# Ranks of a Constrained Hermitian Matrix Expression with Applications

Authors: Shao-Wen Yu
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

We establish the formulas of the maximal and minimal ranks of the quaternion Hermitian matrix expressionC4−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.


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## Body

## 1. Introduction

Throughout this paper, we denote the real number field byℝ, 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 (9) Subject to System (10)

Corollary8 in [10] over Hilbert C*-modules can be changed into the following lemma over ℍ.Lemma 1.
LetA1,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:(1)
the system (10) have a Hermitian solution,(2)
C3=C3*,
(11)A1C2=C1B1,A1C1*=C1A1*,B1*C2=C2*B1,(12)RA1C1=0,RFD=0,RMG=0,(3)
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 [24]).
LetA∈ℍm×n, B∈ℍm×k,C∈ℍl×n,D∈ℍj×k, and E∈ℍl×i. Then the following rank equalities hold:(a)
r(CLA)=r[AC]-r(A),(b)
r[BALC]=r[BA0C]-r(C),(c)
r[CRBA]=r[C0AB]-r(B),(d)
r[ABLDREC0]=r[AB0C0E0D0]-r(D)-r(E).Lemma2 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.
LetA=±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].
Ifℛ(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.
LetA1,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 Lemma1, 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 thatA=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 Lemma2, 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 Theorem4, letting C4 vanish and A4 be I with appropriate size, respectively, we have the following.Corollary 5.
Assume thatA1, 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 Theorem4, assuming that A1,B1,C1, and C2 vanish, we have the following.Corollary 6.
Suppose that the matrix equationA3XA3*=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 (5)

In this section, we use Theorem4 to give a necessary and sufficient condition for the existence of Hermitian solution to system (5) by rank equalities.Theorem 7.
LetA1, 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 thatC3=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 Theorem7, we can also get the following.Corollary 8.
Suppose thatA3, 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 thatA1, 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 (2)

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 Theorem4 to establish the extremal ranks of SA given by (42) with respect to A~ which is a solution to system (2).Theorem 10.
SupposeA1,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 Theorem4 and its proof, letting A3=A3*=C3=A,  A4=B*,andC4=D, we can easily get the proof.In Theorem10, let A1,C1,B1,andC2 vanish. Then we can easily get the following.Corollary 11.
The extreme ranks ofSA 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 (9)

As another application of Theorem4, 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 off(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 ofSA 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].



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