𝑁₁=𝑖_±𝑄𝑃𝑄𝑇_∗𝑇00_∗00=𝑖_±𝐴₃𝐶₃𝐷^∗₃𝐶^∗₁𝐵₃𝐶^∗₂𝐶^∗₃0𝐴^∗₁𝐶00₁𝐷₃𝐴₁𝐵000_∗3000𝐴^∗₂𝐶₂00𝐴₂0−𝑟𝐴₁−𝑟𝐴₂,𝑖_±𝑁₂=𝑖_±𝑃𝑄𝐽^∗𝑄^∗00𝐽00=𝑖_±𝐴₃𝐷^∗₃𝐶₃𝐷₁𝐵₃𝐶^∗₂𝐷₃0𝐵₁𝐷00_∗1𝐶^∗₃𝐵^∗₁𝐵000_∗3000𝐴^∗₂𝐶₂00𝐴₂0−𝑟𝐵₁−𝑟𝐴₂. (46)(47) Substituting (44)-(47) into (38) and (41) yields (31)–(34), respectively.

Corollary 8. Let 𝐴₁, 𝐶₁, 𝐵₁, 𝐷₁, 𝐴₂, 𝐶₂, 𝐴₃, 𝐵₃, 𝐶₃, 𝐷₃, and 𝐸_𝑖, (𝑖=1,2,…,5) be as in Theorem 7, and suppose that the system of matrix equations (13) and (11) is consistent, respectively. Denote the set of all solutions to (13) by 𝑆 and (11) by 𝐺. Then, one has the following:(a)there exist 𝑋∈𝐺 and 𝑌∈𝑆 such that 𝐴₃−𝐵₃𝑋−(𝐵₃𝑋)^∗−𝐶₃𝑌𝐷₃−(𝐶₃𝑌𝐷₃)^∗>0 if and only if 𝑖₊𝐸₁−𝑟𝐴₁−𝑟𝐴₂𝑖≥𝑝,₊𝐸₂−𝑟𝐵₁−𝑟𝐴₂≥𝑝. (48)(b)there exist 𝑋∈𝐺 and 𝑌∈𝑆 such that 𝐴₃−𝐵₃𝑋−(𝐵₃𝑋)^∗−𝐶₃𝑌𝐷₃−(𝐶₃𝑌𝐷₃)^∗<0 if and only if 𝑖₋𝐸₁−𝑟𝐴₁−𝑟𝐴₂𝑖≥𝑝,₋𝐸₂−𝑟𝐵₁−𝑟𝐴₂≥𝑝, (49)(c)there exist 𝑋∈𝐺 and 𝑌∈𝑆 such that 𝐴₃−𝐵₃𝑋−(𝐵₃𝑋)^∗−𝐶₃𝑌𝐷₃−(𝐶₃𝑌𝐷₃)^∗≥0 if and only if 𝑟𝐸₃𝐵−𝑟³𝐴₂+𝑖₋𝐸₁−𝑟𝐸₄𝑟𝐸≤0,₃𝐵−𝑟³𝐴₂+𝑖₋𝐸₂−𝑟𝐸₅≤0. (50)(d)there exist 𝑋∈𝐺 and 𝑌∈𝑆 such that 𝐴₃−𝐵₃𝑋−(𝐵₃𝑋)^∗−𝐶₃𝑌𝐷₃−(𝐶₃𝑌𝐷₃)^∗≤0 if and only if 𝑟𝐸₃𝐵−𝑟³𝐴₂+𝑖₊𝐸₁−𝑟𝐸₄𝑟𝐸≤0,₃𝐵−𝑟³𝐴₂+𝑖₊𝐸₂−𝑟𝐸₅≤0, (51)(e)𝐴₃−𝐵₃𝑋−(𝐵₃𝑋)^∗−𝐶₃𝑌𝐷₃−(𝐶₃𝑌𝐷₃)^∗>0 for all 𝑋∈𝐺 and 𝑌∈𝑆 if and only if 𝑟𝐸₃𝐵−𝑟³𝐴₂+𝑖₊𝐸₁−𝑟𝐸₄𝐸=𝑝𝑜𝑟𝑟₃𝐵−𝑟³𝐴₂+𝑖₊𝐸₂−𝑟𝐸₅=𝑝, (52)(f)𝐴₃−𝐵₃𝑋−(𝐵₃𝑋)^∗−𝐶₃𝑌𝐷₃−(𝐶₃𝑌𝐷₃)^∗<0 for all 𝑋∈𝐺 and 𝑌∈𝑆 if and only if 𝑟𝐸₃𝐵−𝑟³𝐴₂+𝑖₋𝐸₁−𝑟𝐸₄𝐸=𝑝𝑜𝑟𝑟₃𝐵−𝑟³𝐴₂+𝑖₋𝐸₂−𝑟𝐸₅=𝑝, (53)(g)𝐴₃−𝐵₃𝑋−(𝐵₃𝑋)^∗−𝐶₃𝑌𝐷₃−(𝐶₃𝑌𝐷₃)^∗≥0 for all 𝑋∈𝐺 and 𝑌∈𝑆 if and only if 𝑖₋𝐸₁−𝑟𝐴₁−𝑟𝐴₂≤0𝑜𝑟𝑖₋𝐸₂−𝑟𝐵₁−𝑟𝐴₂≤0, (54)(h)𝐴₃−𝐵₃𝑋−(𝐵₃𝑋)^∗−𝐶₃𝑌𝐷₃−(𝐶₃𝑌𝐷₃)^∗≤0 for all 𝑋∈𝐺 and 𝑌∈𝑆 if and only if 𝑖₊𝐸₁−𝑟𝐴₁−𝑟𝐴₂≤0𝑜𝑟𝑖₊𝐸₂−𝑟𝐵₁−𝑟𝐴₂≤0, (55)(i)there exist 𝑋∈𝐺 and 𝑌∈𝑆 such that 𝐴₃−𝐵₃𝑋−(𝐵₃𝑋)^∗−𝐶₃𝑌𝐷₃−(𝐶₃𝑌𝐷₃)^∗ is nonsingular if and only if 𝑟𝐸₁−2𝑟𝐴₁−2𝑟𝐴₂𝑟𝐸≥𝑝,₂−2𝑟𝐵₁−2𝑟𝐴₂𝑟𝐸≥𝑝,₃−2𝑟𝐴₂−𝑟𝐴₁−𝑟𝐵₁≥𝑝. (56)

3. Relations between the Hermitian Part of the Solutions to (13) and (11)

Now we consider the extremal ranks and inertias of the difference between the Hermitian part of the solutions to (13) and (11).

Theorem 9. Let 𝐴₁∈ℂ^𝑚×𝑝, 𝐶₁∈ℂ^𝑚×𝑝, 𝐵₁∈ℂ^𝑝×𝑙, 𝐷₁∈ℂ^𝑝×𝑙, 𝐴₂∈ℂ^𝑡×𝑝, and 𝐶₂∈ℂ^𝑡×𝑝, be given. Suppose that the system of matrix equations (13) and (11) is consistent, respectively. Denote the set of all solutions to (13) by 𝑆 and (11) by 𝐺. Put 𝐻₁=0𝐼𝐶^∗₁−𝐼𝐶^∗₂𝐼0𝐴^∗₁𝐶00₁𝐴₁000−𝐼000𝐴^∗₂𝐶₂00𝐴₂0,𝐻₂=𝑟0𝐼𝐷₁−𝐼𝐶^∗₂𝐼0𝐵₁𝐷00_∗1𝐵^∗₁000−𝐼000𝐴^∗₂𝐶₂00𝐴₂0,𝐻₃=0−𝐼𝐼𝐼𝐶^∗₂−𝐼000𝐴^∗₂𝐷^∗₁00𝐵^∗₁0𝐶₁0𝐴₁𝐶00₂𝐴₂,𝐻000₄=0−𝐼𝐼𝐼𝐶^∗₂𝐶^∗₁−𝐼000𝐴^∗₂0𝐼0000𝐴^∗₁000𝐵^∗₁𝐶00₁0𝐴₁𝐶000₂𝐴₂,𝐻0000₅=0−𝐼𝐼𝐼𝐶^∗₂𝐷₁−𝐼000𝐴^∗₂0𝐼0000𝐵₁𝐷^∗₁00𝐴^∗₁0000𝐴₁𝐶000₂𝐴₂.0000 (57) Then one has the following: max_{𝑋∈𝐺,𝑌∈𝑆}𝑟𝑋+𝑋^∗−𝑌+𝑌^∗=min𝑝,𝑟𝐻₁−2𝑟𝐴₁−2𝑟𝐴₂,𝑟𝐻₂−2𝑟𝐵₁−2𝑟𝐴₂,𝑟𝐻₃−2𝑟𝐴₂−𝑟𝐴₁−𝑟𝐵₁,min_{𝑋∈𝐺,𝑌∈𝑆}𝑟𝑋+𝑋^∗−𝑌+𝑌^∗=2𝑟𝐻₃𝑟𝐻−2𝑝+max₁−2𝑟𝐻₄,𝑟𝐻₂−2𝑟𝐻₅,𝑖₊𝐻₁+𝑖₋𝐻₂−𝑟𝐻₄−𝑟𝐻₅,𝑖₋𝐻₁+𝑖₊𝐻₂−𝑟𝐻₄−𝑟𝐻₅,max_{𝑋∈𝐺,𝑌∈𝑆}𝑖_±𝑋+𝑋^∗−𝑌+𝑌^∗=min𝑖_±𝐻₁−𝑟𝐴₁−𝑟𝐴₂,𝑖_±𝐻₂−𝑟𝐵₁−𝑟𝐴₂,min_{𝑋∈𝐺,𝑌∈𝑆}𝑖_±𝑋+𝑋^∗−𝑌+𝑌^∗=𝑟𝐸₃−𝑝+max𝑖_±𝐻₁−𝑟𝐻₄,𝑖_±𝐻₂−𝑟𝐻₅. (58)

Proof. By letting 𝐴₃=0, 𝐵₃=−𝐼, 𝐶₃=𝐼, and 𝐷₃=𝐼 in Theorem 7, we can get the results.

Corollary 10. Let 𝐴₁, 𝐶₁, 𝐵₁, 𝐷₁, 𝐴₂, 𝐶₂, and 𝐻_𝑖, (𝑖=1,2,…,5) be as in Theorem 9, and suppose that the system of matrix equations (13) and (11) is consistent, respectively. Denote the set of all solutions to (13) by 𝑆 and (11) by 𝐺. Then, one has the following:(a)there exist 𝑋∈𝐺 and 𝑌∈𝑆 such that (𝑋+𝑋^∗)>(𝑌+𝑌^∗) if and only if 𝑖₊𝐻₁−𝑟𝐴₁−𝑟𝐴₂𝑖≥𝑝,₊𝐻₂−𝑟𝐵₁−𝑟𝐴₂≥𝑝. (59)(b)there exist 𝑋∈𝐺 and 𝑌∈𝑆 such that (𝑋+𝑋^∗)<(𝑌+𝑌^∗) if and only if 𝑖₋𝐻₁−𝑟𝐴₁−𝑟𝐴₂𝑖≥𝑝,₋𝐻₂−𝑟𝐵₁−𝑟𝐴₂≥𝑝, (60)(c)there exist 𝑋∈𝐺 and 𝑌∈𝑆 such that (𝑋+𝑋^∗)≥(𝑌+𝑌^∗) if and only if 𝑟𝐻₃−𝑝+𝑖₋𝐻₁−𝑟𝐻₄𝑟𝐻≤0,₃−𝑝+𝑖₋𝐻₁−𝑟𝐻₄≤0, (61)(d)there exist 𝑋∈𝐺 and 𝑌∈𝑆 such that (𝑋+𝑋^∗)≤(𝑌+𝑌^∗) if and only if 𝑟𝐻₃−𝑝+𝑖₊𝐻₁−𝑟𝐻₄𝑟𝐻≤0,₃−𝑝+𝑖₊𝐻₂−𝑟𝐻₅≤0, (62)(e)(𝑋+𝑋^∗)>(𝑌+𝑌^∗) for all 𝑋∈𝐺 and 𝑌∈𝑆 if and only if 𝑟𝐻₃−𝑝+𝑖₊𝐻₁−𝑟𝐻₄𝐻=𝑝𝑜𝑟𝑟₃−𝑝+𝑖₊𝐻₂−𝑟𝐻₅=𝑝, (63)(f)(𝑋+𝑋^∗)<(𝑌+𝑌^∗) for all 𝑋∈𝐺 and 𝑌∈𝑆 if and only if 𝑟𝐻₃−𝑝+𝑖₋𝐻₁−𝑟𝐻₄𝐻=𝑝𝑜𝑟𝑟₃−𝑝+𝑖₋𝐻₂−𝑟𝐻₅=𝑝, (64)(g)(𝑋+𝑋^∗)≥(𝑌+𝑌^∗) for all 𝑋∈𝐺 and 𝑌∈𝑆 if and only if 𝑖₋𝐻₁−𝑟𝐴₁−𝑟𝐴₂≤0𝑜𝑟𝑖₋𝐻₂−𝑟𝐵₁−𝑟𝐴₂≤0, (65)(h)(𝑋+𝑋^∗)≤(𝑌+𝑌^∗) for all 𝑋∈𝐺 and 𝑌∈𝑆 if and only if 𝑖₊𝐻₁−𝑟𝐴₁−𝑟𝐴₂≤0𝑜𝑟𝑖₊𝐻₂−𝑟𝐵₁−𝑟𝐴₂≤0, (66)(i)there exist 𝑋∈𝐺 and 𝑌∈𝑆 such that (𝑋+𝑋^∗)−(𝑌+𝑌^∗) is nonsingular if and only if 𝑟𝐻₁−2𝑟𝐴₁−2𝑟𝐴₂𝑟𝐻≥𝑝,₂−2𝑟𝐵₁−2𝑟𝐴₂𝑟𝐻≥𝑝,₃−2𝑟𝐴₂−𝑟𝐴₁−𝑟𝐵₁≥𝑝. (67)

4. The Solvability Conditions and the General Solution to System (6)

We now turn our attention to (6). We in this section use Theorem 9 to give some necessary and sufficient conditions for the existence of a solution to (6) and present an expression of the general solution to (6). We begin with a lemma which is used in the latter part of this section.

Lemma 11 (see [14]). Let 𝐴₁∈ℂ^𝑚×𝑛1, 𝐵₁∈ℂ^𝑚×𝑛2, 𝐶₁∈ℂ^𝑞×𝑚, and 𝐸₁∈ℂ_ℎ^𝑚×𝑚 be given. Let 𝐴=𝑅_𝐴1𝐵₁, 𝐵=𝐶₁𝑅_𝐴1, 𝐸=𝑅_𝐴1𝐸₁𝑅_𝐴1, 𝑀=𝑅_𝐴𝐵^∗, 𝑁=𝐴^∗𝐿_𝐵, and 𝑆=𝐵^∗𝐿_𝑀. Then the following statements are equivalent:(1)equation (5) is consistent,(2)𝑅_𝑀𝑅_𝐴𝐸=0,𝑅_𝐴𝐸𝑅_𝐴=0,𝐿_𝐵𝐸𝐿_𝐵=0, (68)(3)𝑟𝐸₁𝐵₁𝐶^∗₁𝐴₁𝐴^∗₁𝐵000=𝑟₁𝐶^∗₁𝐴₁+𝑟𝐴₁,𝑟𝐸₁𝐵₁𝐴₁𝐴^∗₁𝐵00_∗1𝐵00=2𝑟₁𝐴₁,𝑟𝐸₁𝐶^∗₁𝐴₁𝐴^∗₁𝐶00₁𝐶00=2𝑟^∗₁𝐴₁. (69)In this case, the general solution of (5) can be expressed as 1𝑌=2𝐴^†𝐸𝐵^†−𝐴^†𝐵^∗𝑀^†𝐸𝐵^†−𝐴^†𝑆𝐵^†∗𝐸𝑁^†𝐴^∗𝐵^†+𝐴^†𝐸𝑀^†∗+𝑁^†∗𝐸𝐵^†𝑆^†𝑆+𝐿_𝐴𝑉₁+𝑉₂𝑅_𝐵+𝑈₁𝐿_𝑆𝐿_𝑀+𝑅_𝑁𝑈^∗₂𝐿_𝑀−𝐴^†𝑆𝑈₂𝑅_𝑁𝐴^∗𝐵^†,𝑋=𝐴^†₁𝐸₁−𝐵₁𝑌𝐶₁−𝐵₁𝑌𝐶₁^∗−12𝐴^†₁𝐸₁−𝐵₁𝑌𝐶₁−𝐵₁𝑌𝐶₁^∗𝐴₁𝐴^†₁−𝐴^†₁𝑊₁𝐴^∗₁+𝑊^∗₁𝐴₁𝐴^†₁+𝐿_𝐴1𝑊₂, (70) where 𝑈₁, 𝑈₂, 𝑉₁, 𝑉₂, 𝑊₁, and 𝑊₂ are arbitrary matrices over ℂ with appropriate sizes.

Now we give the main theorem of this section.

Theorem 12. Let 𝐴_𝑖, 𝐶_𝑖, (𝑖=1,2,3), 𝐵_𝑗, and 𝐷_𝑗, (𝑗=1,3) be given. Set 𝐴=𝐵₃𝐿_𝐴2,𝐵=𝐶₃𝐿_𝐴1,𝐶=𝑅_𝐵1𝐷₃,𝐹=𝑅_𝐴𝐵,𝐺=𝐶𝑅_𝐴,𝑀=𝑅_𝐹𝐺^∗,𝑁=𝐹^∗𝐿_𝐺,𝑆=𝐺^∗𝐿_𝑀,𝐷=𝐴₃−𝐵₃𝐴^†₂𝐶₂−𝐵₃𝐴^†₂𝐶₂^∗−𝐶₃𝐴^†₁𝐶₁+𝐿_𝐴1𝐷₁𝐵^†₁𝐷₃−𝐷^∗₃𝐴^†₁𝐶₁+𝐿_𝐴1𝐷₁𝐵^†₁^∗𝐶^∗₃,𝐸=𝑅_𝐴𝐷𝑅_𝐴. (71)(72)(73) Then the following statements are equivalent:(1)system (6) is consistent,(2)the equalities in (14) and (17) hold, and 𝑅_𝑀𝑅_𝐹𝐸=0,𝑅_𝐹𝐸𝑅_𝐹=0,𝐿_𝐺𝐸𝐿_𝐺=0, (74)(3)the equalities in (15) and (18) hold, and 𝑟𝐴₃𝐶₃𝐷^∗₃𝐵₃𝐶^∗₂𝐵^∗₃000𝐴^∗₂𝐶₁𝐷₃𝐴₁𝐷000_∗1𝐶^∗₃0𝐵^∗₁𝐶00₂00𝐴₂0𝐶=𝑟³𝐷^∗₃𝐵₃𝐴₁000𝐵^∗₁000𝐴₂𝐴+𝑟²𝐵₃,𝑟𝐴₃𝐶₃𝐵₃𝐷^∗₃𝐶^∗₁𝐶^∗₂𝐶^∗₃00𝐴^∗₁0𝐵^∗₃000𝐴^∗₂𝐶₁𝐷₃𝐴₁𝐶000₂0𝐴₂𝐶00=2𝑟³𝐵₃𝐴₁00𝐴₂,𝑟𝐴₃𝐷^∗₃𝐵₃𝐶₃𝐷₁𝐶^∗₂𝐷₃00𝐵₁0𝐵^∗₃000𝐴^∗₂𝐷^∗₁𝐶^∗₃𝐵^∗₁𝐶000₂0𝐴₂𝐷00=2𝑟^∗3𝐵₃𝐵^∗₁00𝐴₂. (75)In this case, the general solution of system (6) can be expressed as 𝑋=𝐴^†₂𝐶₂+𝐿_𝐴2𝑈,𝑌=𝐴^†₁𝐶₁+𝐿_𝐴1𝐷₁𝐵^†₁+𝐿_𝐴1𝑉𝑅_𝐵1, (76) where 1𝑉=2𝐹^†𝐸𝐺^†−𝐹^†𝐺^∗𝑀^†𝐸𝐺^†−𝐹^†𝑆𝐺^†∗𝐸𝑁^†𝐹^∗𝐺^†+𝐹^†𝐸𝑀^†∗+𝑁^†∗𝐸𝐺^†𝑆^†𝑆+𝐿_𝐹𝑉₁+𝑉₂𝑅_𝐺+𝑈₁𝐿_𝑆𝐿_𝑀+𝑅_𝑁𝑈^∗₂𝐿_𝑀−𝐹^†𝑆𝑈₂𝑅_𝑁𝐹^∗𝐺^†,𝑈=𝐴^†𝐷−𝐵𝑉𝐶−(𝐵𝑉𝐶)^∗−12𝐴^†𝐷−𝐵𝑉𝐶−(𝐵𝑉𝐶)^∗𝐴𝐴^†−𝐴^†𝑊₁𝐴^∗+𝑊^∗₁𝐴𝐴^†+𝐿_𝐴𝑊₂, (77) where 𝑈₁, 𝑈₂, 𝑉₁, 𝑉₂, 𝑊₁, and 𝑊₂ are arbitrary matrices over ℂ with appropriate sizes.

Proof. (2) ⇔ (3): Applying Lemma 3 and Lemma 11 gives 𝑅_𝑀𝑅_𝐹𝐸=0⟺𝑟𝑅_𝑀𝑅_𝐹𝐸=0⟺𝑟𝐷𝐵𝐶^∗𝐴𝐴^∗000=𝑟𝐵𝐶^∗𝐴+𝑟(𝐴)⟺𝑟𝐷𝐶₃𝐷^∗₃𝐵₃0𝐵^∗₃000𝐴^∗₂0𝐴₁00000𝐵^∗₁00000𝐴₂0𝐶=𝑟³𝐷^∗₃𝐵₃𝐴₁000𝐵^∗₁000𝐴₂𝐴+𝑟²𝐵₃𝐴⟺𝑟³𝐶₃𝐷^∗₃𝐵₃𝐶^∗₂𝐵^∗₃000𝐴^∗₂𝐶₁𝐷₃𝐴₁𝐷000_∗1𝐶^∗₃0𝐵^∗₁𝐶00₂00𝐴₂0𝐶=𝑟³𝐷^∗₃𝐵₃𝐴₁000𝐵^∗₁000𝐴₂𝐴+𝑟²𝐵₃. (78) By a similar approach, we can obtain that 𝑅_𝐹𝐸𝑅_𝐹𝐴=0⟺𝑟³𝐶₃𝐵₃𝐷^∗₃𝐶^∗₁𝐶^∗₂𝐶^∗₃00𝐴^∗₁0𝐵^∗₃000𝐴^∗₂𝐶₁𝐷₃𝐴₁𝐶000₂0𝐴₂𝐶00=2𝑟³𝐵₃𝐴₁00𝐴₂,𝐿_𝐺𝐸𝐿_𝐺𝐴=0⟺𝑟³𝐷^∗₃𝐵₃𝐶₃𝐷₁𝐶^∗₂𝐷₃00𝐵₁0𝐵^∗₃000𝐴^∗₂𝐷^∗₁𝐶^∗₃𝐵^∗₁𝐶000₂0𝐴₂𝐷00=2𝑟^∗3𝐵₃𝐵^∗₁00𝐴₂. (79) (1) ⇔ (2): We separate the four equations in system (6) into three groups: 𝐴₁𝑌=𝐶₁,𝑌𝐵₁=𝐷₁,𝐴₂𝑋=𝐶₂,𝐵₃𝑋+𝐵₃𝑋^∗+𝐶₃𝑌𝐷₃+𝐶₃𝑌𝐷₃^∗=𝐴₃. (80)(81)(82) By Lemma 1, we obtain that system (80) is solvable if and only if (14), (81) is consistent if and only if (17). The general solutions to system (80) and (81) can be expressed as (16) and (19), respectively. Substituting (16) and (19) into (82) yields 𝐴𝑈+(𝐴𝑈)^∗+𝐵𝑉𝐶+(𝐵𝑉𝐶)^∗=𝐷. (83) Hence, the system (5) is consistent if and only if (80), (81), and (83) are consistent, respectively. It follows from Lemma 11 that (83) is solvable if and only if 𝑅_𝑀𝑅_𝐹𝐸=0,𝑅_𝐹𝐸𝑅_𝐹=0,𝐿_𝐺𝐸𝐿_𝐺=0. (84) We know by Lemma 11 that the general solution of (83) can be expressed as (77).

In Theorem 12, let 𝐴₁ and 𝐷₁ vanish. Then we can obtain the general solution to the following system: 𝐴₂𝑋=𝐶₂,𝑌𝐵₁=𝐷₁,𝐵₃𝑋+𝐵₃𝑋^∗+𝐶₃𝑌𝐷₃+𝐶₃𝑌𝐷₃^∗=𝐴₃. (85)

Corollary 13. Let 𝐴₂, 𝐶₂, 𝐵₁, 𝐷₁, 𝐵₃, 𝐶₃, 𝐷₃, and 𝐴₃=𝐴^∗₃ be given. Set 𝐴=𝐵₃𝐿_𝐴2,𝐶=𝑅_𝐵1𝐷₃,𝐹=𝑅_𝐴𝐶₃,𝐺=𝐶𝑅_𝐴,𝑀=𝑅_𝐹𝐺^∗,𝑁=𝐹^∗𝐿_𝐺,𝑆=𝐺^∗𝐿_𝑀,𝐷=𝐴₃−𝐵₃𝐴^†₂𝐶₂−𝐵₃𝐴^†₂𝐶₂^∗−𝐶₃𝐷₁𝐵^†₁𝐷₃−𝐶₃𝐷₁𝐵^†₁𝐷₃^∗,𝐸=𝑅_𝐴𝐷𝑅_𝐴. (86) Then the following statements are equivalent:(1)system (85) is consistent(2)𝑅_𝐴2𝐶₂=0,𝐷₁𝐿_𝐵1=0,𝑅_𝑀𝑅_𝐹𝑅𝐸=0,_𝐹𝐸𝑅_𝐹=0,𝐿_𝐺𝐸𝐿_𝐺=0, (87)(88)(3)𝑟𝐴₂𝐶₂=𝑟𝐴₂,𝐷₁𝐵₁=𝑟𝐵₁,𝑟𝐴₃𝐶₃𝐷^∗₃𝐵₃𝐶^∗₂𝐵^∗₃000𝐴^∗₂𝐷^∗₁𝐶^∗₃0𝐵^∗₁𝐶00₂00𝐴₂0𝐶=𝑟³𝐷^∗₃𝐵₃0𝐵^∗₁000𝐴₂𝐴+𝑟²𝐵₃,𝑟𝐴₃𝐶₃𝐵₃𝐶^∗₂𝐶^∗₃𝐵000_∗300𝐴^∗₂𝐶₂0𝐴₂0𝐶=2𝑟³𝐵₃0𝐴₂,𝑟𝐴₃𝐷^∗₃𝐵₃𝐶₃𝐷₁𝐶^∗₂𝐷₃00𝐵₁0𝐵^∗₃000𝐴^∗₂𝐷^∗₁𝐶^∗₃𝐵^∗₁𝐶000₂0𝐴₂𝐷00=2𝑟^∗3𝐵₃𝐵^∗₁00𝐴₂. (89)In this case, the general solution of system (6) can be expressed as 𝑋=𝐴^†₂𝐶₂+𝐿_𝐴2𝑈,𝑌=𝐷₁𝐵^†₁+𝑉𝑅_𝐵1, (90) where 1𝑉=2𝐹^†𝐸𝐺^†−𝐹^†𝐺^∗𝑀^†𝐸𝐺^†−𝐹^†𝑆𝐺^†∗𝐸𝑁^†𝐹^∗𝐺^†+𝐹^†𝐸𝑀^†∗+𝑁^†∗𝐸𝐺^†𝑆^†𝑆+𝐿_𝐹𝑉₁+𝑉₂𝑅_𝐺+𝑈₁𝐿_𝑆𝐿_𝑀+𝑅_𝑁𝑈^∗₂𝐿_𝑀−𝐹^†𝑆𝑈₂𝑅_𝑁𝐹^∗𝐺^†,𝑈=𝐴^†𝐷−𝐶₃𝑉𝐶−𝐶₃𝑉𝐶^∗−12𝐴^†𝐷−𝐶₃𝑉𝐶−𝐶₃𝑉𝐶^∗𝐴𝐴^†−𝐴^†𝑊₁𝐴^∗+𝑊^∗₁𝐴𝐴^†+𝐿_𝐴𝑊₂, (91) where 𝑈₁, 𝑈₂, 𝑉₁, 𝑉₂, 𝑊₁, and 𝑊₂ are arbitrary matrices over ℂ with appropriate sizes.

Acknowledgments

The authors would like to thank Dr. Mohamed, Dr. Sivakumar, and a referee very much for their valuable suggestions and comments, which resulted in a great improvement of the original paper. This research was supported by the Grants from the National Natural Science Foundation of China (NSFC (11161008)) and Doctoral Program Fund of Ministry of Education of P.R.China (20115201110002).