We establish the formulas of the maximal and minimal ranks of the quaternion Hermitian matrix expression C4−A4XA4∗ where X is a Hermitian solution to quaternion matrix equations A1X=C1, XB1=C2, and A3XA3*=C3. As applications, we give a new necessary and sufficient condition for the existence of Hermitian solution to the system of matrix equations A1X=C1, XB1=C2, A3XA3*=C3, and A4XA4*=C4, which was investigated by Wang and Wu, 2010, by rank equalities. In addition, extremal ranks of the generalized Hermitian Schur complement C4−A4A3~A4∗ with respect to a Hermitian g-inverse A3~
of A3, which is a common solution to quaternion matrix equations A1X=C1 and XB1=C2, are also considered.

1. Introduction

Throughout this paper, we denote the real number field by ℝ, the complex number field by ℂ, the set of all m×n matrices over the quaternion algebra
(1)ℍ={a0+a1i+a2j+a3k∣i2=j2a=k2=ijk=-1,a0,a1,a2,a3∈ℝ}
by ℍm×n, the identity matrix with the appropriate size by I, the column right space, the row left space of a matrix A over ℍ by ℛ(A), 𝒩(A), respectively, the dimension of ℛ(A) by dimℛ(A), a Hermitian g-inverse of a matrix A by X=A∽ which satisfies AA∽A=A and X=X*, and the Moore-Penrose inverse of matrix A over ℍ by A† which satisfies four Penrose equations AA†A=A,A†AA†=A†,(AA†)*=AA†,and(A†A)*=A†A. In this case A† is unique and (A†)*=(A*)†. Moreover, RA and LA stand for the two projectors LA=I-A†A, RA=I-AA†induced by A. Clearly, RA and LA are idempotent, Hermitian and RA=LA*. By [1], for a quaternion matrix A, dimℛ(A)=dim𝒩(A). dimℛ(A) is called the rank of a quaternion matrix A and denoted by r(A).

Mitra [2] investigated the system of matrix equations
(2)A1X=C1,XB1=C2.

Khatri and Mitra [3] gave necessary and sufficient conditions for the existence of the common Hermitian solution to (2) and presented an explicit expression for the general Hermitian solution to (2) by generalized inverses. Using the singular value decomposition (SVD), Yuan [4] investigated the general symmetric solution of (2) over the real number field ℝ. By the SVD, Dai and Lancaster [5] considered the symmetric solution of equation
(3)AXA*=C
over ℝ, which was motivated and illustrated with an inverse problem of vibration theory. Groß [6], Tian and Liu [7] gave the solvability conditions for Hermitian solution and its expressions of (3) over ℂ in terms of generalized inverses, respectively. Liu, Tian and Takane [8] investigated ranks of Hermitian and skew-Hermitian solutions to the matrix equation (3). By using the generalized SVD, Chang and Wang [9] examined the symmetric solution to the matrix equations
(4)A3XA3*=C3,A4XA4*=C4
over ℝ. Note that all the matrix equations mentioned above are special cases of
(5)A1X=C1,XB1=C2,A3XA3*=C3,A4XA4*=C4.
Wang and Wu [10] gave some necessary and sufficient conditions for the existence of the common Hermitian solution to (5) for operators between Hilbert C*-modules by generalized inverses and range inclusion of matrices. In view of the complicated computations of the generalized inverses of matrices, we naturally hope to establish a more practical, necessary, and sufficient condition for system (5) over quaternion algebra to have Hermitian solution by rank equalities.

As is known to us, solutions to matrix equations and ranks of solutions to matrix equations have been considered previously by many authors [10–34], and extremal ranks of matrix expressions can be used to characterize their rank invariance, nonsingularity, range inclusion, and solvability conditions of matrix equations. Tian and Cheng [35] investigated the maximal and minimal ranks of A-BXC with respect to X with applications; Tian [36] gave the maximal and minimal ranks of A1-B1XC1 subject to a consistent matrix equation B2XC2=A2.Tian and Liu [7] established the solvability conditions for (4) to have a Hermitian solution over ℂ by the ranks of coefficient matrices. Wang and Jiang [20] derived extreme ranks of (skew)Hermitian solutions to a quaternion matrix equation AXA*+BYB*=C. Wang, Yu and Lin [31] derived the extremal ranks of C4-A4XB4 subject to a consistent system of matrix equations
(6)A1X=C1,XB1=C2,A3XB3=C3
over ℍ and gave a new solvability condition to system
(7)A1X=C1,XB1=C2,A3XB3=C3,A4XB4=C4.

In matrix theory and its applications, there are many matrix expressions that have symmetric patterns or involve Hermitian (skew-Hermitian) matrices. For example,
(8)A-BXB*,A-BX±X*B*,A-BXB*-CYC*,A-BXC±(BXC)*,
where A=±A*,B, and C are given and X and Y are variable matrices. In recent papers [7, 8, 37, 38], Liu and Tian considered some maximization and minimization problems on the ranks of Hermitian matrix expressions (8).

Define a Hermitian matrix expression
(9)f(X)=C4-A4XA4*,
where C4=C4*; we have an observation that by investigating extremal ranks of (9), where X is a Hermitian solution to a system of matrix equations
(10)A1X=C1,XB1=C2,A3XA3*=C3.
A new necessary and sufficient condition for system (5) to have Hermitian solution can be given by rank equalities, which is more practical than one given by generalized inverses and range inclusion of matrices.

It is well known that Schur complement is one of the most important matrix expressions in matrix theory; there have been many results in the literature on Schur complements and their applications [39–41]. Tian [36, 42] has investigated the maximal and minimal ranks of Schur complements with applications.

Motivated by the work mentioned above, we in this paper investigate the extremal ranks of the quaternion Hermitian matrix expression (9) subject to the consistent system of quaternion matrix equations (10) and its applications. In Section 2, we derive the formulas of extremal ranks of (9) with respect to Hermitian solution of (10). As applications, in Section 3, we give a new, necessary, and sufficient condition for the existence of Hermitian solution to system (5) by rank equalities. In Section 4, we derive extremal ranks of generalized Hermitian Schur complement subject to (2). We also consider the rank invariance problem in Section 5.

2. Extremal Ranks of (<xref ref-type="disp-formula" rid="EEq1.6">9</xref>) Subject to System (<xref ref-type="disp-formula" rid="EEq1.7">10</xref>)

Corollary 8 in [10] over Hilbert C*-modules can be changed into the following lemma over ℍ.

Lemma 1.

Let A1,C1∈ℍm×n, B1,C2∈ℍn×s,A3∈ℍr×n,C3∈ℍr×r be given, and F=B1*LA1,M=SLF,S=A3LA1,D=C2*-B1*A1†C1,J=A1†C1+F†D,G=C3-A3(J+LA1LF*J*)A3*; then the following statements are equivalent:

C3=C3*; the equalities in (11) hold and
(13)r[A1C1]=r(A1),r[A1C1B1*C2*]=r[A1B1*],r[A1C1A3*B1*C2*A3*A3C3]=r[A1B1*A3].

In that case, the general Hermitian solution of (10) can be expressed as
(14)X=J+LA1LFJ*+LA1LFM†G(M†)*LFLA1+LA1LFLMVLFLA1+LA1LFV*LMLFLA1,
where V is Hermitian matrix over ℍ with compatible size.

Lemma 2 (see Lemma 2.4 in [<xref ref-type="bibr" rid="B30">24</xref>]).

Let A∈ℍm×n, B∈ℍm×k,C∈ℍl×n,D∈ℍj×k, and E∈ℍl×i. Then the following rank equalities hold:

r(CLA)=r[AC]-r(A),

r[BALC]=r[BA0C]-r(C),

r[CRBA]=r[C0AB]-r(B),

r[ABLDREC0]=r[AB0C0E0D0]-r(D)-r(E).

Lemma 2 plays an important role in simplifying ranks of various block matrices.

Liu and Tian [38] has given the following lemma over a field. The result can be generalized to ℍ.

Lemma 3.

Let A=±A*∈ℍm×m, B∈ℍm×n, and C∈ℍp×m be given; then
(15)maxX∈ℍn×pr[A-BXC∓(BXC)*]=min{r[ABC*],r[ABB*0],r[AC*C0]},minX∈ℍn×pr[A-BXC∓(BXC)*]=2r[ABC*]+max{s1,s2},
where
(16)s1=r[ABB*0]-2r[ABC*B*00],s2=r[AC*C0]-2r[ABC*C00].

If ℛ(B)⊆ℛ(C*),
(17)maxXr[A-BXC-(BXC)*]=min{r[AC*],r[ABB*0]},maxXr[A-BXC-(BXC)*]=min{r[AC*],r[ABB*0]}.

Now we consider the extremal ranks of the matrix expression (9) subject to the consistent system (10).

Theorem 4.

Let A1,C1,B1,C2,A3,andC3 be defined as Lemma 1, C4∈ℍt×t,andA4∈ℍt×n. Then the extremal ranks of the quaternion matrix expression f(X) defined as (9) subject to system (10) are the following:
(18)maxr[f(X)]=min{a,b},
where
(19)a=r[C4A4C2*A4*B1*C1A4*A1]-r[B1*A1],b=r[0A4*A3*B1A1*A4C4000A30-C3-A3C2-A3C1*B1*0-C2*A3*-C2*B1-C2*A1*A10-C1A3*-C1B1-C1A1*]-2r[A3B1*A1],(20)minr[f(X)]=2r[C4A4C2*A4*B1*C1A4*A1]+r[0A4*A3*B1A1*A4C4000A30-C3-A3C2-A3C1*B1*0-C2*A3*-C2*B1-C2*A1*A10-C1A3*-C1B1-C1A1*]-2r[0A4*B1A1*A4C400A30-A3C2-A3C1*B1*0-C2*B1-C2*A1*A10-C1B1-C1A1*].

Proof.

By Lemma 1, the general Hermitian solution of the system (10) can be expressed as
(21)X=J+LA1LFJ*+LA1LFM†G(M†)*LFLA1+LA1LFLMVLFLA1+LA1LFV*LMLFLA1,
where V is Hermitian matrix over ℍ with appropriate size. Substituting (21) into (9) yields
(22)f(X)=C4-A4(+LA1LFM†G(M†)*LFJ+LA1LFJ*mC4-A4+LA1LFM†G(M†)*LFLA1)A4*-A4LA1LFLMVLFLA1A4*-A4LA1LFV*LMLFLA1A4*.
Put
(23)C4-A4(J+LA1LFJ*+LA1LFM†G(M†)*LFLA1)A4*=A,J+LA1LFJ*+LA1LFM†G(M†)*LFLA1=J′,A4LA1LFLM=N,LFLA1A4*=P;
then
(24)f(X)=A-NVP-(NVP)*.

Note that A=A* and ℛ(N)⊆ℛ(P*). Thus, applying (17) to (24), we get the following:
(25)maxr[f(X)]=maxVr(A-NVP-(NVP)*)=min{r[AP*],r[ANN*0]},minr[f(X)]=minVr(A-NVP-(NVP)*)=2r[AP*]+r[ANN*0]-2r[ANP0].

Now we simplify the ranks of block matrices in (25).

In view of Lemma 2, block Gaussian elimination, (11), (12), and (23), we have the following:
(26)r(F)=r(B1*LA1)=r[B1*A1]-r(A1),r(M)=r(SLF)=r[SF]-r(F)=r[A3LA1B1*LA1]-r(F)=r[A3B1*A1]-r(A1)-r(F),r[AP*]=r[C4-A4JA4*P*]=r[C4-A4JA4*A4LA10F]-r(F)=r[C4-A4JA4*A40B1*0A1]-r(F)-r(A1)=r[C4A4C2*A4*B1*C1A4*A1]-r[B1*A1],r[ANN*0]=r[C4-A4J′A4*A4LA1LFLMRM*RF*RA1*A4*0]=r[C4-A4J′A4*A4000A4*0A3*B1A1*0A30000B1*0000A1000]-2r(M)-2r(F)-2r(A1)=r[C4A4000A4*0A3*B1A1*A3J′A4*A3000B1*J′A4*B1*000A1J′A4*A1000]-2r[A3B1*A1]=r[C4A4000A4*0A3*B1A1*0A3-C3-A3C2-A3C1*0B1*-C2*A3*-C2*B1-C2*A1*0A1-C1A3*-C1B1-C1A1*]-2r[A3B1*A1]=r[0A4*A3*B1A1*A4C4000A30-C3-A3C2-A3C1*B1*0-C2*A3*-C2*B1-C2*A1*A10-C1A3*-C1B1-C1A1*]-2r[A3B1*A1],r[ANP0]=r[C4-A4J′A4*A4LA1LFLMRF*RA1*A4*0]=r[C4A400A4*0B1A1*0A3-A3C2-A3C1*0B1*-C2*B1-C2*A1*0A1-C1B1-C1A1*]-r[A3B1*A1]-r[B1*A1]=r[0A4*B1A1*A4C400A30-A3C2-A3C1*B1*0-C2*B1-C2*A1*A10-C1B1-C1A1*]-r[A3B1*A1]-r[B1*A1].

Substituting (26) into (25) yields (18) and (20).

In Theorem 4, letting C4 vanish and A4 be I with appropriate size, respectively, we have the following.

Corollary 5.

Assume that A1, C1∈ℍm×n, B1,C2∈ℍn×s,A3∈ℍr×n,andC3∈ℍr×r are given; then the maximal and minimal ranks of the Hermitian solution X to the system (10) can be expressed as
(27)maxr(X)=min{a,b},
where
(28)a=n+r[C2*C1]-r[B1*A1],b=2n+r[C3A3C2A3C1*C2*A3*C2*B1C2*A1*C1A3*C1B1C1A1*]-2r[A3B1*A1],minr(X)=2r[C2*C1]+r[C3A3C2A3C1*C2*A3*C2*B1C2*A1*C1A3*C1B1C1A1*]-2r[A3C2A3C1*C2*B1C2*A1*C1B1C1A1*].

In Theorem 4, assuming that A1,B1,C1, and C2 vanish, we have the following.

Corollary 6.

Suppose that the matrix equation A3XA3*=C3 is consistent; then the extremal ranks of the quaternion matrix expression f(X) defined as (9) subject to A3XA3*=C3 are the following:
(29)maxr[f(X)]=min{r[C4A4],r[0A4*A3*A4C40A30-C3]-2r(A3)},r[C4A4],r[0A4*A3*A4C40A30-C3]minr[f(X)]=2r[C4A4]+r[0A4*A3*A4C40A30-C3]-2r[0A4*A4C4A30].

3. A Practical Solvability Condition for Hermitian Solution to System (<xref ref-type="disp-formula" rid="EEq1.4">5</xref>)

In this section, we use Theorem 4 to give a necessary and sufficient condition for the existence of Hermitian solution to system (5) by rank equalities.

Theorem 7.

Let A1, C1∈ℍm×n, B1,C2∈ℍn×s, A3∈ℍr×n,C3∈ℍr×r,A4∈ℍt×n, and C4∈ℍt×tbe given; then the system (5) have Hermitian solution if and only ifC3=C3*, (11), (13) hold, and the following equalities are all satisfied:
(30)r[A4C4]=r(A4),(31)r[C4A4C2*A4*B1*C1A4*A1]=r[A4B1*A1],(32)r[0A4*A3*B1A1*A4C4000A30-C3-A3C2-A3C1*B1*0-C2*A3*-C2*B1-C2*A1*A10-C1A3*-C1B1-C1A1*]=2r[A4A3B1*A1].

Proof.

It is obvious that the system (5) have Hermitian solution if and only if the system (10) have Hermitian solution and
(33)minr[f(X)]=0,
where f(X) is defined as (9) subject to system (10). Let X0 be a Hermitian solution to the system (5); then X0 is a Hermitian solution to system (10) and X0 satisfies A4X0A4*=C4. Hence, Lemma 1 yields C3=C3*, (11), (13), and (30). It follows from
(34)[I00000I000A3X00I00B1*X000I0A1X0000I]×[0A4*A3*B1A1*A4C4000A30-C3-A3C2-A3C1*B1*0-C2*A3*-C2*B1-C2*A1*A10-C1A3*-C1B1-C1A1*]×[I-X0A4*0000I00000I00000I00000I]=[0A4*A3*B1A1*A40000A30000B1*0000A10000]
that (32) holds. Similarly, we can obtain (31).

Conversely, assume that C3=C3*, (11), (13) hold; then by Lemma 1, system (10) have Hermitian solution. By (20), (31)-(32), and
(35)r[0A4*B1A1*A4C400A30-A3C2-A3C1*B1*0-C2*B1-C2*A1*A10-C1B1-C1A1*]≥r[A4A3B1*A1]+r[A4B1*A1]
we can get
(36)minr[f(X)]≤0.
However,
(37)minr[f(X)]≥0.

Hence (33) holds, implying that the system (5) have Hermitian solution.

By Theorem 7, we can also get the following.

Corollary 8.

Suppose that A3, C3, A4, and C4 are those in Theorem 7; then the quaternion matrix equations A3XA3*=C3 and A4XA4*=C4 have common Hermitian solution if and only if (30) hold and the following equalities are satisfied:
(38)r[A3C3]=r(A3),r[0A4*A3*A4C40A30-C3]=2r[A3A4].

Corollary 9.

Suppose that A1, C1∈ℍm×n, B1, C2∈ℍn×s,and A, B∈ℍn×n are Hermitian. Then A and B have a common Hermitian g-inverse which is a solution to the system (2) if and only if (11) holds and the following equalities are all satisfied:
(39)r[A1C1AB1*C2*AAA]=r[A1B1*A],r[A1C1BB1*C2*BBB]=r[A1B1*B],(40)r[0BAB1A1*BB000A0-A-AC2-AC1*B1*0-C2*A-C2*B1-C2*A1*A10-C1A-C1B1-C1A1*]=2r[BAB1*A1].

4. Extremal Ranks of Schur Complement Subject to (<xref ref-type="disp-formula" rid="EEq1.1">2</xref>)

As is well known, for a given block matrix
(41)M=[ABB*D],
where A and D are Hermitian quaternion matrices with appropriate sizes, then the Hermitian Schur complement of A in M is defined as
(42)SA=D-B*A~B,
where A~ is a Hermitian g-inverse of A, that is, A~∈{X∣AXA=A,X=X*}.

Now we use Theorem 4 to establish the extremal ranks of SA given by (42) with respect to A~ which is a solution to system (2).

Theorem 10.

Suppose A1,C1∈ℍm×n, B1,C2∈ℍn×s,D∈ℍt×t, B∈ℍn×t,andA∈ℍn×n are given and system (2) is consistent; then the extreme ranks of SA given by (42) with respect to A~ which is a solution of (2) are the following:
(43)maxA1A~=C1A~B1=C2r(SA)=min{a,b},
where
(44)a=r[DB*C2*BB1*C1BA1]-r[B1*A1],b=r[0BAB1A1*B*D000A0-A-AC2-AC1*B1*0-C2*A-C2*B1-C2*A1*A10-C1A-C1B1-C1A1*]-2r[AB1*A1],minA1A~=C1A~B1=C2r(SA)=2r[DB*C2*BB1*C1BA1]m+r[0BAB1A1*B*D000A0-A-AC2-AC1*B1*0-C2*A-C2*B1-C2*A1*A10-C1A-C1B1-C1A1*]m-2r[0BB1A1*B*D00A0-AC2-AC1*B1*0-C2*B1-C2*A1*A10-C1B1-C1A1*].

Proof.

It is obvious that
(45)maxA1A~=C1,A~B1=C2r(D-B*A~B)=maxA1X=C1,XB1=C2,AXA=Ar(D-B*XB),minA1A~=C1,A~B1=C2r(D-B*A~B)=minA1X=C1,XB1=C2,AXA=Ar(D-B*XB).

Thus in Theorem 4 and its proof, letting A3=A3*=C3=A, A4=B*,andC4=D, we can easily get the proof.

In Theorem 10, let A1,C1,B1,andC2 vanish. Then we can easily get the following.

Corollary 11.

The extreme ranks of SA given by (42) with respect to A~ are the following:
(46)maxA~r(SA)=min{r[DB*],r[0BAB*D0A0-A]-2r(A)},minA~r(SA)=2r[DB*]+r[0BAB*D0A0-A]-2r[0BB*DA0].

5. The Rank Invariance of (<xref ref-type="disp-formula" rid="EEq1.6">9</xref>)

As another application of Theorem 4, we in this section consider the rank invariance of the matrix expression (9) with respect to the Hermitian solution of system (10).

Theorem 12.

Suppose that (10) have Hermitian solution; then the rank of f(X) defined by (9) with respect to the Hermitian solution of (10) is invariant if and only if
(47)r[0A4*B1A1*A4C400A30-A3C2-A3C1*B1*0-C2*B1-C2*A1*A10-C1B1-C1A1*]vv=r[C4A4C2*A4*B1*C1A4*A1]+r[A3B1*A1],r[0A4*A3*B1A1*A4C4000A30-C3-A3C2-A3C1*B1*0-C2*A3*-C2*B1-C2*A1*A10-C1A3*-C1B1-C1A1*]+r[B1*A1]vv=r[0A4*B1A1*A4C400A30-A3C2-A3C1*B1*0-C2*B1-C2*A1*A10-C1B1-C1A1*]+r[A3B1*A1],
or
(48)r[0A4*B1A1*A4C400A30-A3C2-A3C1*B1*0-C2*B1-C2*A1*A10-C1B1-C1A1*]vv=r[C4A4C2*A4*B1*C1A4*A1]+r[A3B1*A1].

Proof.

It is obvious that the rank of f(X) with respect to Hermitian solution of system (10) is invariant if and only if
(49)maxr[f(X)]-minr[f(X)]=0.

By (49), Theorem 4, and simplifications, we can get (47) and (48).

Corollary 13.

The rank of SA defined by (42) with respect to A~ which is a solution to system (2) is invariant if and only if
(50)r[0BB1A1*B*D00A0-AC2-AC1*B1*0-C2*B1-C2*A1*A10-C1B1-C1A1*]=r[DB*C2*BB1*C1BA1]+r[AB1*A1],r[0BAB1A1*B*D000A0-A-AC2-AC1*B1*0-C2*A-C2*B1-C2*A1*A10-C1A-C1B1-C1A1*]+r[B1*A1]vv=r[0BB1A1*B*D00A0-AC2-AC1*B1*0-C2*B1-C2*A1*A10-C1B1-C1A1*]+r[AB1*A1],
or
(51)r[0BB1A1*B*D00A0-AC2-AC1*B1*0-C2*B1-C2*A1*A10-C1B1-C1A1*]vv=r[DB*C2*BB1*C1BA1]+r[AB1*A1].

Acknowledgments

This research was supported by the National Natural Science Foundation of China, Tian Yuan Foundation (11226067), the Fundamental Research Funds for the Central Universities (WM1214063), and China Postdoctoral Science Foundation (2012M511014).

HungerfordT. W.MitraS. K.The matrix equations AX=C, XB=DKhatriC. G.MitraS. K.Hermitian and nonnegative definite solutions of linear matrix equationsYuanY. X.On the symmetric solutions of matrix equation (AX,XC)=(B,D)DaiH.LancasterP.Linear matrix equations from an inverse problem of vibration theoryGroßJ.A note on the general Hermitian solution to AXA∗=BTianY. G.LiuY. H.Extremal ranks of some symmetric matrix expressions with applicationsLiuY. H.TianY. G.TakaneY.Ranks of Hermitian and skew-Hermitian solutions to the matrix equation AXA∗=BChangX. W.WangJ. S.The symmetric solution of the matrix equations AX+YA=C, AXAT+BYBT=C and (ATXA,BTXB)=(C,D)WangQ.-W.WuZ.-C.Common Hermitian solutions to some operator equations on Hilbert C∗-modulesFaridF. O.MoslehianM. S.WangQ.-W.WuZ.-C.On the Hermitian solutions to a system of adjointable operator equationsHeZ.-H.WangQ.-W.Solutions to optimization problems on ranks and inertias of a matrix function with applicationsWangQ.-W.The general solution to a system of real quaternion matrix equationsWangQ.-W.Bisymmetric and centrosymmetric solutions to systems of real quaternion matrix equationsWangQ. W.A system of four matrix equations over von Neumann regular rings and its applicationsWangQ.-W.A system of matrix equations and a linear matrix equation over arbitrary regular rings with identityWangQ.-W.ChangH.-X.NingQ.The common solution to six quaternion matrix equations with applicationsWangQ.-W.ChangH.-X.LinC.-Y.P-(skew)symmetric common solutions to a pair of quaternion matrix equationsWangQ. W.HeZ. H.Some matrix equations with applicationsWangQ. W.JiangJ.Extreme ranks of (skew-)Hermitian solutions to a quaternion matrix equationWangQ.-W.LiC.-K.Ranks and the least-norm of the general solution to a system of quaternion matrix equationsWangQ.-W.LiuX.YuS.-W.The common bisymmetric nonnegative definite solutions with extreme ranks and inertias to a pair of matrix equationsWangQ.-W.QinF.LinC.-Y.The common solution to matrix equations over a regular ring with applicationsWangQ.-W.SongG.-J.LinC.-Y.Extreme ranks of the solution to a consistent system of linear quaternion matrix equations with an applicationWangQ.-W.SongG.-J.LinC.-Y.Rank equalities related to the generalized inverse AT,S(2) with applicationsWangQ. W.SongG. J.LiuX.Maximal and minimal ranks of the common solution of some linear matrix equations over an arbitrary division ring with applicationsWangQ.-W.WuZ.-C.LinC.-Y.Extremal ranks of a quaternion matrix expression subject to consistent systems of quaternion matrix equations with applicationsWangQ. W.YuS. W.Ranks of the common solution to some quaternion matrix equations with applicationsWangQ. W.YuS. W.XieW.Extreme ranks of real matrices in solution of the quaternion matrix equation AXB=C with applicationsWangQ.-W.YuS.-W.ZhangQ.The real solutions to a system of quaternion matrix equations with applicationsWangQ.-W.YuS.-W.LinC.-Y.Extreme ranks of a linear quaternion matrix expression subject to triple quaternion matrix equations with applicationsWangQ.-W.ZhangF.The reflexive re-nonnegative definite solution to a quaternion matrix equationWangQ. W.ZhangX.HeZ. H.On the Hermitian structures of the solution to a pair of matrix equationsZhangX.WangQ.-W.LiuX.Inertias and ranks of some Hermitian matrix functions with applicationsTianY. G.ChengS. Z.The maximal and minimal ranks of A−BXC with applicationsTianY. G.Upper and lower bounds for ranks of matrix expressions using generalized inversesLiuY. H.TianY. G.More on extremal ranks of the matrix expressions A−BX±X∗B∗ with statistical applicationsLiuY. H.TianY. G.Max-min problems on the ranks and inertias of the matrix expressions A−BXC±(BXC)∗ with applicationsAndoT.Generalized Schur complementsCarlsonD.HaynsworthE.MarkhamT.A generalization of the Schur complement by means of the Moore-Penrose inverseFiedlerM.Remarks on the Schur complementTianY. G.More on maximal and minimal ranks of Schur complements with applications