Equalities and inequalities for Hermitian solutions and Hermitian definite solutions of the two matrix equations $AX = B$ and $AXA^* = B$

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by  Yongge Tian

Equalities and inequalities for Hermitian solutions and Her-
mitian definite solutions of the two matrix equations AX = B
and AXA∗ = B
YONGGE TIAN
Abstract. This paper studies algebraic properties of Hermitian solutions and Hermitian definite
solutionsofthetwotypesofmatrixequationAX =B andAXA∗ =B. Wefirstestablishavariety
ofrankandinertia formulasforcalculatingthe maximalandminimalranksandinertiasofHermi-
tian solutions and Hermitian definite solutions of the matrix equations AX =B and AXA∗ =B,
and then use them to characterize many qualities and inequalities for Hermitian solutions and
3
Hermitian definite solutions of the two matrix equations and their variations.
1
0
2 Mathematics Subject Classifications. 15A03, 15A09, 15A24, 15B57.

n Keywords. Matrixequation,Hermitiansolution,Hermitiandefinite solution,generalizedinverse,
a
rank, inertia, matrix equality, matrix inequality, Lo¨wner partial ordering.
J

7
1 1 Introduction

]
A Consider the following two well-known linear matrix equations
R
AX =B (1.1)
.
h
t and
a
m AXA∗ =B, (1.2)
[ both of which are simplest cases of various types of linear matrix equation (with symmetric pat-

1 terns), and are the starting point of many advanced study on complicated matrix equations. A
v huge amount of results about the two equations and applications were given in the literature. In
8 particular,manyproblemsonalgebraicpropertiesofsolutionsofthetwomatrixequationswereex-
3
plicitlycharacterizedbyusingformulasforranksandinertiasofmatrices. Inthispaper,theauthor
2
focuses on Hermitian solutions or Hermitian definite solutions of (1.1) and (1.2), and studies the
4
. following optimization problems on the ranks and inertias of Hermitian solutions and Hermitian
1
definite solutions of (1.1) and (1.2):
0
3
1 Problem 1.1 Let A, B ∈ Cm×n be given, and assume that (1.1) has a Hermitian solution or
: HermitiandefinitesolutionX ∈Cn×n. Inthiscase,establishformulasforcalculatingtheextremal
v
ranks and inertias of
i
X X −P, (1.3)
r
a where P ∈ Cn×n is a given Hermitian matrix, and use the formulas to characterize behaviors
of these Hermitian solution and definite solution, in particular, to give necessary and sufficient
conditions for the four inequalities
X ≻P, X <P, X ≺P, X 4P (1.4)
in the Lo¨wner partial ordering to hold, respectively.
Problem 1.2 Let A, B, C, D ∈ Cm×n be given, X, Y ∈ Cn×n be two unknown matrices, and
assume that the two linear matrix equations
AX =B, CY =D (1.5)
haveHermitiansolutions,respectively. Inthis case,establishformulasforcalculatingtheextremal
ranks and inertias of the difference
X −Y (1.6)
ThisworkwassupportedpartiallybyNationalNaturalScienceFoundationofChina(GrantNo. 11271384).
1
of the Hermitian solutions, and use the formulas to derive necessary and sufficient conditions for
X ≻Y, X <Y, X ≺Y, X 4Y (1.7)
to hold in the Lo¨wner partial ordering, respectively.
Problem 1.3 Let A ∈ Cm×n and B ∈ Cm be given, and assume that (1.2) has a Hermitian
H
solution. Pre- and post-multiplying a matrix T ∈ Cp×n and its conjugate transpose T∗ on both
sides of (1.2) yields a transformed equation as follows
TAXA∗T∗ =TBT∗. (1.8)
Further define
S ={X ∈Cn | AXA∗ =B}, (1.9)
H
T ={Y ∈Cn | TAYA∗T∗ =TBT∗}. (1.10)
H
In this case, give necessary and sufficient conditions for S = T to hold, as well as necessary and
sufficient conditions for X ≻ Y, X < Y, X ≺ Y and X 4 Y to hold for X ∈ S and Y ∈ T,
respectively.
Problem 1.4 Assume that (1.2) has a Hermitian solution, and define
S ={X ∈Cn |AXA∗ =B}, (1.11)
H
T ={(X +X )/2|T AX A∗T∗ =T BT∗, T AX A∗T∗ =T BT∗, X , X ∈Cn }. (1.12)
1 2 1 1 1 1 1 2 2 2 2 2 1 2 H
In this case, give necessary and sufficient conditions for S =T to hold.
Problem 1.5 Denote the sets of all least-squares solutions and least-rank Hermitian solutions of
(1.2) as
S ={X ∈Cn | kB−AXA∗k =min}, (1.13)
H F
T ={Y ∈Cn | r(B−AYA∗)=min}. (1.14)
H
In this case, establish necessary and sufficient conditions for X ≻ Y, X < Y, X ≺ Y and X 4 Y
to hold for X ∈S and Y ∈T, respectively.
Matrix equations have been a prominent concerns in matrix theory and applications. As is
known to all, two key tasks in solving a matrix equation is to give identifying condition for the
existence of a solution of the equation, and to give general solution of the equation. Once general
solutionis given,the subsequent workis to describe behaviorsof solutions of the matrix equation,
such as, the uniqueness of solutions; the norms of solutions; the ranks and ranges of solutions, the
definitenessofsolutions,equalitiesandinequalitiesofsolutions,etc. Problems1.1and1.2describe
the inequalities for solutions of (1.1), as well as relations between solutions of two linear matrix
equations.
Throughoutthis paper, Cm×n andCm standfor the sets ofall m×n complex matricesand all
H
m×m complex Hermitian matrices, respectively; the symbols A∗, r(A) and R(A) stand for the
transpose,conjugatetranspose,rankandrange(columnspace)ofamatrixA∈Cm×n,respectively;
I denotes the identity matrix oforderm; [A, B]denotes a rowblock matrix consistingofA and
m
B. The Moore–Penroseinverseof a matrix A∈Cm×n, denoted by A†, is defined to be the unique
matrix X ∈Cn×m satisfying the following four matrix equations
(i) AXA=A, (ii) XAX =X, (iii) (AX)∗ =AX, (iv) (XA)∗ =XA.
Further, let E = I −AA† and F = I −A†A, which ranks are given by r(E ) = m−r(A)
A m A n A
and r(F )=n−r(A). A well-known property of the Moore–Penroseinverse is (A†)∗ =(A∗)†. In
A
particular, both (A†)∗ =A† and AA† =A†A hold if A is Hermitian, i.e., A=A∗. A<0 (A≻0)
meansthatAisHermitianpositivesemi-definite(Hermitianpositivedefinite). TwoA, B ∈Cm are
H
saidto satisfy the inequality A<B (A≻B) in the Lo¨wner partialorderingif A−B is Hermitian
2
positivesemi-definite(Hermitianpositivedefinite). i (A)denotesthenumbersofthe positiveand
±
negative eigenvalues of a Hermitian matrix A counted with multiplicities, respectively.
The results on ranks and inertias of matrices in Lemmas 1.6 and 1.7 below are obvious or
well-known (see also [10, 11] for their references), while the closed-formformulas for matrix ranks
and inertias in Lemmas 1.9 and 1.12–1.15 were established by the present author, which we shall
use in the latter part of this paper to derive analytical solutions to Problems 1.1–1.5.
Lemma 1.6 Let S, S and S be three sets consisting of (square) matrices over Cm×n, and let H
1 2
be a set consisting of Hermitian matrices over Cm. Then, the following hold.
H
(a) Under m=n, S has a nonsingular matrix if and only if max r(X)=m.
X∈S
(b) Under m=n, all X ∈S are nonsingular if and only if min r(X)=m.
X∈S
(c) 0∈S if and only if min r(X)=0.
X∈S
(d) S ={0} if and only if max r(X)=0.
X∈S
(e) H has a matrix X ≻0 (X ≺0) if and only if max i (X)=m (max i (X)=m).
X∈H + X∈H −
(f) All X ∈H satisfy X ≻0 (X ≺0) if and only if min i (X)=m (min i (X)=m).
X∈H + X∈H −
(g) H has a matrix X <0 (X 40) if and only if min i (X)=0 (min i (X)=0).
X∈H − X∈H +
(h) All X ∈H satisfy X <0 (X 40) if and only if max i (X)=0 (max i (X)=0).
X∈H − X∈H +
(i) The following hold
S ∩S 6=∅ ⇔ min r(X −X )=0, (1.15)
1 2 1 2
X1∈S1,X2∈S2
S ⊆S ⇔ max min r(X −X )=0, (1.16)
1 2 1 2
X1∈S1X2∈S2
S ⊇S ⇔ max min r(X −X )=0, (1.17)
1 2 1 2
X2∈S2X1∈S1
there exist X ∈S and X ∈S such that X ≻X ⇔ max i (X −X )=m,
1 1 2 2 1 2 + 1 2
X1∈S1,X2∈S2
(1.18)
there exist X ∈S and X ∈S such that X <X ⇔ min i (X −X )=0.
1 1 2 2 1 2 − 1 2
X1∈S1,X2∈S2
(1.19)
Lemma 1.7 Let A∈Cm, B ∈Cn, Q∈Cm×n, and assume that P ∈Cm×m is nonsingular. Then,
H H
i (PAP∗)=i (A), (1.20)
± ±
i (A) if λ>0
i (λA)= ± , (1.21)
± i (A) if λ<0
∓
(cid:26)
A 0
i =i (A)+i (B), (1.22)
± 0 B ± ±
(cid:20) (cid:21)
0 Q 0 Q
i =i =r(Q). (1.23)
+ Q∗ 0 − Q∗ 0
(cid:20) (cid:21) (cid:20) (cid:21)
Lemma 1.8 ([7]) Let A∈Cm×n, B ∈Cm×k, C ∈Cl×n and D ∈Cl×k. Then,
r[A, B]=r(A)+r(E B)=r(B)+r(E A), (1.24)
A B
A
r =r(A)+r(CF )=r(C)+r(AF ), (1.25)
C A C
(cid:20) (cid:21)
A B
r =r(B)+r(C)+r(E AF ). (1.26)
C 0 B C
(cid:20) (cid:21)
3
Lemma 1.9 ([10]) Let A∈Cm, B ∈Cm×n, D ∈Cn, and let
H H
A B A B
M = , M = . (1.27)
1 B∗ 0 2 B∗ D
(cid:20) (cid:21) (cid:20) (cid:21)
Then, the following expansion formulas hold
i (M )=r(B)+i (E AE ), r(M )=2r(B)+r(E AE ), (1.28)
± 1 ± B B 1 B B
0 E B 0 E B
i (M )=i (A)+i A , r(M )=r(A)+r A . (1.29)
± 2 ± ± B∗E D−B∗A†B 2 B∗E D−B∗A†B
A A
(cid:20) (cid:21) (cid:20) (cid:21)
Under the condition A<0,
i (M )=r[A, B], i (M )=r(B), r(M )=r[A, B]+r(B). (1.30)
+ 1 − 1 1
Under the condition R(B)⊆R(A),
i (M )=i (A)+i (D−B∗A†B), r(M )=r(A)+r(D−B∗A†B). (1.31)
± 2 ± ± 2
Some general rank and inertia expansion formulas derived from (1.24)–(1.29) are given below
A B A B 0
r =r −r(P), (1.32)
E C 0 C 0 P
P
(cid:20) (cid:21) (cid:20) (cid:21)
A B
A BF
r Q =r C 0 −r(Q), (1.33)
C 0  
(cid:20) (cid:21) 0 Q
 A B 0
A BF
r Q =r C 0 P −r(P)−r(Q), (1.34)
(cid:20)EPC 0 (cid:21)  0 Q 0 
 A B 0
A BF
i P =i B∗ 0 P∗ −r(P), (1.35)
±(cid:20)FPB∗ 0 (cid:21) ± 0 P 0 
 A B Q
E AE E B
i Q Q Q =i B∗ D 0 −r(Q). (1.36)
±(cid:20) B∗EQ D (cid:21) ±Q∗ 0 0 
 
WeshallusethemtosimplifyranksandinertiasofblockmatricesinvolvingMoore–Penroseinverses
of matrices.
Lemma 1.10 ([2]) Let A, B ∈Cm×n be given. Then, the following hold.
(a) Eq. (1.1) has a solution X ∈Cn if and only if R(B)⊆R(A) and AB∗ =BA∗. In this case,
H
the general Hermitian solution of (1.1) can be written as
X =A†B+(A†B)∗−A†BA†A+F UF , (1.37)
A A
where U ∈Cn is arbitrary.
H
(b) The matrix equation in (1.1) has a solution 0 4 X ∈ Cn if and only if R(B) ⊆ R(A),
H
AB∗ <0 and R(AB∗)=R(BA∗)=R(B). In this case, the general solution 04X ∈Cn of
H
(1.1) can be written as
X =B∗(AB∗)†B+F UF , (1.38)
A A
where 04U ∈Cn is arbitrary.
H
Lemma 1.11 Eq. (1.2) has a solution X ∈ Cn if and only if R(B) ⊆ R(A), or equivalently,
H
AA†B =B. In this case, the general Hermitian solution of AXA∗ =B can be written as
X =A†B(A†)∗+F U +U∗F , (1.39)
A A
where U ∈Cn×n is arbitrary.
4
Lemma 1.12 ([11]) Let Aj ∈Cmj×n and Bj ∈CmHj be given, j =1, 2, and assume that
A X A∗ =B and A X A∗ =B (1.40)
1 1 1 1 2 2 2 2
are solvable for X , X ∈Cn. Also define
1 2 H
B 0 A
1 1
S = X ∈Cn | A X A∗ =B , j =1, 2, M = 0 −B A . (1.41)
j j H j j j j  2 2
A∗ A∗ 0
(cid:8) (cid:9) 1 2
 
Then,
max r(X −X )=min{n, r(M)+2n−2r(A )−2r(A )}, (1.42)
1 2 1 2
X1∈S1,X2∈S2
min r(X −X )=r(M)−2r[A∗, A∗], (1.43)
1 2 1 2
X1∈S1,X2∈S2
max i (X −X )=i (M)+n−r(A )−r(A ), (1.44)
± 1 2 ± 1 2
X1∈S1,X2∈S2
min i (X −X )=i (M)−r[A∗, A∗]. (1.45)
± 1 2 ± 1 2
X1∈S1,X2∈S2
Consequently, the following hold.
(a) There exist X ∈ S and X ∈ S such that X −X is nonsingular if and only if r(M) >
1 1 2 2 1 2
2r(A )+2r(A )−n.
1 2
(b) X −X is nonsingular for all X ∈S and X ∈S if and only if r(M)=2r[A∗, A∗]+n.
1 2 1 1 2 2 1 2
(c) There exist X ∈ S and X ∈ S such that X = X if and only if R(B ) ⊆ R(A ) and
1 1 2 2 1 2 j j
r(M)=2r[A∗, A∗], j =1, 2.
1 2
(d) The rank of X − X is invariant for all X ∈ S and X ∈ S if and only if r(M) =
1 2 1 1 2 2
2r[A∗, A∗]−n or r(A )=r(A )=n.
1 2 1 2
(e) There exist X ∈ S and X ∈ S such that X ≻ X (X ≺ X ) if and only if i (M) =
1 1 2 2 1 2 1 2 +
r(A )+r(A ) (i (M)=r(A )+r(A )).
1 2 − 1 2
(f) X ≻ X (X ≺ X ) for all X ∈ S and X ∈ S if and only if i (M) = r[A∗, A∗]+n
1 2 1 2 1 1 2 2 + 1 2
(i (M)=r[A∗, A∗]+n).
− 1 2
(g) There exist X ∈ S and X ∈ S such that X < X (X 4 X ) if and only if i (M) =
1 1 2 2 1 2 1 2 −
r[A∗, A∗] (i (M)=r[A∗, A∗]).
1 2 + 1 2
(h) X <X (X 4X ) for all X ∈S and X ∈S if and only if i (M)=r(A )+r(A )−n
1 2 1 2 1 1 2 2 − 1 2
(i (M)=r(A )+r(A )−n).
+ 1 2
(i) i (X −X ) is invariant for all X ∈S and X ∈S ⇔ i (X −X ) is invariant for all
+ 1 2 1 1 2 2 − 1 2
X ∈S and X ∈S ⇔ r(A )=r(A )=n.
1 1 2 2 1 2
A B
Lemma 1.13 Let A ∈ Cm and B ∈ Cm×n be given, and denote M = . Then, the
H B∗ 0
(cid:20) (cid:21)
following hold.
(a) [10, 21] The extremal ranks and inertias of A−BXB∗ subject to X ∈Cn are given by
H
max r(A−BXB∗)=r[A, B], (1.46)
X∈CnH
min r(A−BXB∗)=2r[A, B]−r(M), (1.47)
X∈CnH
max i (A−BXB∗)=i (M), (1.48)
± ±
X∈CnH
min i (A−BXB∗)=r[A, B]−i (M). (1.49)
± ∓
X∈CnH
5
(b) [15] The extremal ranks and inertias of A±BXB∗ subject to 04X ∈Cn are given by
H
maxr(A+BXB∗)=r[A, B], min r(A+BXB∗)=i (A)+r[A, B]−i (M), (1.50)
+ +
04X∈CnH 04X∈CnH
maxi (A+BXB∗)=i (M), min i (A+BXB∗)=i (A), (1.51)
+ + + +
04X∈CnH 04X∈CnH
maxi (A+BXB∗)=i (A), min i (A+BXB∗)=r[A, B]−i (M), (1.52)
− − − +
04X∈CnH 04X∈CnH
maxr(A−BXB∗)=r[A, B], min r(A−BXB∗)=i (A)+r[A, B]−i (M), (1.53)
− −
04X∈CnH 04X∈CnH
maxi (A−BXB∗)=i (A), min i (A−BXB∗)=r[A, B]−i (M), (1.54)
+ + + −
04X∈CnH 04X∈CnH
maxi (A−BXB∗)=i (M), min i (A−BXB∗)=i (A). (1.55)
− − − −
04X∈CnH 04X∈CnH
Lemma 1.14 ([4, 10]) Let A∈Cm, B ∈Cm×n and C ∈Cm×k be given. Then,
H
max r(A−BXB∗−CYC∗)=r[A, B, C], (1.56)
X∈CnH, Y∈CkH
A B A B C
min r(A−BXB∗−CYC∗)=2r[A, B, C]+r −r
X∈CnH, Y∈CkH (cid:20)C∗ 0 (cid:21) (cid:20)B∗ 0 0 (cid:21)
A B C
−r , (1.57)
C∗ 0 0
(cid:20) (cid:21)
A B C
max i (A−BXB∗−CYC∗)=i B∗ 0 0 , (1.58)
± ±
X∈CnH, Y∈CkH C∗ 0 0 
  A B C
min i (A−BXB∗−CYC∗)=r[A, B, C]−i B∗ 0 0 . (1.59)
± ∓
X∈CnH, Y∈CkH C∗ 0 0 
Lemma 1.15 ([5, 11]) Let A ∈ Cm, B ∈ Cm×n and C ∈Cp×m be given and assume that
H
R(B)⊆R(C∗). Then,
A B
max r[A−BXC−(BXC)∗]=min r[A, C∗], r , (1.60)
X∈Cn×p B∗ 0
(cid:26) (cid:20) (cid:21)(cid:27)
A B A B
min r[A−BXC−(BXC)∗]=2r[A, C∗]+r −2r , (1.61)
X∈Cn×p B∗ 0 C 0
(cid:20) (cid:21) (cid:20) (cid:21)
A B
max i [A−BXC−(BXC)∗]=i , (1.62)
X∈Cn×p ± ± B∗ 0
(cid:20) (cid:21)
A B A B
min i [A−BXC−(BXC)∗]=r[A, C∗]+i −r , (1.63)
X∈Cn×p ± ± B∗ 0 C 0
(cid:20) (cid:21) (cid:20) (cid:21)
and
A B
max r[A−BX −(BX)∗]=min m, r , (1.64)
X∈Cn×m B∗ 0
(cid:26) (cid:20) (cid:21)(cid:27)
A B
min r[A−BX −(BX)∗]=r −2r(B), (1.65)
X∈Cn×m B∗ 0
(cid:20) (cid:21)
A B
max i [A−BX −(BX)∗]=i , (1.66)
X∈Cn×m ± ± B∗ 0
(cid:20) (cid:21)
A B
min i [A−BX −(BX)∗]=i −r(B). (1.67)
X∈Cn×m ± ± B∗ 0
(cid:20) (cid:21)
2 Properties of Hermitian solutions and Hermitian definite
solutions of AX = B
Someformulasforcalculatingthe ranksandinertiasofHermitiansolutionsandHermitiandefinite
solutionsofthe matrixequationin(1.1)wereestablishedin[11]. Inthis section,wereconsiderthe
6
ranks and inertias of these solutions and give a group of complete results.
Theorem 2.1 Assume that (1.1) has a Hermitian solution, and let P ∈Cn. Also, define
H
S ={X ∈Cn | AX =B}. (2.1)
H
Then,
maxr(X −P )=r(B−AP )−r(A)+n, (2.2)
X∈S
minr(X −P )=2r(B−AP )−r(BA∗−APA∗), (2.3)
X∈S
maxi (X −P )=i (BA∗−APA∗)−r(A)+n, (2.4)
± ±
X∈S
mini (X −P )=r(B−AP )−i (BA∗−APA∗). (2.5)
± ∓
X∈S
In consequence, the following hold.
(a) There exists an X ∈S such that X −P is nonsingular if and only if R(AP −B)=R(A).
(b) X−P is nonsingular for all X ∈S if and only if 2r(B−AP )=r(BA∗−APA∗)+n.
(c) There exists an X ∈S such that X ≻P (X ≺P) if and only if
R(BA∗−APA∗)=R(A) and BA∗ <APA∗
(R(BA∗−APA∗)=R(A) and BA∗ 4APA∗).
(d) X ≻P (X ≺P) holds for all X ∈S if and only if
r(B−AP )=n and BA∗ <APA∗ (r(B−AP )=n and AB∗ 4APA∗).
(e) There exists an X ∈S such that X <P (X 4P) if and only if
R(B−AP )=R(BA∗−APA∗) and BA∗ <APA∗
(R(B−AP )=R(BA∗−APA∗) and BA∗ 4APA∗).
(f) X <P (X 4P) holds for all X ∈S if and only if
BA∗ <APA∗ and r(A)=n (BA∗ 4APA∗ and r(A)=n).
In particular, the following hold.
(g) There exists an X ∈S such that X is nonsingular if and only if R(B)=R(A).
(h) X is nonsingular for all X ∈S if and only if r(B)=n.
(i) There exists an X ∈S such that X ≻0 (X ≺0) if and only if
R(B)=R(A) and BA∗ <0 (R(B)=R(A) and BA∗ 40).
(j) X ≻0 (X ≺0) holds for all X ∈S if and only if
r(B)=n and BA∗ <0 (r(B)=n and AB∗ 40).
(k) There exists an X ∈S such that X <0 (X 40) if and only if
R(B)=R(BA∗) and BA∗ <0 (R(B)=R(BA∗) and BA∗ 40).
(l) X <0 (X 40) holds for all X ∈S if and only if
BA∗ <0 and r(A)=n (BA∗ 40 and r(A)=n).
7
Proof By Lemma 1.10(a), X −P can be written as
X−P =X −P +F UF , (2.6)
0 A A
where X =A†B+(A†B)∗−A†BA†A andU ∈Cn is arbitrary. Applying Lemma 1.13(a)to (2.6)
0 H
gives
maxr(X −P )= max r(X −P +F UF )=r[X −P, F ], (2.7)
0 A A 0 A
X∈S U∈CnH
X −P F
minr(X −P )= min r(X −P +F UF )=2r[X −P, F ]−r 0 A , (2.8)
X∈S U∈CnH 0 A A 0 A (cid:20) FA 0 (cid:21)
X −P F
maxi (X −P )= min i (X −P +F UF )=i 0 A , (2.9)
X∈S ± U∈CnH ± 0 A A ±(cid:20) FA 0 (cid:21)
X −P F
mini (X −P )= min i (X −P +F UF )=r[X −P, F ]−i 0 A . (2.10)
X∈S ± U∈CnH ± 0 A A 0 A ∓(cid:20) FA 0 (cid:21)
Applying (1.24) and (1.28) to the block matrices in (2.7)–(2.10) and simplifying by (1.20) and
elementary block matrix operations, we obtain
r[X −P, F ]=r(A†AX −A†AP )+r(F )=r(B−AP )+n−r(A), (2.11)
0 A 0 A
X −P F
i 0 A =r(F )+i [A†A(X −P )A†A]=n−r(A)+i (BA∗−APA∗), (2.12)
± F 0 A ± 0 ±
A
(cid:20) (cid:21)
X −P F
r 0 A =2r(F )+r[A†A(X −P)A†A]=2n−2r(A)+r(BA∗−APA∗). (2.13)
F 0 A 0
A
(cid:20) (cid:21)
Substituting (2.11)–(2.13) into (2.7)–(2.10) and simplifying leads to (2.2)–(2.5). Results (a)–(l)
follow from applying Lemma 1.6 to (2.2)–(2.5). (cid:3)
Theorem 2.2 Assume that (1.1) has a Hermitian solution X < 0, and let 0 4 P ∈ Cn. Also,
H
define
AB∗ B
S ={04X ∈Cn | AX =B}, M = . (2.14)
H B∗ P
(cid:20) (cid:21)
Then,
maxr(X −P)=r(B−AP )−r(A)+n, (2.15)
X∈S
minr(X −P )=i (M)+r(B−AP )−i (BA∗−APA∗), (2.16)
− +
X∈S
maxi (X −P )=i (BA∗−APA∗)−r(A)+n, (2.17)
+ +
X∈S
mini (X −P )=i (M), (2.18)
+ −
X∈S
maxi (X −P )=i (M)−r(B), (2.19)
− +
X∈S
mini (X −P )=r(B−AP )−i (BA∗−APA∗). (2.20)
− +
X∈S
Consequently, the following hold.
(a) There exists an X ∈S such that X −P is nonsingular if and only if R(B−AP )=R(A).
(b) X−P isnonsingularforallX ∈S ifandonlyifi (M)+r(B−AP )=i (BA∗−APA∗)+n.
− +
(c) There exists an X ∈ S such that X ≻ P if and only if R(BA∗ −APA∗) = R(A) and
BA∗ <APA∗.
(d) X ≻P holds for all X ∈S if and only if i (M)=n.
−
(e) There exists an X ∈S such that X ≺P if and only if i (M)=r(B)+n.
+
8
(f) X ≺P holds for all X ∈S if and only if r(B−AP )=n and BA∗ 4APA∗.
(g) There exists an X ∈S such that X <P if and only if R(B−AP )=R(BA∗−APA∗) and
BA∗ <APA∗.
(h) X <P holds for all X ∈S if and only if i (M)=r(B).
+
(i) There exists an X ∈S such that X 4P if and only if M <0.
(j) X 4P holds for all X ∈S if and only if i (BA∗−APA∗)=n−r(A).
+
In particular, the following hold.
(k) There exists an X ∈S such that X −I is nonsingular if and only if R(B−A)=R(A).
n
(l) X−I is nonsingular for all X ∈S if and only if i (BA∗−BB∗)+r(B−A)=i (BA∗−
n − +
AA∗)+n.
(m) There exists an X ∈ S such that X ≻ I if and only if R(BA∗ − AA∗) = R(A) and
n
BA∗ <AA∗.
(n) X ≻I holds for all X ∈S if and only if BA∗ ≺BB∗.
n
(o) There exists an X ∈ S such that X ≺ I if and only if R(BA∗ − BB∗) = R(B) and
n
BA∗ <BB∗.
(p) X ≺I holds for all X ∈S if and only if r(B−A)=n and BA∗ 4AA∗.
n
(q) There exists an X ∈ S such that X < I if and only if R(B−A) = R(BA∗−AA∗) and
n
BA∗ <AA∗.
(r) X <I holds for all X ∈S if and only if r(B)=n and BA∗ 4BB∗.
n
(s) There exists an X ∈S such that X 4I if and only if BA∗ <BB∗.
n
(t) X 4I holds for all X ∈S if and only if i (BA∗−AA∗)=n−r(A).
n +
Proof By Lemma 1.10(b), X −P can be written as
X−P =X −P +F UF , (2.21)
0 A A
where X =B∗(AB∗)†B and 04U ∈Cn is arbitrary. Applying Lemma 1.13(b) to (2.21) gives
0 H
maxr(X −P )= max r(X −P +F UF )=r[X −P, F ], (2.22)
0 A A 0 A
X∈S 04U∈CnH
minr(X −P )= min r(X −P +F UF )=i (X −P )+r[X −P, F ]
0 A A + 0 0 A
X∈S 04U∈CnH
X −P F
−i 0 A , (2.23)
+ F 0
A
(cid:20) (cid:21)
X −P F
maxi (X −P )= min i (X −P +F UF )=i 0 A , (2.24)
X∈S + 04U∈CnH + 0 A A +(cid:20) FA 0 (cid:21)
mini (X −P )= min i (X −P +F UF )=i (X −P ), (2.25)
+ + 0 A A + 0
X∈S 04U∈CnH
maxi (X −P )= min i (X −P +F UF )=i (X −P ), (2.26)
− − 0 A A − 0
X∈S 04U∈CnH
X −P F
mini (X −P )= min i (X −P +F UF )=r[X −P, F ]−i 0 A . (2.27)
X∈S − 04U∈CnH ± 0 A A 0 A +(cid:20) FA 0 (cid:21)
Applying (1.21) and (1.31) to X −P gives
0
AB∗ B∗
i (X −P )=i [P −B∗(AB∗)†B∗]=i −i (AB∗)=i (M)−i (AB∗),
± 0 ∓ ∓ B P ∓ ∓ ∓
(cid:20) (cid:21)
9
so that
i (X −P)=i (M), i (X −P )=i (M)−r(B). (2.28)
+ 0 − + 0 +
Substituting (2.11)–(2.13) and (2.28) into (2.22)–(2.27) and simplifying leads to (2.15)–(2.20).
Results (a)–(t) follow from applying Lemma 1.6 to (2.15)–(2.20). (cid:3)
A generalproblem related to Hermitian solutions and Hermitian definite solutions of AX =B
is to establish formulas for calculating the extremal ranks and inertias of P −QXQ∗ subject to
the Hermitian solutions and Hermitiandefinite solutions ofAX =B. The results obtained canbe
used to solve optimization problems of P −QXQ∗ subject to AX =B.
3 Relations between Hermitian solutions of AX = B and
CY = D
In order to compare Hermitian solutions of matrix equations, we first establish some fundamental
formulas for calculating the extremal ranks and inertias of difference of Hermitian solutions of
the two matrix equations AX = B and CY = D, and then use them to characterize relationship
between the Hermitian solutions.
Theorem 3.1 Assume that each of the matrix equations in (1.5) has a Hermitian solution, and
let
S ={X ∈Cn | AX =B}, T ={Y ∈Cn | CY =D}. (3.1)
H H
Also denote
AB∗ 0 A
A B
M = 0 −CD∗ C , N = .
  C D
A∗ C∗ 0 (cid:20) (cid:21)
 
Then,
max r(X −Y )=n+r(N)−r(A)−r(C), (3.2)
X∈S,Y∈T
A BA∗ A BC∗
min r(X −Y )=2r(N)+r(AD∗−BC∗)−r −r , (3.3)
X∈S,Y∈T C DA∗ C DC∗
(cid:20) (cid:21) (cid:20) (cid:21)
max i (X −Y )=n+i (M)−r(A)−r(C), (3.4)
± ±
X∈S,Y∈T
min i (X −Y )=r(N)−i (M). (3.5)
± ∓
X∈S,Y∈T
In consequence, the following hold.
(a) There exist X ∈ S and Y ∈ T such that X − Y is nonsingular if and only if r(N) =
r(A)+r(C).
(b) X−Y is nonsingular for all X ∈S and Y ∈T if and only if
A BA∗ A BC∗
2r(N)+r(AD∗−BC∗)=r +r +n.
C DA∗ C DC∗
(cid:20) (cid:21) (cid:20) (cid:21)
B A A B
(c) S∩T 6=∅ if and only if R ⊆R , [B∗, D∗]= [A∗, C∗].
D C C D
(cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21)
(d) There exist X ∈S and Y ∈T such that X ≻Y (X ≺Y) if and only if i (M)=r(A)+r(C)
+
(i (M)=r(A)+r(C)).
−
(e) X ≻ Y (X ≺ Y) holds for all X ∈ S and Y ∈ T if and only if i (M) = r(N) − n
−
(i (M)=r(N)−n).
+
(f) There exist X ∈S and Y ∈T such that X <Y (X 4Y) holds if and only if i (M)=r(N)
+
(i (M)=r(N)).
−
10