vol20_pp552-573. 195KB Jun 04 2011 12:06:09 AM
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 20, pp. 552-573, August 2010
ELA
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EXTREME RANKS OF (SKEW-)HERMITIAN SOLUTIONS TO A
QUATERNION MATRIX EQUATION∗
QING WEN WANG† AND JING JIANG‡
Abstract. The extreme ranks, i.e., the maximal and minimal ranks, are established for the
general Hermitian solution as well as the general skew-Hermitian solution to the classical matrix
equation AXA∗ + BY B ∗ = C over the quaternion algebra. Also given in this paper are the formulas
of extreme ranks of real matrices Xi , Yi , i = 1, · · · , 4, in a pair (skew-)Hermitian solution X =
X1 + X2 i + X3 j + X4 k, Y = Y1 + Y2 i + Y3 j + Y4 k. Moreover, the necessary and sufficient conditions
for the existence of a real (skew-)symmetric solution, a complex (skew-)Hermitian solution, and a
pure imaginary (skew-)Hermitian solution to the matrix equation mentioned above are presented in
this paper. Also established are expressions of such solutions to the equation when corresponding
solvability conditions are satisfied. The findings of this paper widely extend the known results in the
literature.
Key words. Quaternion matrix equation, Minimal rank, Maximal rank, Hermitian solution,
Skew-Hermitian solution.
AMS subject classifications. 15A03, 15A09, 15A24, 15A33.
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
H = {a0 + a1 i + a2 j + a3 k | i2 = j 2 = k 2 = ijk = −1, a0 , a1 , a2 , a3 ∈ R}
by Hm×n , the identity matrix with the appropriate size by I. For a quaternion matrix
A, we denote the column right space, the row left space of A by R (A), N (A) , respectively, the dimension of R (A) by dim R (A) . By [9], dim R (A) = dim N (A) , which
is called the rank of A, and denoted by r(A). The Moore-Penrose inverse of a matrix
A ∈ Hm×n , denoted by A† , is defined to be the unique matrix X ∈ Hn×m satisfying
∗ Received
by the editors February 14, 2010. Accepted for publication July 31, 2010. Handling
Editor: Michael Neumann.
† Department
of Mathematics, Shanghai University, Shanghai 200444, P.R. China
(wqw858@yahoo.com.cn). Supported by the grants from the Ph.D. Programs Foundation of
Ministry of Education of China (20093108110001), the Scientific Research Innovation Foundation
of Shanghai Municipal Education Commission (09YZ13), Natural Science Foundation of China
(60672160), Singapore MoE Tier 1 Research Grant RG60/07, and Shanghai Leading Academic
Discipline Project (J50101).
‡ Department of Mathematics, Shanghai University, Shanghai 200444, P.R. China, and Department of Mathematics and Applied Mathematics, QiLu Normal University, Jinan 250013, P.R. China.
552
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 20, pp. 552-573, August 2010
ELA
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Extreme Ranks of (Skew-) Hermitian Solutions
553
the following four matrix equations
(1) AXA = A, (2) XAX = X, (3) (AX)∗ = AX, (4) (XA)∗ = XA.
Moreover, LA and RA stand for the two projectors LA = I − A† A, RA = I − AA†
induced by A.
We know that matrix equation is one of the topics of very active research in
matrix theory and applications, and a large number of papers have presented several
methods for solving several matrix equations (e.g. [4]-[6], [10], [23]-[26], [29]-[31], [37],
[39], [42]). As a very classical linear matrix equation,
(1.1)
AXA∗ + BY B ∗ = C
has been investigated by many authors from different aspects. For example, using generalized singular value decomposition, Chang and Wang [2] derived the expressions for
the general symmetric and minimum-2-norm symmetric solutions to (1.1) within the
real setting. Xu, Wei, and Zheng [38] obtained the general form of all least-squares
Hermitian (skew-Hermitian) solutions to (1.1). Liao and Bai [11] used generalized
singular value decomposition to investigate the symmetric positive semidefinite solution to (1.1). Zhang [40] gave necessary and sufficient conditions for the existence of a
Hermitian nonnegative-definite solution to (1.1). Wang and Zhang [32] gave a necessary and sufficient condition for the existence and an expression for the re-nonnegative
definite solution to (1.1) over H by using the decomposition of pairwise matrices.
Research on extreme ranks, i.e., maximal and minimal ranks, of solutions to linear
matrix equations have been actively ongoing for more than 30 years (see [15]-[17],
[19], [20], [27], [28], [33]-[35]). It is worthy to say that minimal and maximal ranks
of a general solution to a matrix equation are very useful in linear programming
computations (see [15]-[17]). In 2009, Liu, Tian and Takane [13] presented formulas
for the maximal and minimal ranks of a Hermitian solution and a skew-Hermitian
solution to the special case of the matrix equation (1.1) in which B = 0. The following
matrix equation:
(1.2)
AXA∗ = C
over C, has been well examined by many authors (see [1], [3], [7], [8], [32], [36], [41]).
Note that, to our knowledge, there has been little information on extreme ranks
of the (skew-)Hermitian solution to the matrix equation (1.1). This paper aims to
consider the formulas of extremal ranks of the general (skew-)Hermitian solution to
(1.1).
The paper is organized as follows. In Section 2, we first give an expression of
the general Hermitian solution to (1.1) by using the generalized inverses of the coefficient matrices of this equation, then derive the formulas of extremal ranks of the
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 20, pp. 552-573, August 2010
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554
Q.W. Wang and J. Jiang
general Hermitian solution to (1.1). We also give the corresponding results on the
skew-Hermitian solution to (1.1). In Section 3, we first present the maximal and
minimal ranks of the real matrices X1 , X2 , X3 , X4 and Y1 , Y2 , Y3 , Y4 in a Hermitian
solution X = X1 + X2 i + X3 j + X4 k, Y = Y1 + Y2 i + Y3 j + Y4 k to (1.1), then give
some necessary and sufficient conditions for (1.1) to have a real symmetric solution,
a complex Hermitian solution, and a pure imaginary Hermitian solution. The corresponding results on the skew-Hermitian solution to (1.1) are also considered. Some
special cases of (1.1) are also considered in Section 4.
2. Ranks of the general Hermitian solution to (1.1). In this section, we
consider the maximal and minimal ranks of the general (skew-)Hermitian solution to
(1.1) over H.
We begin with the following lemma that is due to Tian [18], and can be generalized
to H.
Lemma 2.1. Suppose that the matrix equation
(2.1)
AXB + CY D = E
is consistent over H, where X ∈ Hp×q , Y ∈ Hs×t unknown. Then the general solution
of (2.1) can be expressed by
X = X0 + S1 LG U RH T1 + LA V1 + V2 RB ,
Y = Y0 − S2 LG U RH T2 + LC W1 + W2 RD ,
where X0 and Y0 are a special pair solution of (2.1),
∗
∗
S1 = (Ip , 0) , S2 = (0, Is ) , T1 = (Iq , 0) , T2 = (0, It ) , G = (A, C), H =
µ
B
D
¶
;
U, V1 , V2 , W1 , and W2 are arbitrary quaternion matrices with suitable sizes.
Lemma 2.2. Consider the matrix equation (1.1) where A ∈ Hm×n , B ∈ Hm×p ,
C ∈ Hm×m are given, and X ∈ Hn×n , Y ∈ Hp×p unknown.
1. If C = C ∗ , and (1.1) has a Hermitian solution, then the general Hermitian
solution to (1.1) can be expressed as
X = X0 + S1 LG ZLG S1∗ + LA V + V ∗ LA ,
(2.2)
Y = Y0 − S2 LG ZLG S2∗ + LB W + W ∗ LB ,
where X0 and Y0 are a special pair Hermitian solution of (1.1),
(2.3)
S1 = (In , 0) , S2 = (0, Ip ) , G = (A, B) ;
Z is an arbitrary Hermitian quaternion matrix with consistent size, and V
and W are arbitrary quaternion matrices with suitable sizes.
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 20, pp. 552-573, August 2010
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Extreme Ranks of (Skew-) Hermitian Solutions
555
2. If C = −C ∗ , and (1.1) has a skew-Hermitian solution, then the general skewHermitian solution can be expressed as
X = X0 + S1 LG ZLG S1∗ + LA V − V ∗ LA ,
Y = Y0 − S2 LG ZLG S2∗ + LB W − W ∗ LB ,
where X0 and Y0 are a special pair skew-Hermitian solution of (1.1), and S1 ,
S2 , and G are the same as (2.3); Z is an arbitrary skew-Hermitian quaternion
matrix with consistent size, and V and W are arbitrary quaternion matrices
with suitable sizes.
Proof. We here only prove 1. The proof of 2 can be similarly finished.
Suppose that X1 = X1∗ , Y1 = Y1∗ are an arbitrary pair Hermitian solution of
(1.1). It follows from Lemma 2.1 and R(A,B)∗ = L(A,B) , RA∗ = LA , RB ∗ = LB that
˜ 0 + S1 LG U LG S1∗ + LA V1 + V2 LA ,
X1 = X
Y1 = Y˜0 − S2 LG U LG S2∗ + LB W1 + W2 LB ,
˜ 0 and Y˜0 are a special pair solution of (1.1), and U, V1 , V2 , W1 , and W2 are
where X
³
´
X1 +X1∗ Y1 +Y1∗
arbitrary matrices with suitable sizes. Since
is also a pair Hermitian
, 2
2
solution of (1.1), we get that
1 ˜
1
1
1
∗
∗
∗
˜∗
(X1 + X1∗ ) = (X
0 + X0 ) + S1 LG (U + U )LG S1 + LA (V1 + V2 )
2
2
2
2
1
+ (V1 + V2∗ )∗ LA ,
2
1
1
1
1
Y1 = (Y1 + Y1∗ ) = (Y˜0 + Y˜0∗ ) − S2 LG (U + U ∗ )LG S2∗ + LB (W1 + W2∗ )
2
2
2
2
1
+ (W1 + W2∗ )∗ LB .
2
X1 =
Putting
1 ˜
˜ 0∗ ), Y0 = 1 (Y˜0 + Y˜0∗ ), V = 1 (V1 + V2∗ ),
(X0 + X
2
2
2
1
1
∗
∗
W = (W1 + W2 ), Z = (U + U ),
2
2
X0 =
and noting that X0 and Y0 are a special pair Hermitian solution of (1.1), Z is an
arbitrary Hermitian matrix, we get that any pair Hermitian solution (X1 , Y1 ) of (1.1)
has the form of (2.2).
Conversely, it can be verified that a pair of matrices having the form of (2.2) are
a pair Hermitian solution of (1.1). Therefore, (2.2) is an expression of the general
Hermitian solution to (1.1).
Electronic Journal of Linear Algebra ISSN 1081-3810
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Q.W. Wang and J. Jiang
Tian and Liu in [21] and [12] gave the following extremal ranks of matrix expressions A − BXB ∗ and A − BX − X ∗ B ∗ over a field. The results can be generalized to
H.
Lemma 2.3. Let A ∈ Hm×m , B ∈ Hm×n .
1. If A = A∗ , then
max
X=X ∗ ∈Hn×n
r(A − BXB ∗ ) = r[A, B],
¸
A B
;
min
r(A − BXB ) = 2r[A, B] − r
B∗ 0
X=X ∗ ∈Hn×n
¸¾
½
·
A B
,
max r(A + BX + X ∗ B ∗ ) = min m, r
n×m
B∗ 0
X∈H
¸
·
A B
− 2r(B).
min r(A + BX + X ∗ B ∗ ) = r
B∗ 0
X∈Hn×m
∗
·
2. If A = −A∗ , then
max
X=−X ∗ ∈Hn×n
r(A − BXB ∗ ) = r[A, B],
¸
·
A
B
;
r(A − BXB ∗ ) = 2r[A, B] − r
−B ∗ 0
X=−X ∗ ∈Hn×n
¸¾
½
·
A
B
,
max
r(A + BX − X ∗ B ∗ ) = min m, r
∗
n×m
−B
0
X∈H
¸
·
A
B
− 2r(B).
min r(A + BX − X ∗ B ∗ ) = r
−B ∗ 0
X∈Hn×m
min
The following lemma is due to Marsaglia and Styan [14], which can be generalized
to H.
Lemma 2.4. Let A ∈ Hm×n , B ∈ Hm×k , C ∈ Hl×n , D ∈ Hj×k and E ∈ Hl×i .
Then they satisfy the following rank equalities:
¸
·
A
− r(A),
(a) r(CLA ) = r
C
¸
·
£
¤
B A
− r(C),
(b) r B ALC = r
0 C
¸
¸
·
·
C 0
C
− r(B).
=r
(c) r
A B
RB A
Now we give one of the main theorems in this paper.
Theorem 2.5. Suppose that the matrix equation (1.1), where A ∈ Hm×n , B ∈
, C ∈ Hm×m , C = C ∗ , X ∈ Hn×n , and Y ∈ Hp×p , has a Hermitian solution.
m×p
H
(a) The maximal and minimal ranks of the general Hermitian solution to (1.1)
Electronic Journal of Linear Algebra ISSN 1081-3810
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Extreme Ranks of (Skew-) Hermitian Solutions
are given by:
(2.4)
max
AXA∗ +BY B ∗ =C
r(X) = min{n, r[B, C] + 2n − r(A) − r(G)},
X=X ∗
(2.5)
min
AXA∗ +BY B ∗ =C
r(X) = 2r[B, C] − r
X=X ∗
(2.6)
max
AXA∗ +BY B ∗ =C
min
AXA∗ +BY B ∗ =C
C
B∗
B
0
¸
;
r(Y ) = min{p, r[A, C] + 2p − r(B) − r(G)},
Y =Y ∗
(2.7)
·
r(Y ) = 2r[A, C] − r
Y =Y ∗
·
C
A∗
A
0
¸
.
(b) The rank of the general Hermitian solution X to (1.1) is invariant if and only
if
¸
·
C B
=n
2r[B, C] − r
B∗ 0
or
r[B, C] + r(A) + r(G) = r
·
C
B∗
B
0
¸
+ 2n.
The rank of the general Hermitian solution Y to (1.1) is invariant if and only
if
¸
·
C A
=p
2r[A, C] − r
A∗ 0
or
r[A, C] + r(B) + r(G) = r
·
C
A∗
A
0
¸
+ 2p.
Proof. (a) Applying Lemma 2.2 and Lemma 2.3 to X of (2.2), we get that
max
AXA∗ +BY B ∗ =C
X=X ∗
(2.8)
r(X) = max
r(X0 + S1 LG ZLG S1∗ + LA V + V ∗ LA )
∗
Z=Z ,V
·
½
¸¾
X0 + S1 LG ZLG S1∗ LA
= min n, max∗ r
Z=Z
LA
0
½
µ·
¸ ·
¸
X0 LA
S1 LG
= min n, max∗ r
+
Z[ LG S1∗
Z=Z
LA 0
0
½
·
¸¾
X0 LA S1 LG
= min n, r
;
LA 0
0
¶¾
0 ]
Electronic Journal of Linear Algebra ISSN 1081-3810
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Q.W. Wang and J. Jiang
min
AXA∗ +BY B ∗ =C
r(X) = min
r(X0 + S1 LG ZLG S1∗ + LA V + V ∗ LA )
∗
Z=Z ,V
X=X ∗
·
¸
X0 + S1 LG ZLG S1∗ LA
= min∗ r
− 2r(LA )
Z=Z
LA
0
µ·
¸ ·
¸
¶
X0 LA
S1 LG
= min∗ r
+
Z [ LG S1∗ 0 ] − 2r(LA )
Z=Z
LA 0
0
·
¸
X0
LA S1 LG
X0 LA S1 LG
= 2r
− r LA
0
0
LA 0
0
∗
0
0
LG S1
− 2r(LA ).
(2.9)
By Lemma 2.4, block Gaussian elimination, and AX0 A∗ + BY0 B ∗ = C, we have that
r(LA ) = n − r(A),
X0 In S1 0
·
¸
In 0
0 A∗
X0 S1 LG LA
− r(A) − r(A∗ ) − r(G)
r
= r
0
A 0
0
LA
0
0
0
0 G 0
r
X0
LA
LG S1∗
LA
0
0
S1 LG
0
0
= r[B, C] + 2n − r(A) − r(G),
X0 In S1 0
0
I
0 A∗ 0
n 0
∗
= r S1 0
0
0 G∗ − r(A) − r(A∗ ) − 2r(G)
0
A 0
0
0
0
0 G 0
0
¸
·
C B
+ 2n − 2r(G).
=r
B∗ 0
Substituting above two equalities into (2.8) and (2.9) yields (2.4) and (2.5), respectively.
Similarly, we can get the corresponding results on Y.
(b) The ranks of X, Y, expressed as (2.2), in the general pair Hermitian solution
to (1.1) are invariant if and only if
(2.10)
max r(X) − min r(X) = 0, max r(Y ) − min r(Y ) = 0.
Hence result (b) follows from (2.4)-(2.7), and (2.10).
Similarly, we can get the following.
Theorem 2.6. Let A ∈ Hm×n , B ∈ Hm×p , C ∈ Hm×m , C = −C ∗ , X ∈ Hn×n ,
and Y ∈ Hp×p . Suppose that (1.1) has a skew-Hermitian solution over H.
Electronic Journal of Linear Algebra ISSN 1081-3810
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Extreme Ranks of (Skew-) Hermitian Solutions
(a) The maximal and minimal ranks of the general skew-Hermitian solution to
(1.1) are given by
max
AXA∗ +BY B ∗ =C
r(X) = min{n, r[B, C] + 2n − r(A) − r(G)},
X=−X ∗
min
∗
AXA +BY B ∗ =C
r(X) = 2r[B, C] − r
X=−X ∗
max
AXA∗ +BY B ∗ =C
min
C
−B ∗
B
0
¸
.
r(Y ) = min{p, r[A, C] + 2p − r(B) − r(G)};
Y =−Y ∗
AXA∗ +BY B ∗ =C
·
r(Y ) = 2r[A, C] − r
Y =−Y ∗
·
C
−A∗
A
0
¸
.
(b) The rank of the general skew-Hermitian solution X to (1.1) is invariant if
and only if
¸
·
C
B
= n,
2r[B, C] − r
−B ∗ 0
or r[B, C] + r(A) + r(G) = r
·
C
−B ∗
B
0
¸
+ 2n.
The rank of the general skew-Hermitian solution Y to (1.1) is invariant if
and only if
¸
¸
·
·
C
A
C
A
+2p.
= p, or r[A, C]+r(B)+r(G) = r
2r[A, C]−r
−A∗ 0
−A∗ 0
3. Extreme ranks of the real matrices in a Hermitian solution to (1.1)
over H. In this section, we consider the maximal and minimal ranks of real matrices
Xi , Yi in a pair Hermitian solution X = X1 + X2 i + X3 j + X4 k ∈ Hn×n and Y =
Y1 + Y2 i + Y3 j + Y4 k ∈ Hp×p to (1.1) where
A = A1 + A2 i + A3 j + A4 k, B = B1 + B2 i + B3 j + B4 k, C = C1 + C2 i + C3 j + C4 k,
Ai , Bi , Ci , i = 1, · · · , 4, are real matrices with suitable sizes.
For an arbitrary quaternion matrix M = M1 + M2 i + M3 j + M4 k, we now define
a map φ(·), from Hm×n to R4m×4n , by
M1 −M2 −M3 −M4
M2 M1 −M4 M3
.
(3.1)
φ(M ) =
M3 M4
M1 −M2
M4
−M3
M2
M1
By (3.1), it is easy to verify that φ(·) satisfies the following properties:
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Q.W. Wang and J. Jiang
(a)
(b)
(c)
(d)
M = N ⇐⇒ φ(M ) = φ(N ).
φ(M + N ) = φ(M ) + φ(N ), φ(M N ) = φ(M )φ(N ), φ(kM ) = kφ(M ), k ∈ R.
φ(M ∗ ) = φT (M ), φ(M † ) = φ† (M ).
−1
−1
−1
φ(M ) = Tm
φ(M )Tn = Rm
φ(M )Rn = Sm
φ(M )Sn , where for t = m, n,
Rt =
·
0
I2t
−I2t
0
¸
0
0
, St =
0
It
0
It
Tt =
0
0
−It
0
0
0
0
0
0
−It
0
0
−It
0
0
It
0
0
−It
0
,
0
0
0
0
.
It
0
(e) r [φ(M )] = 4r(M ).
(f) M ∗ = M ⇔ φT (M ) = φ(M ), and M ∗ = −M ⇔ φT (M ) = −φ(M ).
In the following theorems and corollaries, X0 , Y0 , S1 , S2 , and G are defined as in
Lemma 2.2.
Theorem 3.1. The matrix equation (1.1) has a Hermitian solution over H if
and only if the matrix equation
(3.2)
φ(A) (Xij )4×4 φT (A) + φ(B) (Yij )4×4 φT (B) = φ(C), i, j = 1, 2, 3, 4,
has a symmetric solution over R. In this case, the general Hermitian solution of (1.1)
over H can be written as:
(3.3)
(3.4)
X = X1 + X2 i + X3 j + X4 k
1
1 T
T
= (X11 + X22 + X33 + X44 ) + (X12
− X12 + X34
− X34 )i
4
4
1 T
1 T
T
T
− X13 + X24 − X24
)j + (X14
− X14 + X23
− X23 )k,
+ (X13
4
4
Y = Y1 + Y2 i + Y3 j + Y4 k
1
1 T
T
= (Y11 + Y22 + Y33 + Y44 ) + (Y12
− Y12 + Y34
− Y34 )i
4
4
1 T
1 T
T
T
− Y13 + Y24 − Y24
)j + (Y14
− Y14 + Y23
− Y23 )k,
+ (Y13
4
4
T
T
T
T
where Xtt = Xtt
, Ytt = YttT , t = 1, 2, 3, 4; X1j
= Xj1 , Y1j
= Yj1 , j = 2, 3, 4; X2j
=
T
T
T
= Y43 are the general solutions of (3.2) over
= X43 , Y34
Xj2 , Y2j
= Yj2 , j = 3, 4; X34
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561
R. Written in explicit forms, Xi , Yi , i = 1, 2, 3, 4, in (3.3) and (3.4) are
1
1
1
1
P1 φ(X0 )P1T + P2 φ(X0 )P2T + P3 φ(X0 )P3T + P4 φ(X0 )P4T
4
4
4
4
1
T
T
+ [P1 , P2 , P3 , P4 ] E1 ZE1 [P1 , P2 , P3 , P4 ]
4
T
V1
V1
V2 V2 T
T
+ [P1 , P2 , P3 , P4 ] Lφ(A)
V3 + V3 Lφ(A) [P1 , P2 , P3 , P4 ] ,
V4
V4
X1 =
(3.5)
1
1
1
1
P2 φ(X0 )P1T − P1 φ(X0 )P2T + P4 φ(X0 )P3T − P3 φ(X0 )P4T
4
4
4
4
1
T
− [P1 , −P2 , P3 , −P4 ] E1 ZE1T [P2 , P1 , P4 , P3 ]
4
T
V1
V1
V2 V2 T
T
− [−P2 , P1 , −P4 , P3 ] Lφ(A)
V3 + V3 Lφ(A) [−P2 , P1 , −P4 , P3 ] ,
X2 =
(3.6)
V4
V4
1
1
1
1
P3 φ(X0 )P1T − P1 φ(X0 )P3T + P2 φ(X0 )P4T − P4 φ(X0 )P2T
4
4
4
4
1
T
− [P1 , −P2 , −P3 , P4 ]E1 ZE1T [P3 , P4 , P1 , P2 ]
4
T
V1
V1
V2 V2 T
T
− [−P3 , P1 , −P2 , P4 ] Lφ(A)
V3 + V3 Lφ(A) [−P3 , P1 , −P2 , P4 ] ,
X3 =
(3.7)
V4
V4
1
1
1
1
X4 = P4 φ(X0 )P1T − P1 φ(X0 )P4T + P3 φ(X0 )P2T − P2 φ(X0 )P3T
4
4
4
4
1
T
T
− [−P1, − P2 , P3 , P4 ]E1 ZE1 [P4 , P3 , P2 , P1 ]
4
T
V1
V1
V2 V2 T
T
(3.8)
− [−P4 , P1 , −P3 , P2 ] Lφ(A)
V3 + V3 Lφ(A) [−P4 , P1 , −P3 , P2 ] ,
V4
V4
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1
1
1
1
Q1 φ(Y0 )QT1 + Q2 φ(Y0 )QT2 + Q3 φ(Y0 )QT3 + Q4 φ(Y0 )QT4
4
4
4
4
1
T
T
− [Q1 , Q2 , Q3, Q4 ]E2 ZE2 [Q1 , Q2 , Q3 , Q4 ]
4
T
W1
W1
W2 W2 T
T
+ [Q1 , Q2 , Q3 , Q4 ] Lφ(B)
W3 + W3 Lφ(B) [Q1 , Q2 , Q3 , Q4 ] ,
W4
W4
Y1 =
(3.9)
1
1
1
1
Q2 φ(Y0 )QT1 − Q1 φ(Y0 )QT2 + Q4 φ(Y0 )QT3 − Q3 φ(Y0 )QT4
4
4
4
4
1
T
T
+ [Q1 , −Q2 , Q3 , −Q4 ]E2 ZE2 [Q2 , Q1 , Q4 , Q3 ]
4
(3.10)
T
W1
W1
W2 W2 T
T
− [−Q2 , Q1 , −Q4 , Q3 ] Lφ(B)
W3 + W3 Lφ(B) [−Q2 , Q1 , −Q4 , Q3 ] ,
Y2 =
W4
W4
1
1
1
1
Q3 φ(Y0 )QT1 − Q1 φ(Y0 )QT3 + Q2 φ(Y0 )QT4 − Q4 φ(Y0 )QT2
4
4
4
4
1
T
T
+ [Q1 , −Q2 , −Q3 , Q4 ]E2 ZE2 [Q3 , Q4 , Q1 , Q2 ]
4
(3.11)
T
W1
W1
W2 W2 T
T
− [−Q3 , Q1 , −Q2 , Q4 ] Lφ(B)
W3 + W3 Lφ(B) [−Q3 , Q1 , −Q2 , Q4 ] ,
W4
W4
Y3 =
1
1
1
1
Q4 φ(Y0 )QT1 − Q1 φ(Y0 )QT4 + Q3 φ(Y0 )QT2 − Q2 φ(Y0 )QT3
4
4
4
4
1
T
+ [−Q1 , −Q2 , Q3 , Q4 ]E2 ZE2T [Q4 , Q3 , Q2 , Q1 ]
4
(3.12)
T
W1
W1
W2 W2 T
T
− [−Q4 , Q1 , −Q3 , Q2 ] Lφ(B)
W3 + W3 Lφ(B) [−Q4 , Q1 , −Q3 , Q2 ] ,
W4
W4
Y4 =
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Extreme Ranks of (Skew-) Hermitian Solutions
where
P1 = [In , 0, 0, 0] , P2 = [0, In , 0, 0] , P3 = [0, 0, In , 0] , P4 = [0, 0, 0, In ] ,
Q1 = [Ip , 0, 0, 0] , Q2 = [0, Ip , 0, 0] , Q3 = [0, 0, Ip , 0] , Q4 = [0, 0, 0, Ip ] ,
E1 = φ(S1 )Lφ(G) , E2 = φ(S2 )Lφ(G) ;
Z is arbitrary real symmetric matrix, and V1 , V2 , V3 , V4 , W1 , W2 , W3 , and W4 are
arbitrary real matrices with compatible sizes.
Proof. Suppose that (1.1) has a Hermitian solution
X = X1 + X2 i + X3 j + X4 k, Y = Y1 + Y2 i + Y3 j + Y4 k
over H. Applying properties (a) and (b) of φ(·) to (1.1) yields
φ(A)φ(X)φT (A) + φ(B)φ(Y )φT (B) = φ(C), φT (X) = φ(X), φT (Y ) = φ(Y ),
which implies that φ(X), φ(Y ) are real symmetric solutions to (3.2). Conversely,
suppose that (3.2) has a real symmetric solution
X11
X21
X=
X31
X41
i.e.
X12
X22
X32
X42
X13
X23
X33
X43
Y11
X14
Y21
X24
, Y =
Y31
X24
Y41
X44
Y12
Y22
Y32
Y42
Y13
Y23
Y33
Y43
Y14
Y24
,
Y24
Y44
φ(A)XφT (A) + φ(B)Y φT (B) = φ(C), X T = X, Y T = Y.
By property (d) of φ(·), we have that
−1
−1
−1
Tm
φ(A)Tn XTn−1 φT (A)Tm + Tm
φ(B)Tp Y Tp−1 φT (B)Tm = Tm
φ(C)Tm ,
−1
−1
−1
Rm
φ(A)Rn XRn−1 φT (A)Rm + Rm
φ(B)Rp Y Rp−1 φT (B)Rm = Rm
φ(C)Rm ,
−1
−1
−1
Sm
φ(A)Sn XSn−1 φT (A)Sm + Sm
φ(B)Sp Y Sp−1 φT (B)Sm = Sm
φ(C)Sm .
Hence
φ(A)Tn XTn−1 φT (A) + φ(B)Tp Y Tp−1 φT (B) = φ(C),
φ(A)Rn XRn−1 φT (A) + φ(B)Rp Y Rp−1 φT (B) = φ(C),
φ(A)Sn XSn−1 φT (A) + φ(B)Sp Y Sp−1 φT (B) = φ(C),
implying Tn XTn−1 , Tp Y Tp−1 , Rn XRn−1 , Rp Y Rp−1 , Sn XSn−1 , and Sp Y Sp−1 are also
symmetric solutions of (3.2) over R. Thus,
1
1
(X + Tn XTn−1 + Rn XRn−1 + Sn XSn−1 ), (Y + Tp Y Tp−1 + Rp Y Rp−1 + Sp Y Sp−1 )
4
4
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are symmetric solutions of (3.2), where
g
X + Tp XTp−1 + Rp XRp−1 + Sp XSp−1 = (X
ij )4×4 , i, j = 1, 2, 3, 4
and
g
X
11 = X11 + X22 + X33 + X44 ,
T
T
g
X13 = X13 − X13
+ X24
− X24 ,
T
T
g
X21 = X12 − X12 + X34 − X34 ,
T
T
g
X
12 = X12 − X12 + X34 − X34 ,
T
T
g
X
14 = X14 − X14 + X23 − X23 ,
g
X
22 = X11 + X22 + X33 + X44 ,
T
T
g
X
41 = X14 − X14 + X23 − X23 ,
T
T
g
X
43 = X12 − X12 + X34 − X34 ,
T
T
g
X
42 = X13 − X13 + X24 − X24 ,
g
X
44 = X11 + X22 + X33 + X44 .
T
T
g
X
23 = X14 − X14 + X23 − X23 ,
T
T
g
X
31 = X13 − X13 + X24 − X24 ,
g
X
33 = X11 + X22 + X33 + X44 ,
T
T
g
X
24 = X13 − X13 + X24 − X24 ,
T
T
g
X
32 = X14 − X14 + X23 − X23 ,
T
T
g
X
34 = X12 − X12 + X34 − X34 ,
g
Y + Tp Y Tp−1 + Rp Y Rp−1 + Sp Y Sp−1 has a form similar to (X
ij )4×4 . We omit it here
for simplicity.
Let
ˆ = 1 (X11 + X22 + X33 + X44 ) + 1 (X T − X12 + X T − X34 )i
X
34
4
4 12
1 T
1 T
T
T
+ (X13
− X13 + X24 − X24
)j + (X14
− X14 + X23
− X23 )k,
4
4
1
1 T
T
Yˆ = (Y11 + Y22 + Y33 + Y44 ) + (Y12
− Y12 + Y34
− Y34 )i
4
4
1 T
1 T
T
T
− Y13 + Y24 − Y24
)j + (Y14
− Y14 + Y23
− Y23 )k.
+ (Y13
4
4
Then by (3.1),
ˆ = 1 (X + Tn XT −1 + Rn XR−1 + Sn XS −1 ),
φ(X)
n
n
n
4
1
φ(Yˆ ) = (Y + Tp Y Tp−1 + Rp Y Rp−1 + Sp Y Sp−1 ),
4
ˆ
ˆ
we have that X, Y are a pair Hermitian solution of (1.1) by the property (a). Observe
that Xij and Yij , i, j = 1, 2, 3, 4 in (3.2) can be written as
Xij = Pi XPjT , Yij = Qi Y QTj .
From Lemma 2.2, the general solutions to (3.2) can be written as
X = φ(X0 ) + φ(S1 )Lφ(G) ZLφ(G) φT (S1 ) + 4Lφ(A) [V1 , V2 , V3 , V4 ]
T
+ 4 [V1 , V2 , V3 , V4 ] Lφ(A) ,
Y = φ(Y0 ) − φ(S2 )Lφ(G) ZLφ(G) φT (S2 ) + 4Lφ(B) [W1 , W2 , W3 , W4 ]
T
+ 4 [W1 , W2 , W3 , W4 ] Lφ(B) ,
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Extreme Ranks of (Skew-) Hermitian Solutions
where Z is an arbitrary real symmetric matrix and V1 , V2 , V3 , V4 , W1 , W2 , W3 , and
W4 are arbitrary with suitable sizes. Hence, for i, j = 1, 2, 3, 4,
Xij = Pi φ(X0 )PjT + Pi φ(S1 )Lφ(G) ZLφ(G) φT (S1 )PjT + 4Pi Lφ(A) Vj + 4ViT Lφ(A) PjT ,
Yij = Qi φ(Y0 )QTj − Qi φ(S2 )Lφ(G) ZLφ(G) φT (S2 )QTj + 4Qi Lφ(B) Wj + 4WiT Lφ(B) QTj .
Substituting them into (3.3) and (3.4) yields real matrices Xi and Yi , i = 1, 2, 3, 4 in
(3.5)-(3.12).
Now we consider the maximal and minimal ranks of real matrices Xi , Yi in Hermitian solutions X = X1 + X2 i + X3 j+ X4 k and Y = Y1 + Y2 i + Y3 j + Y4 k to
(1.1).
Theorem 3.2. Suppose the matrix equation (1.1) has a Hermitian solution over
H, and for i, j = 1, 2, 3, 4,
¯
½
¾
¯
A(X1 + X2 i + X3 j + X4 k)A∗
n×n ¯
Ji = Xi ∈ R
,
¯ +B(Y1 + Y2 i + Y3 j + Y4 k)B ∗ = C
½
Tj = Yj ∈ Rp×p
Then we have the following:
¯
¾
¯
A(X1 + X2 i + X3 j + X4 k)A∗
¯
.
¯ +B(Y1 + Y2 i + Y3 j + Y4 k)B ∗ = C
(a) The maximal and minimal ranks of Xi in the general Hermitian solution
X = X1 + X2 i + X3 j + X4 k to (1.1) are given by
(
"
#
)
0
0
Ai
max r(Xi ) = min n, r
ei φ(B) φ(C) + 2n − 4r(A) − 4r(G) ;
Xi ∈Ji
A
min r(Xi ) = 2r
Xi ∈Ji
"
0
e
Ai
Ai
0
φ(B) φ(C)
−A2
A1
−2r
A4
−A3
where
A2
−A1
e1 =
A
A4
A3
A3
−A4
−A1
−A2
−A3
−A4
A1
A2
#
0
−r 0
ei
A
−A4
A3
,
−A2
A1
−A1
−A4
−A2
−A3
e2 =
, A
−A3
A2
A4
−A1
Ai
0
0
φT (B)
φ(B) φ(C)
A3
−A4
−A1
−A2
−A4
−A3
,
A2
−A1
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−A1
−A2
e3 =
A
−A3
A4
−A2
−A1
A1 =
−A4
A3
−A1
A2
A3 =
−A3
−A4
A2
−A1
A4
A3
−A1
−A4
−A2
−A3
e4 =
, A
−A3
A2
A4
−A1
A2
−A1
A4
A3
A3
−A4
,
−A1
−A2
A3
−A4
−A1
A2
T
A4
−A3
, A2 =
−A2
−A1
−A1
A2
−A3
A4
A3
−A4
−A1
A2
T
A4
−A3
,
−A2
−A1
−A2
−A1
−A4
A3
T
A4
−A1
A2
−A3
, A4 =
−A3
−A2
−A1
−A4
−A2
−A1
−A4
A3
T
A3
A4
.
−A1
A2
(b) The maximal and minimal ranks of Yj in a Hermitian solution Y1 + Y2 i +
Y3 j + Y4 k to (1.1) are given by
#
)
(
"
Bj
0
0
+ 2p − 4r(B) − 4r(G) ;
max r(Yj ) = min p, r e
Yj ∈Tj
Bj φ(A) φ(C)
min r(Yj ) = 2r
Yj ∈Tj
"
0
e
Bj
0
φ(A)
−B2
B1
−2r
B4
−B3
Bj
φ(C)
−B3
−B4
B1
B2
#
0
−r 0
ej
B
0
0
φ(A)
−B4
B3
,
−B2
B1
Bj
φT (A)
φ(C)
where
B2
−B1
e1 =
B
B4
B3
−B1
−B2
e3 =
B
−B3
B4
B3
−B4
−B1
−B2
−B1
−B4
−B2
−B3
e2 =
, B
−B3
B2
B4
−B1
B3
−B4
−B1
−B2
−B4
−B3
,
B2
−B1
B2
−B1
B4
B3
−B1
−B4
−B2
−B3
e4 =
, B
−B3
B2
B4
−B1
B2
−B1
B4
B3
B3
−B4
,
−B1
−B2
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Extreme Ranks of (Skew-) Hermitian Solutions
−B2
−B1
B1 =
−B4
B3
−B1
B2
B3 =
−B3
−B4
B3
−B4
−B1
B2
T
B4
−B1
B2
−B3
, B2 =
−B3
−B2
−B1
−B4
B3
−B4
−B1
B2
T
B4
−B3
,
−B2
−B1
−B2
−B1
−B4
B3
T
B4
−B1
−B3
B2
, B4 =
−B3
−B2
−B1
−B4
−B2
−B1
−B4
B3
T
B3
B4
.
−B1
B2
Proof. We only derive the maximal and minimal ranks of the matrix X1 . The
others can be established similarly. Applying Lemma 2.3 to (3.5), we get the following
1
max r(X1 ) = max r(M + Pˆ Z Pˆ T + P V + V T P T )
4
Z=Z T ,V
½
¾
= min n, max r(U )
X1 ∈J1
Z=Z T
¸¾
½
·
M P Pˆ
,
= min n, r
PT 0 0
1
min r(X1 ) = min r(M + Pˆ Z Pˆ T + P V + V T P T )
X1 ∈J1
4
Z=Z T ,V
= min r(U ) − 2r(P )
Z=Z T
= 2r
·
M
PT
P
0
Pˆ
0
¸
M
− r PT
Pˆ T
P
0
0
Pˆ
0 − 2r(P ),
0
where
¸ h
·
¸
i
1 Pˆ
P
Z Pˆ T 0 ,
+
0
4 0
1
1
1
1
M = P1 φ(X0 )P1T + P2 φ(X0 )P2T + P3 φ(X0 )P3T + P4 φ(X0 )P4T ,
4
4
4
4
P = [P1 , P2 , P3 , P4 ]Lφ(A) , Pˆ = [P1 , P2 , P3 , P4 ]E1 .
U=
·
M + 41 Pˆ Z Pˆ T
PT
P
0
¸
=
·
M
PT
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By Lemma 2.4, block Gaussian elimination, and AX0 A∗ + BY0 B ∗ = C, we have that
M
D1
D2
0
¸
·
D1T
0
0
Ψ(A∗ )
M P Pˆ
− 4r(φ(A))
=
r
r
0 Ψ(A)
0
0
PT 0 0
0
0
Ψ(G)
0
where
− 4r(φ(A∗ )) − 4r(φ(G))
"
#
0
0
A1
=r
e1 φ(B) φ(C) + 2n − 4r(A) − 4r(G),
A
D1 = [P1 , P2 , P3 , P4 ] , D2 = [P1 , P2 , P3 , P4 ] φ(S1 ),
φ(V )
0
0
0
0
φ(V )
0
0
, V = A, A∗ , G.
Ψ(V ) =
0
0
φ(V )
0
0
0
0
φ(V )
Similarly, we can simplify the following
0
0
A1
M P Pˆ
0
φT (B) + 2n − 4r(G) − 4r(G∗ ),
r PT 0 0 = r 0
e1 φ(B) φ(C)
Pˆ T 0 0
A
−A2
A1
r(P ) = r
A4
−A3
−A3
−A4
A1
A2
−A4
A3
+ n − 4r(A).
−A2
A1
Thus we have the results for extreme ranks of the matrix X1 in (a). Similarly, applying
Lemma 2.3 and Lemma 2.4 to (3.6)–(3.12) yields the other results in (a) and (b).
Corollary 3.3. Suppose that the matrix equation (1.1) has a Hermitian solution
over H, then we have the following:
(a) (1.1) has a real symmetric solution X if and only if, for i = 2, 3, 4,
#
"
0
0
Ai
Ai
0
0
0
φT (B)
2r
ei φ(B) φ(C) = r 0
A
ei φ(B) φ(C)
A
−A2 −A3 −A4
A1 −A4 A3
.
+ 2r
A4
A1 −A2
−A3 A2
A1
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Extreme Ranks of (Skew-) Hermitian Solutions
In that case, the real symmetric solution X can be expressed as X = X1 in
(3.5).
(b) All the solutions of (1.1) for X are real symmetric if and only if
r
"
0
e
Ai
0
Ai
φ(B) φ(C)
#
+ 2n = 4r(A) + 4r(G), i = 2, 3, 4.
In that case, the real symmetric solution X can be expressed as X = X1 in
(3.5).
(c) (1.1) has a complex Hermitian solution X if and only if, for i = 3, 4,
2r
"
0
ei
A
Ai
0
φ(B) φ(C)
#
0
0
Ai
0
φT (B)
= r 0
ei φ(B) φ(C)
A
−A2 −A3 −A4
A1 −A4 A3
+ 2r
A4
A1 −A2
−A3 A2
A1
.
In that case, the complex Hermitian solution X can be expressed as X =
X1 + X2 i, where X1 , X2 are expressed as (3.5) and (3.6).
(d) All the solutions of (1.1) for X are complex Hermitian if and only if
r
"
0
e
Ai
Ai
0
φ(B) φ(C)
#
+ 2n = 4r(A) + 4r(G), i = 3, 4.
In that case, the complex Hermitian solution X can be expressed as X =
X1 + X2 i, where X1 , X2 are expressed as (3.5) and (3.6).
(e) (1.1) has a pure imaginary Hermitian solution X if and only if
2r
"
0
e
A1
0
A1
φ(B) φ(C)
#
A1
0
0
0
φT (B)
= r 0
e1 φ(B) φ(C)
A
−A2 −A3 −A4
A1 −A4 A3
+ 2r
A4
A1 −A2
−A3 A2
A1
.
In that case, the pure imaginary Hermitian solution X can be expressed as
X = X2 i + X3 j + X4 k, where X2 , X3 , and X4 are expressed as (3.6), (3.7),
and (3.8).
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Q.W. Wang and J. Jiang
(f ) All the solutions of (1.1) for X are pure imaginary Hermitian if and only if
#
"
A1
0
0
r
e1 φ(B) φ(C) + 2n = 4r(A) + 4r(G).
A
In that case, the pure imaginary Hermitian solution X can be expressed as
X = X2 i + X3 j + X4 k, where X2 , X3 , and X4 are expressed as (3.6), (3.7),
and (3.8).
Using the same method, we can get the corresponding results on Y.
Remark 3.4. Similarly, we can get the corresponding results on the skewHermitian solution of (1.1).
4. Ranks of Hermitian solution to some special cases of (1.1). In this
section, we consider some special cases of (1.1) over C. When B vanishes, (1.1)
becomes (1.2) where A ∈ Cm×n , C ∈ Cm×m . We get the corresponding results on
(1.2) as follows.
Corollary 4.1. Let A = A1 + A2 i ∈ Cm×n , C = C ∗ = C1 + C2 i ∈ Cm×m be
given. Then
(a) The matrix equation (1.2) has a Hermitian solution if and only if the real
matrix equation
·
¸ ·
¸·
¸
¸·
A1 −A2
C1 −C2
X11 X12
AT2
AT1
(4.1)
=
A2 A1
C2 C1
−AT2 AT1
X21 X22
has a symmetric solution over R. In this case, the general Hermitian solution
of (1.2) over C can be written as
(4.2)
X = X1 + X2 i =
1
1 T
(X11 + X22 ) + (X12
− X12 )i,
2
2
T
T
where Xtt = Xtt
, t = 1, 2; and X12
= X21 are the general solutions of (4.1)
over R. Written in an explicit form, X1 , X2 in (4.2) are
·
¸
1
1
V1
T
T
X1 = P1 φ(X0 )P1 + P2 φ(X0 )P2 + [P1 , P2 ] Lφ(A)
V2
2
2
·
¸T
V1
T
+
LTφ(A) [P1 , P2 ] ,
V2
·
¸
1
1
V1
T
T
X2 = P2 φ(X0 )P1 − P1 φ(X0 )P2 − [−P2 , P1 ] Lφ(A)
V2
2
2
·
¸T
V1
T
+
LTφ(A) [−P2 , P1 ] ,
V2
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571
where X0 = A† C(A† )∗ , P1 = [In , 0] , P2 = [0, In ] , and V1 and V2 are arbitrary
real matrices with compatible sizes.
(b) Put
©
ª
J1 = X1 ∈ Rn×n | A(X1 + X2 i)A∗ = C ,
©
ª
J2 = X2 ∈ Rn×n | A(X1 + X2 i)A∗ = C .
Then we have the following:
(i) The maximal and minimal ranks of X1 in the Hermitian solution X =
X1 + X2 i to (1.2) are given by
C1 −C2 A1
max r(X1 ) = min n, r C2
C1 A2 + 2n − 4r(A) ,
X1 ∈J1
0
AT2
AT1
·
¸
C1 −C2 A1
A1
min r(X1 ) = r C2
.
C1 A2 − 2r
X1 ∈J1
A2
0
AT2
AT1
(ii) The maximal and minimal ranks of X2 in the Hermitian solution X =
X1 + X2 i to (1.2) are given by
C1 −C2 A1
max r(X2 ) = min n, r C2
C1
A2 + 2n − 4r(A) ,
X2 ∈J2
0
AT2 −AT1
·
¸
C1 −C2 A1
A1
min r(X2 ) = r C2
.
C1
A2 − 2r
X2 ∈J2
A2
0
AT2 −AT1
Remark 4.2. Corollary 4.1 is Theorem 2.2 of [13]. Similarly, Theorem 3.2 of
[13] can be regarded as a special case of (1.1) with skew-Hermitian solutions.
Acknowledgment. The authors would like to thank Professor Michael Neumann
and a referee very much for their valuable suggestions and comments, which resulted
in a great improvement of the original manuscript.
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EXTREME RANKS OF (SKEW-)HERMITIAN SOLUTIONS TO A
QUATERNION MATRIX EQUATION∗
QING WEN WANG† AND JING JIANG‡
Abstract. The extreme ranks, i.e., the maximal and minimal ranks, are established for the
general Hermitian solution as well as the general skew-Hermitian solution to the classical matrix
equation AXA∗ + BY B ∗ = C over the quaternion algebra. Also given in this paper are the formulas
of extreme ranks of real matrices Xi , Yi , i = 1, · · · , 4, in a pair (skew-)Hermitian solution X =
X1 + X2 i + X3 j + X4 k, Y = Y1 + Y2 i + Y3 j + Y4 k. Moreover, the necessary and sufficient conditions
for the existence of a real (skew-)symmetric solution, a complex (skew-)Hermitian solution, and a
pure imaginary (skew-)Hermitian solution to the matrix equation mentioned above are presented in
this paper. Also established are expressions of such solutions to the equation when corresponding
solvability conditions are satisfied. The findings of this paper widely extend the known results in the
literature.
Key words. Quaternion matrix equation, Minimal rank, Maximal rank, Hermitian solution,
Skew-Hermitian solution.
AMS subject classifications. 15A03, 15A09, 15A24, 15A33.
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
H = {a0 + a1 i + a2 j + a3 k | i2 = j 2 = k 2 = ijk = −1, a0 , a1 , a2 , a3 ∈ R}
by Hm×n , the identity matrix with the appropriate size by I. For a quaternion matrix
A, we denote the column right space, the row left space of A by R (A), N (A) , respectively, the dimension of R (A) by dim R (A) . By [9], dim R (A) = dim N (A) , which
is called the rank of A, and denoted by r(A). The Moore-Penrose inverse of a matrix
A ∈ Hm×n , denoted by A† , is defined to be the unique matrix X ∈ Hn×m satisfying
∗ Received
by the editors February 14, 2010. Accepted for publication July 31, 2010. Handling
Editor: Michael Neumann.
† Department
of Mathematics, Shanghai University, Shanghai 200444, P.R. China
(wqw858@yahoo.com.cn). Supported by the grants from the Ph.D. Programs Foundation of
Ministry of Education of China (20093108110001), the Scientific Research Innovation Foundation
of Shanghai Municipal Education Commission (09YZ13), Natural Science Foundation of China
(60672160), Singapore MoE Tier 1 Research Grant RG60/07, and Shanghai Leading Academic
Discipline Project (J50101).
‡ Department of Mathematics, Shanghai University, Shanghai 200444, P.R. China, and Department of Mathematics and Applied Mathematics, QiLu Normal University, Jinan 250013, P.R. China.
552
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Extreme Ranks of (Skew-) Hermitian Solutions
553
the following four matrix equations
(1) AXA = A, (2) XAX = X, (3) (AX)∗ = AX, (4) (XA)∗ = XA.
Moreover, LA and RA stand for the two projectors LA = I − A† A, RA = I − AA†
induced by A.
We know that matrix equation is one of the topics of very active research in
matrix theory and applications, and a large number of papers have presented several
methods for solving several matrix equations (e.g. [4]-[6], [10], [23]-[26], [29]-[31], [37],
[39], [42]). As a very classical linear matrix equation,
(1.1)
AXA∗ + BY B ∗ = C
has been investigated by many authors from different aspects. For example, using generalized singular value decomposition, Chang and Wang [2] derived the expressions for
the general symmetric and minimum-2-norm symmetric solutions to (1.1) within the
real setting. Xu, Wei, and Zheng [38] obtained the general form of all least-squares
Hermitian (skew-Hermitian) solutions to (1.1). Liao and Bai [11] used generalized
singular value decomposition to investigate the symmetric positive semidefinite solution to (1.1). Zhang [40] gave necessary and sufficient conditions for the existence of a
Hermitian nonnegative-definite solution to (1.1). Wang and Zhang [32] gave a necessary and sufficient condition for the existence and an expression for the re-nonnegative
definite solution to (1.1) over H by using the decomposition of pairwise matrices.
Research on extreme ranks, i.e., maximal and minimal ranks, of solutions to linear
matrix equations have been actively ongoing for more than 30 years (see [15]-[17],
[19], [20], [27], [28], [33]-[35]). It is worthy to say that minimal and maximal ranks
of a general solution to a matrix equation are very useful in linear programming
computations (see [15]-[17]). In 2009, Liu, Tian and Takane [13] presented formulas
for the maximal and minimal ranks of a Hermitian solution and a skew-Hermitian
solution to the special case of the matrix equation (1.1) in which B = 0. The following
matrix equation:
(1.2)
AXA∗ = C
over C, has been well examined by many authors (see [1], [3], [7], [8], [32], [36], [41]).
Note that, to our knowledge, there has been little information on extreme ranks
of the (skew-)Hermitian solution to the matrix equation (1.1). This paper aims to
consider the formulas of extremal ranks of the general (skew-)Hermitian solution to
(1.1).
The paper is organized as follows. In Section 2, we first give an expression of
the general Hermitian solution to (1.1) by using the generalized inverses of the coefficient matrices of this equation, then derive the formulas of extremal ranks of the
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Q.W. Wang and J. Jiang
general Hermitian solution to (1.1). We also give the corresponding results on the
skew-Hermitian solution to (1.1). In Section 3, we first present the maximal and
minimal ranks of the real matrices X1 , X2 , X3 , X4 and Y1 , Y2 , Y3 , Y4 in a Hermitian
solution X = X1 + X2 i + X3 j + X4 k, Y = Y1 + Y2 i + Y3 j + Y4 k to (1.1), then give
some necessary and sufficient conditions for (1.1) to have a real symmetric solution,
a complex Hermitian solution, and a pure imaginary Hermitian solution. The corresponding results on the skew-Hermitian solution to (1.1) are also considered. Some
special cases of (1.1) are also considered in Section 4.
2. Ranks of the general Hermitian solution to (1.1). In this section, we
consider the maximal and minimal ranks of the general (skew-)Hermitian solution to
(1.1) over H.
We begin with the following lemma that is due to Tian [18], and can be generalized
to H.
Lemma 2.1. Suppose that the matrix equation
(2.1)
AXB + CY D = E
is consistent over H, where X ∈ Hp×q , Y ∈ Hs×t unknown. Then the general solution
of (2.1) can be expressed by
X = X0 + S1 LG U RH T1 + LA V1 + V2 RB ,
Y = Y0 − S2 LG U RH T2 + LC W1 + W2 RD ,
where X0 and Y0 are a special pair solution of (2.1),
∗
∗
S1 = (Ip , 0) , S2 = (0, Is ) , T1 = (Iq , 0) , T2 = (0, It ) , G = (A, C), H =
µ
B
D
¶
;
U, V1 , V2 , W1 , and W2 are arbitrary quaternion matrices with suitable sizes.
Lemma 2.2. Consider the matrix equation (1.1) where A ∈ Hm×n , B ∈ Hm×p ,
C ∈ Hm×m are given, and X ∈ Hn×n , Y ∈ Hp×p unknown.
1. If C = C ∗ , and (1.1) has a Hermitian solution, then the general Hermitian
solution to (1.1) can be expressed as
X = X0 + S1 LG ZLG S1∗ + LA V + V ∗ LA ,
(2.2)
Y = Y0 − S2 LG ZLG S2∗ + LB W + W ∗ LB ,
where X0 and Y0 are a special pair Hermitian solution of (1.1),
(2.3)
S1 = (In , 0) , S2 = (0, Ip ) , G = (A, B) ;
Z is an arbitrary Hermitian quaternion matrix with consistent size, and V
and W are arbitrary quaternion matrices with suitable sizes.
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Extreme Ranks of (Skew-) Hermitian Solutions
555
2. If C = −C ∗ , and (1.1) has a skew-Hermitian solution, then the general skewHermitian solution can be expressed as
X = X0 + S1 LG ZLG S1∗ + LA V − V ∗ LA ,
Y = Y0 − S2 LG ZLG S2∗ + LB W − W ∗ LB ,
where X0 and Y0 are a special pair skew-Hermitian solution of (1.1), and S1 ,
S2 , and G are the same as (2.3); Z is an arbitrary skew-Hermitian quaternion
matrix with consistent size, and V and W are arbitrary quaternion matrices
with suitable sizes.
Proof. We here only prove 1. The proof of 2 can be similarly finished.
Suppose that X1 = X1∗ , Y1 = Y1∗ are an arbitrary pair Hermitian solution of
(1.1). It follows from Lemma 2.1 and R(A,B)∗ = L(A,B) , RA∗ = LA , RB ∗ = LB that
˜ 0 + S1 LG U LG S1∗ + LA V1 + V2 LA ,
X1 = X
Y1 = Y˜0 − S2 LG U LG S2∗ + LB W1 + W2 LB ,
˜ 0 and Y˜0 are a special pair solution of (1.1), and U, V1 , V2 , W1 , and W2 are
where X
³
´
X1 +X1∗ Y1 +Y1∗
arbitrary matrices with suitable sizes. Since
is also a pair Hermitian
, 2
2
solution of (1.1), we get that
1 ˜
1
1
1
∗
∗
∗
˜∗
(X1 + X1∗ ) = (X
0 + X0 ) + S1 LG (U + U )LG S1 + LA (V1 + V2 )
2
2
2
2
1
+ (V1 + V2∗ )∗ LA ,
2
1
1
1
1
Y1 = (Y1 + Y1∗ ) = (Y˜0 + Y˜0∗ ) − S2 LG (U + U ∗ )LG S2∗ + LB (W1 + W2∗ )
2
2
2
2
1
+ (W1 + W2∗ )∗ LB .
2
X1 =
Putting
1 ˜
˜ 0∗ ), Y0 = 1 (Y˜0 + Y˜0∗ ), V = 1 (V1 + V2∗ ),
(X0 + X
2
2
2
1
1
∗
∗
W = (W1 + W2 ), Z = (U + U ),
2
2
X0 =
and noting that X0 and Y0 are a special pair Hermitian solution of (1.1), Z is an
arbitrary Hermitian matrix, we get that any pair Hermitian solution (X1 , Y1 ) of (1.1)
has the form of (2.2).
Conversely, it can be verified that a pair of matrices having the form of (2.2) are
a pair Hermitian solution of (1.1). Therefore, (2.2) is an expression of the general
Hermitian solution to (1.1).
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Q.W. Wang and J. Jiang
Tian and Liu in [21] and [12] gave the following extremal ranks of matrix expressions A − BXB ∗ and A − BX − X ∗ B ∗ over a field. The results can be generalized to
H.
Lemma 2.3. Let A ∈ Hm×m , B ∈ Hm×n .
1. If A = A∗ , then
max
X=X ∗ ∈Hn×n
r(A − BXB ∗ ) = r[A, B],
¸
A B
;
min
r(A − BXB ) = 2r[A, B] − r
B∗ 0
X=X ∗ ∈Hn×n
¸¾
½
·
A B
,
max r(A + BX + X ∗ B ∗ ) = min m, r
n×m
B∗ 0
X∈H
¸
·
A B
− 2r(B).
min r(A + BX + X ∗ B ∗ ) = r
B∗ 0
X∈Hn×m
∗
·
2. If A = −A∗ , then
max
X=−X ∗ ∈Hn×n
r(A − BXB ∗ ) = r[A, B],
¸
·
A
B
;
r(A − BXB ∗ ) = 2r[A, B] − r
−B ∗ 0
X=−X ∗ ∈Hn×n
¸¾
½
·
A
B
,
max
r(A + BX − X ∗ B ∗ ) = min m, r
∗
n×m
−B
0
X∈H
¸
·
A
B
− 2r(B).
min r(A + BX − X ∗ B ∗ ) = r
−B ∗ 0
X∈Hn×m
min
The following lemma is due to Marsaglia and Styan [14], which can be generalized
to H.
Lemma 2.4. Let A ∈ Hm×n , B ∈ Hm×k , C ∈ Hl×n , D ∈ Hj×k and E ∈ Hl×i .
Then they satisfy the following rank equalities:
¸
·
A
− r(A),
(a) r(CLA ) = r
C
¸
·
£
¤
B A
− r(C),
(b) r B ALC = r
0 C
¸
¸
·
·
C 0
C
− r(B).
=r
(c) r
A B
RB A
Now we give one of the main theorems in this paper.
Theorem 2.5. Suppose that the matrix equation (1.1), where A ∈ Hm×n , B ∈
, C ∈ Hm×m , C = C ∗ , X ∈ Hn×n , and Y ∈ Hp×p , has a Hermitian solution.
m×p
H
(a) The maximal and minimal ranks of the general Hermitian solution to (1.1)
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Extreme Ranks of (Skew-) Hermitian Solutions
are given by:
(2.4)
max
AXA∗ +BY B ∗ =C
r(X) = min{n, r[B, C] + 2n − r(A) − r(G)},
X=X ∗
(2.5)
min
AXA∗ +BY B ∗ =C
r(X) = 2r[B, C] − r
X=X ∗
(2.6)
max
AXA∗ +BY B ∗ =C
min
AXA∗ +BY B ∗ =C
C
B∗
B
0
¸
;
r(Y ) = min{p, r[A, C] + 2p − r(B) − r(G)},
Y =Y ∗
(2.7)
·
r(Y ) = 2r[A, C] − r
Y =Y ∗
·
C
A∗
A
0
¸
.
(b) The rank of the general Hermitian solution X to (1.1) is invariant if and only
if
¸
·
C B
=n
2r[B, C] − r
B∗ 0
or
r[B, C] + r(A) + r(G) = r
·
C
B∗
B
0
¸
+ 2n.
The rank of the general Hermitian solution Y to (1.1) is invariant if and only
if
¸
·
C A
=p
2r[A, C] − r
A∗ 0
or
r[A, C] + r(B) + r(G) = r
·
C
A∗
A
0
¸
+ 2p.
Proof. (a) Applying Lemma 2.2 and Lemma 2.3 to X of (2.2), we get that
max
AXA∗ +BY B ∗ =C
X=X ∗
(2.8)
r(X) = max
r(X0 + S1 LG ZLG S1∗ + LA V + V ∗ LA )
∗
Z=Z ,V
·
½
¸¾
X0 + S1 LG ZLG S1∗ LA
= min n, max∗ r
Z=Z
LA
0
½
µ·
¸ ·
¸
X0 LA
S1 LG
= min n, max∗ r
+
Z[ LG S1∗
Z=Z
LA 0
0
½
·
¸¾
X0 LA S1 LG
= min n, r
;
LA 0
0
¶¾
0 ]
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min
AXA∗ +BY B ∗ =C
r(X) = min
r(X0 + S1 LG ZLG S1∗ + LA V + V ∗ LA )
∗
Z=Z ,V
X=X ∗
·
¸
X0 + S1 LG ZLG S1∗ LA
= min∗ r
− 2r(LA )
Z=Z
LA
0
µ·
¸ ·
¸
¶
X0 LA
S1 LG
= min∗ r
+
Z [ LG S1∗ 0 ] − 2r(LA )
Z=Z
LA 0
0
·
¸
X0
LA S1 LG
X0 LA S1 LG
= 2r
− r LA
0
0
LA 0
0
∗
0
0
LG S1
− 2r(LA ).
(2.9)
By Lemma 2.4, block Gaussian elimination, and AX0 A∗ + BY0 B ∗ = C, we have that
r(LA ) = n − r(A),
X0 In S1 0
·
¸
In 0
0 A∗
X0 S1 LG LA
− r(A) − r(A∗ ) − r(G)
r
= r
0
A 0
0
LA
0
0
0
0 G 0
r
X0
LA
LG S1∗
LA
0
0
S1 LG
0
0
= r[B, C] + 2n − r(A) − r(G),
X0 In S1 0
0
I
0 A∗ 0
n 0
∗
= r S1 0
0
0 G∗ − r(A) − r(A∗ ) − 2r(G)
0
A 0
0
0
0
0 G 0
0
¸
·
C B
+ 2n − 2r(G).
=r
B∗ 0
Substituting above two equalities into (2.8) and (2.9) yields (2.4) and (2.5), respectively.
Similarly, we can get the corresponding results on Y.
(b) The ranks of X, Y, expressed as (2.2), in the general pair Hermitian solution
to (1.1) are invariant if and only if
(2.10)
max r(X) − min r(X) = 0, max r(Y ) − min r(Y ) = 0.
Hence result (b) follows from (2.4)-(2.7), and (2.10).
Similarly, we can get the following.
Theorem 2.6. Let A ∈ Hm×n , B ∈ Hm×p , C ∈ Hm×m , C = −C ∗ , X ∈ Hn×n ,
and Y ∈ Hp×p . Suppose that (1.1) has a skew-Hermitian solution over H.
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Extreme Ranks of (Skew-) Hermitian Solutions
(a) The maximal and minimal ranks of the general skew-Hermitian solution to
(1.1) are given by
max
AXA∗ +BY B ∗ =C
r(X) = min{n, r[B, C] + 2n − r(A) − r(G)},
X=−X ∗
min
∗
AXA +BY B ∗ =C
r(X) = 2r[B, C] − r
X=−X ∗
max
AXA∗ +BY B ∗ =C
min
C
−B ∗
B
0
¸
.
r(Y ) = min{p, r[A, C] + 2p − r(B) − r(G)};
Y =−Y ∗
AXA∗ +BY B ∗ =C
·
r(Y ) = 2r[A, C] − r
Y =−Y ∗
·
C
−A∗
A
0
¸
.
(b) The rank of the general skew-Hermitian solution X to (1.1) is invariant if
and only if
¸
·
C
B
= n,
2r[B, C] − r
−B ∗ 0
or r[B, C] + r(A) + r(G) = r
·
C
−B ∗
B
0
¸
+ 2n.
The rank of the general skew-Hermitian solution Y to (1.1) is invariant if
and only if
¸
¸
·
·
C
A
C
A
+2p.
= p, or r[A, C]+r(B)+r(G) = r
2r[A, C]−r
−A∗ 0
−A∗ 0
3. Extreme ranks of the real matrices in a Hermitian solution to (1.1)
over H. In this section, we consider the maximal and minimal ranks of real matrices
Xi , Yi in a pair Hermitian solution X = X1 + X2 i + X3 j + X4 k ∈ Hn×n and Y =
Y1 + Y2 i + Y3 j + Y4 k ∈ Hp×p to (1.1) where
A = A1 + A2 i + A3 j + A4 k, B = B1 + B2 i + B3 j + B4 k, C = C1 + C2 i + C3 j + C4 k,
Ai , Bi , Ci , i = 1, · · · , 4, are real matrices with suitable sizes.
For an arbitrary quaternion matrix M = M1 + M2 i + M3 j + M4 k, we now define
a map φ(·), from Hm×n to R4m×4n , by
M1 −M2 −M3 −M4
M2 M1 −M4 M3
.
(3.1)
φ(M ) =
M3 M4
M1 −M2
M4
−M3
M2
M1
By (3.1), it is easy to verify that φ(·) satisfies the following properties:
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(a)
(b)
(c)
(d)
M = N ⇐⇒ φ(M ) = φ(N ).
φ(M + N ) = φ(M ) + φ(N ), φ(M N ) = φ(M )φ(N ), φ(kM ) = kφ(M ), k ∈ R.
φ(M ∗ ) = φT (M ), φ(M † ) = φ† (M ).
−1
−1
−1
φ(M ) = Tm
φ(M )Tn = Rm
φ(M )Rn = Sm
φ(M )Sn , where for t = m, n,
Rt =
·
0
I2t
−I2t
0
¸
0
0
, St =
0
It
0
It
Tt =
0
0
−It
0
0
0
0
0
0
−It
0
0
−It
0
0
It
0
0
−It
0
,
0
0
0
0
.
It
0
(e) r [φ(M )] = 4r(M ).
(f) M ∗ = M ⇔ φT (M ) = φ(M ), and M ∗ = −M ⇔ φT (M ) = −φ(M ).
In the following theorems and corollaries, X0 , Y0 , S1 , S2 , and G are defined as in
Lemma 2.2.
Theorem 3.1. The matrix equation (1.1) has a Hermitian solution over H if
and only if the matrix equation
(3.2)
φ(A) (Xij )4×4 φT (A) + φ(B) (Yij )4×4 φT (B) = φ(C), i, j = 1, 2, 3, 4,
has a symmetric solution over R. In this case, the general Hermitian solution of (1.1)
over H can be written as:
(3.3)
(3.4)
X = X1 + X2 i + X3 j + X4 k
1
1 T
T
= (X11 + X22 + X33 + X44 ) + (X12
− X12 + X34
− X34 )i
4
4
1 T
1 T
T
T
− X13 + X24 − X24
)j + (X14
− X14 + X23
− X23 )k,
+ (X13
4
4
Y = Y1 + Y2 i + Y3 j + Y4 k
1
1 T
T
= (Y11 + Y22 + Y33 + Y44 ) + (Y12
− Y12 + Y34
− Y34 )i
4
4
1 T
1 T
T
T
− Y13 + Y24 − Y24
)j + (Y14
− Y14 + Y23
− Y23 )k,
+ (Y13
4
4
T
T
T
T
where Xtt = Xtt
, Ytt = YttT , t = 1, 2, 3, 4; X1j
= Xj1 , Y1j
= Yj1 , j = 2, 3, 4; X2j
=
T
T
T
= Y43 are the general solutions of (3.2) over
= X43 , Y34
Xj2 , Y2j
= Yj2 , j = 3, 4; X34
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Extreme Ranks of (Skew-) Hermitian Solutions
561
R. Written in explicit forms, Xi , Yi , i = 1, 2, 3, 4, in (3.3) and (3.4) are
1
1
1
1
P1 φ(X0 )P1T + P2 φ(X0 )P2T + P3 φ(X0 )P3T + P4 φ(X0 )P4T
4
4
4
4
1
T
T
+ [P1 , P2 , P3 , P4 ] E1 ZE1 [P1 , P2 , P3 , P4 ]
4
T
V1
V1
V2 V2 T
T
+ [P1 , P2 , P3 , P4 ] Lφ(A)
V3 + V3 Lφ(A) [P1 , P2 , P3 , P4 ] ,
V4
V4
X1 =
(3.5)
1
1
1
1
P2 φ(X0 )P1T − P1 φ(X0 )P2T + P4 φ(X0 )P3T − P3 φ(X0 )P4T
4
4
4
4
1
T
− [P1 , −P2 , P3 , −P4 ] E1 ZE1T [P2 , P1 , P4 , P3 ]
4
T
V1
V1
V2 V2 T
T
− [−P2 , P1 , −P4 , P3 ] Lφ(A)
V3 + V3 Lφ(A) [−P2 , P1 , −P4 , P3 ] ,
X2 =
(3.6)
V4
V4
1
1
1
1
P3 φ(X0 )P1T − P1 φ(X0 )P3T + P2 φ(X0 )P4T − P4 φ(X0 )P2T
4
4
4
4
1
T
− [P1 , −P2 , −P3 , P4 ]E1 ZE1T [P3 , P4 , P1 , P2 ]
4
T
V1
V1
V2 V2 T
T
− [−P3 , P1 , −P2 , P4 ] Lφ(A)
V3 + V3 Lφ(A) [−P3 , P1 , −P2 , P4 ] ,
X3 =
(3.7)
V4
V4
1
1
1
1
X4 = P4 φ(X0 )P1T − P1 φ(X0 )P4T + P3 φ(X0 )P2T − P2 φ(X0 )P3T
4
4
4
4
1
T
T
− [−P1, − P2 , P3 , P4 ]E1 ZE1 [P4 , P3 , P2 , P1 ]
4
T
V1
V1
V2 V2 T
T
(3.8)
− [−P4 , P1 , −P3 , P2 ] Lφ(A)
V3 + V3 Lφ(A) [−P4 , P1 , −P3 , P2 ] ,
V4
V4
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Q.W. Wang and J. Jiang
1
1
1
1
Q1 φ(Y0 )QT1 + Q2 φ(Y0 )QT2 + Q3 φ(Y0 )QT3 + Q4 φ(Y0 )QT4
4
4
4
4
1
T
T
− [Q1 , Q2 , Q3, Q4 ]E2 ZE2 [Q1 , Q2 , Q3 , Q4 ]
4
T
W1
W1
W2 W2 T
T
+ [Q1 , Q2 , Q3 , Q4 ] Lφ(B)
W3 + W3 Lφ(B) [Q1 , Q2 , Q3 , Q4 ] ,
W4
W4
Y1 =
(3.9)
1
1
1
1
Q2 φ(Y0 )QT1 − Q1 φ(Y0 )QT2 + Q4 φ(Y0 )QT3 − Q3 φ(Y0 )QT4
4
4
4
4
1
T
T
+ [Q1 , −Q2 , Q3 , −Q4 ]E2 ZE2 [Q2 , Q1 , Q4 , Q3 ]
4
(3.10)
T
W1
W1
W2 W2 T
T
− [−Q2 , Q1 , −Q4 , Q3 ] Lφ(B)
W3 + W3 Lφ(B) [−Q2 , Q1 , −Q4 , Q3 ] ,
Y2 =
W4
W4
1
1
1
1
Q3 φ(Y0 )QT1 − Q1 φ(Y0 )QT3 + Q2 φ(Y0 )QT4 − Q4 φ(Y0 )QT2
4
4
4
4
1
T
T
+ [Q1 , −Q2 , −Q3 , Q4 ]E2 ZE2 [Q3 , Q4 , Q1 , Q2 ]
4
(3.11)
T
W1
W1
W2 W2 T
T
− [−Q3 , Q1 , −Q2 , Q4 ] Lφ(B)
W3 + W3 Lφ(B) [−Q3 , Q1 , −Q2 , Q4 ] ,
W4
W4
Y3 =
1
1
1
1
Q4 φ(Y0 )QT1 − Q1 φ(Y0 )QT4 + Q3 φ(Y0 )QT2 − Q2 φ(Y0 )QT3
4
4
4
4
1
T
+ [−Q1 , −Q2 , Q3 , Q4 ]E2 ZE2T [Q4 , Q3 , Q2 , Q1 ]
4
(3.12)
T
W1
W1
W2 W2 T
T
− [−Q4 , Q1 , −Q3 , Q2 ] Lφ(B)
W3 + W3 Lφ(B) [−Q4 , Q1 , −Q3 , Q2 ] ,
W4
W4
Y4 =
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Extreme Ranks of (Skew-) Hermitian Solutions
where
P1 = [In , 0, 0, 0] , P2 = [0, In , 0, 0] , P3 = [0, 0, In , 0] , P4 = [0, 0, 0, In ] ,
Q1 = [Ip , 0, 0, 0] , Q2 = [0, Ip , 0, 0] , Q3 = [0, 0, Ip , 0] , Q4 = [0, 0, 0, Ip ] ,
E1 = φ(S1 )Lφ(G) , E2 = φ(S2 )Lφ(G) ;
Z is arbitrary real symmetric matrix, and V1 , V2 , V3 , V4 , W1 , W2 , W3 , and W4 are
arbitrary real matrices with compatible sizes.
Proof. Suppose that (1.1) has a Hermitian solution
X = X1 + X2 i + X3 j + X4 k, Y = Y1 + Y2 i + Y3 j + Y4 k
over H. Applying properties (a) and (b) of φ(·) to (1.1) yields
φ(A)φ(X)φT (A) + φ(B)φ(Y )φT (B) = φ(C), φT (X) = φ(X), φT (Y ) = φ(Y ),
which implies that φ(X), φ(Y ) are real symmetric solutions to (3.2). Conversely,
suppose that (3.2) has a real symmetric solution
X11
X21
X=
X31
X41
i.e.
X12
X22
X32
X42
X13
X23
X33
X43
Y11
X14
Y21
X24
, Y =
Y31
X24
Y41
X44
Y12
Y22
Y32
Y42
Y13
Y23
Y33
Y43
Y14
Y24
,
Y24
Y44
φ(A)XφT (A) + φ(B)Y φT (B) = φ(C), X T = X, Y T = Y.
By property (d) of φ(·), we have that
−1
−1
−1
Tm
φ(A)Tn XTn−1 φT (A)Tm + Tm
φ(B)Tp Y Tp−1 φT (B)Tm = Tm
φ(C)Tm ,
−1
−1
−1
Rm
φ(A)Rn XRn−1 φT (A)Rm + Rm
φ(B)Rp Y Rp−1 φT (B)Rm = Rm
φ(C)Rm ,
−1
−1
−1
Sm
φ(A)Sn XSn−1 φT (A)Sm + Sm
φ(B)Sp Y Sp−1 φT (B)Sm = Sm
φ(C)Sm .
Hence
φ(A)Tn XTn−1 φT (A) + φ(B)Tp Y Tp−1 φT (B) = φ(C),
φ(A)Rn XRn−1 φT (A) + φ(B)Rp Y Rp−1 φT (B) = φ(C),
φ(A)Sn XSn−1 φT (A) + φ(B)Sp Y Sp−1 φT (B) = φ(C),
implying Tn XTn−1 , Tp Y Tp−1 , Rn XRn−1 , Rp Y Rp−1 , Sn XSn−1 , and Sp Y Sp−1 are also
symmetric solutions of (3.2) over R. Thus,
1
1
(X + Tn XTn−1 + Rn XRn−1 + Sn XSn−1 ), (Y + Tp Y Tp−1 + Rp Y Rp−1 + Sp Y Sp−1 )
4
4
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Q.W. Wang and J. Jiang
are symmetric solutions of (3.2), where
g
X + Tp XTp−1 + Rp XRp−1 + Sp XSp−1 = (X
ij )4×4 , i, j = 1, 2, 3, 4
and
g
X
11 = X11 + X22 + X33 + X44 ,
T
T
g
X13 = X13 − X13
+ X24
− X24 ,
T
T
g
X21 = X12 − X12 + X34 − X34 ,
T
T
g
X
12 = X12 − X12 + X34 − X34 ,
T
T
g
X
14 = X14 − X14 + X23 − X23 ,
g
X
22 = X11 + X22 + X33 + X44 ,
T
T
g
X
41 = X14 − X14 + X23 − X23 ,
T
T
g
X
43 = X12 − X12 + X34 − X34 ,
T
T
g
X
42 = X13 − X13 + X24 − X24 ,
g
X
44 = X11 + X22 + X33 + X44 .
T
T
g
X
23 = X14 − X14 + X23 − X23 ,
T
T
g
X
31 = X13 − X13 + X24 − X24 ,
g
X
33 = X11 + X22 + X33 + X44 ,
T
T
g
X
24 = X13 − X13 + X24 − X24 ,
T
T
g
X
32 = X14 − X14 + X23 − X23 ,
T
T
g
X
34 = X12 − X12 + X34 − X34 ,
g
Y + Tp Y Tp−1 + Rp Y Rp−1 + Sp Y Sp−1 has a form similar to (X
ij )4×4 . We omit it here
for simplicity.
Let
ˆ = 1 (X11 + X22 + X33 + X44 ) + 1 (X T − X12 + X T − X34 )i
X
34
4
4 12
1 T
1 T
T
T
+ (X13
− X13 + X24 − X24
)j + (X14
− X14 + X23
− X23 )k,
4
4
1
1 T
T
Yˆ = (Y11 + Y22 + Y33 + Y44 ) + (Y12
− Y12 + Y34
− Y34 )i
4
4
1 T
1 T
T
T
− Y13 + Y24 − Y24
)j + (Y14
− Y14 + Y23
− Y23 )k.
+ (Y13
4
4
Then by (3.1),
ˆ = 1 (X + Tn XT −1 + Rn XR−1 + Sn XS −1 ),
φ(X)
n
n
n
4
1
φ(Yˆ ) = (Y + Tp Y Tp−1 + Rp Y Rp−1 + Sp Y Sp−1 ),
4
ˆ
ˆ
we have that X, Y are a pair Hermitian solution of (1.1) by the property (a). Observe
that Xij and Yij , i, j = 1, 2, 3, 4 in (3.2) can be written as
Xij = Pi XPjT , Yij = Qi Y QTj .
From Lemma 2.2, the general solutions to (3.2) can be written as
X = φ(X0 ) + φ(S1 )Lφ(G) ZLφ(G) φT (S1 ) + 4Lφ(A) [V1 , V2 , V3 , V4 ]
T
+ 4 [V1 , V2 , V3 , V4 ] Lφ(A) ,
Y = φ(Y0 ) − φ(S2 )Lφ(G) ZLφ(G) φT (S2 ) + 4Lφ(B) [W1 , W2 , W3 , W4 ]
T
+ 4 [W1 , W2 , W3 , W4 ] Lφ(B) ,
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Extreme Ranks of (Skew-) Hermitian Solutions
where Z is an arbitrary real symmetric matrix and V1 , V2 , V3 , V4 , W1 , W2 , W3 , and
W4 are arbitrary with suitable sizes. Hence, for i, j = 1, 2, 3, 4,
Xij = Pi φ(X0 )PjT + Pi φ(S1 )Lφ(G) ZLφ(G) φT (S1 )PjT + 4Pi Lφ(A) Vj + 4ViT Lφ(A) PjT ,
Yij = Qi φ(Y0 )QTj − Qi φ(S2 )Lφ(G) ZLφ(G) φT (S2 )QTj + 4Qi Lφ(B) Wj + 4WiT Lφ(B) QTj .
Substituting them into (3.3) and (3.4) yields real matrices Xi and Yi , i = 1, 2, 3, 4 in
(3.5)-(3.12).
Now we consider the maximal and minimal ranks of real matrices Xi , Yi in Hermitian solutions X = X1 + X2 i + X3 j+ X4 k and Y = Y1 + Y2 i + Y3 j + Y4 k to
(1.1).
Theorem 3.2. Suppose the matrix equation (1.1) has a Hermitian solution over
H, and for i, j = 1, 2, 3, 4,
¯
½
¾
¯
A(X1 + X2 i + X3 j + X4 k)A∗
n×n ¯
Ji = Xi ∈ R
,
¯ +B(Y1 + Y2 i + Y3 j + Y4 k)B ∗ = C
½
Tj = Yj ∈ Rp×p
Then we have the following:
¯
¾
¯
A(X1 + X2 i + X3 j + X4 k)A∗
¯
.
¯ +B(Y1 + Y2 i + Y3 j + Y4 k)B ∗ = C
(a) The maximal and minimal ranks of Xi in the general Hermitian solution
X = X1 + X2 i + X3 j + X4 k to (1.1) are given by
(
"
#
)
0
0
Ai
max r(Xi ) = min n, r
ei φ(B) φ(C) + 2n − 4r(A) − 4r(G) ;
Xi ∈Ji
A
min r(Xi ) = 2r
Xi ∈Ji
"
0
e
Ai
Ai
0
φ(B) φ(C)
−A2
A1
−2r
A4
−A3
where
A2
−A1
e1 =
A
A4
A3
A3
−A4
−A1
−A2
−A3
−A4
A1
A2
#
0
−r 0
ei
A
−A4
A3
,
−A2
A1
−A1
−A4
−A2
−A3
e2 =
, A
−A3
A2
A4
−A1
Ai
0
0
φT (B)
φ(B) φ(C)
A3
−A4
−A1
−A2
−A4
−A3
,
A2
−A1
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−A1
−A2
e3 =
A
−A3
A4
−A2
−A1
A1 =
−A4
A3
−A1
A2
A3 =
−A3
−A4
A2
−A1
A4
A3
−A1
−A4
−A2
−A3
e4 =
, A
−A3
A2
A4
−A1
A2
−A1
A4
A3
A3
−A4
,
−A1
−A2
A3
−A4
−A1
A2
T
A4
−A3
, A2 =
−A2
−A1
−A1
A2
−A3
A4
A3
−A4
−A1
A2
T
A4
−A3
,
−A2
−A1
−A2
−A1
−A4
A3
T
A4
−A1
A2
−A3
, A4 =
−A3
−A2
−A1
−A4
−A2
−A1
−A4
A3
T
A3
A4
.
−A1
A2
(b) The maximal and minimal ranks of Yj in a Hermitian solution Y1 + Y2 i +
Y3 j + Y4 k to (1.1) are given by
#
)
(
"
Bj
0
0
+ 2p − 4r(B) − 4r(G) ;
max r(Yj ) = min p, r e
Yj ∈Tj
Bj φ(A) φ(C)
min r(Yj ) = 2r
Yj ∈Tj
"
0
e
Bj
0
φ(A)
−B2
B1
−2r
B4
−B3
Bj
φ(C)
−B3
−B4
B1
B2
#
0
−r 0
ej
B
0
0
φ(A)
−B4
B3
,
−B2
B1
Bj
φT (A)
φ(C)
where
B2
−B1
e1 =
B
B4
B3
−B1
−B2
e3 =
B
−B3
B4
B3
−B4
−B1
−B2
−B1
−B4
−B2
−B3
e2 =
, B
−B3
B2
B4
−B1
B3
−B4
−B1
−B2
−B4
−B3
,
B2
−B1
B2
−B1
B4
B3
−B1
−B4
−B2
−B3
e4 =
, B
−B3
B2
B4
−B1
B2
−B1
B4
B3
B3
−B4
,
−B1
−B2
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Extreme Ranks of (Skew-) Hermitian Solutions
−B2
−B1
B1 =
−B4
B3
−B1
B2
B3 =
−B3
−B4
B3
−B4
−B1
B2
T
B4
−B1
B2
−B3
, B2 =
−B3
−B2
−B1
−B4
B3
−B4
−B1
B2
T
B4
−B3
,
−B2
−B1
−B2
−B1
−B4
B3
T
B4
−B1
−B3
B2
, B4 =
−B3
−B2
−B1
−B4
−B2
−B1
−B4
B3
T
B3
B4
.
−B1
B2
Proof. We only derive the maximal and minimal ranks of the matrix X1 . The
others can be established similarly. Applying Lemma 2.3 to (3.5), we get the following
1
max r(X1 ) = max r(M + Pˆ Z Pˆ T + P V + V T P T )
4
Z=Z T ,V
½
¾
= min n, max r(U )
X1 ∈J1
Z=Z T
¸¾
½
·
M P Pˆ
,
= min n, r
PT 0 0
1
min r(X1 ) = min r(M + Pˆ Z Pˆ T + P V + V T P T )
X1 ∈J1
4
Z=Z T ,V
= min r(U ) − 2r(P )
Z=Z T
= 2r
·
M
PT
P
0
Pˆ
0
¸
M
− r PT
Pˆ T
P
0
0
Pˆ
0 − 2r(P ),
0
where
¸ h
·
¸
i
1 Pˆ
P
Z Pˆ T 0 ,
+
0
4 0
1
1
1
1
M = P1 φ(X0 )P1T + P2 φ(X0 )P2T + P3 φ(X0 )P3T + P4 φ(X0 )P4T ,
4
4
4
4
P = [P1 , P2 , P3 , P4 ]Lφ(A) , Pˆ = [P1 , P2 , P3 , P4 ]E1 .
U=
·
M + 41 Pˆ Z Pˆ T
PT
P
0
¸
=
·
M
PT
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Q.W. Wang and J. Jiang
By Lemma 2.4, block Gaussian elimination, and AX0 A∗ + BY0 B ∗ = C, we have that
M
D1
D2
0
¸
·
D1T
0
0
Ψ(A∗ )
M P Pˆ
− 4r(φ(A))
=
r
r
0 Ψ(A)
0
0
PT 0 0
0
0
Ψ(G)
0
where
− 4r(φ(A∗ )) − 4r(φ(G))
"
#
0
0
A1
=r
e1 φ(B) φ(C) + 2n − 4r(A) − 4r(G),
A
D1 = [P1 , P2 , P3 , P4 ] , D2 = [P1 , P2 , P3 , P4 ] φ(S1 ),
φ(V )
0
0
0
0
φ(V )
0
0
, V = A, A∗ , G.
Ψ(V ) =
0
0
φ(V )
0
0
0
0
φ(V )
Similarly, we can simplify the following
0
0
A1
M P Pˆ
0
φT (B) + 2n − 4r(G) − 4r(G∗ ),
r PT 0 0 = r 0
e1 φ(B) φ(C)
Pˆ T 0 0
A
−A2
A1
r(P ) = r
A4
−A3
−A3
−A4
A1
A2
−A4
A3
+ n − 4r(A).
−A2
A1
Thus we have the results for extreme ranks of the matrix X1 in (a). Similarly, applying
Lemma 2.3 and Lemma 2.4 to (3.6)–(3.12) yields the other results in (a) and (b).
Corollary 3.3. Suppose that the matrix equation (1.1) has a Hermitian solution
over H, then we have the following:
(a) (1.1) has a real symmetric solution X if and only if, for i = 2, 3, 4,
#
"
0
0
Ai
Ai
0
0
0
φT (B)
2r
ei φ(B) φ(C) = r 0
A
ei φ(B) φ(C)
A
−A2 −A3 −A4
A1 −A4 A3
.
+ 2r
A4
A1 −A2
−A3 A2
A1
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Extreme Ranks of (Skew-) Hermitian Solutions
In that case, the real symmetric solution X can be expressed as X = X1 in
(3.5).
(b) All the solutions of (1.1) for X are real symmetric if and only if
r
"
0
e
Ai
0
Ai
φ(B) φ(C)
#
+ 2n = 4r(A) + 4r(G), i = 2, 3, 4.
In that case, the real symmetric solution X can be expressed as X = X1 in
(3.5).
(c) (1.1) has a complex Hermitian solution X if and only if, for i = 3, 4,
2r
"
0
ei
A
Ai
0
φ(B) φ(C)
#
0
0
Ai
0
φT (B)
= r 0
ei φ(B) φ(C)
A
−A2 −A3 −A4
A1 −A4 A3
+ 2r
A4
A1 −A2
−A3 A2
A1
.
In that case, the complex Hermitian solution X can be expressed as X =
X1 + X2 i, where X1 , X2 are expressed as (3.5) and (3.6).
(d) All the solutions of (1.1) for X are complex Hermitian if and only if
r
"
0
e
Ai
Ai
0
φ(B) φ(C)
#
+ 2n = 4r(A) + 4r(G), i = 3, 4.
In that case, the complex Hermitian solution X can be expressed as X =
X1 + X2 i, where X1 , X2 are expressed as (3.5) and (3.6).
(e) (1.1) has a pure imaginary Hermitian solution X if and only if
2r
"
0
e
A1
0
A1
φ(B) φ(C)
#
A1
0
0
0
φT (B)
= r 0
e1 φ(B) φ(C)
A
−A2 −A3 −A4
A1 −A4 A3
+ 2r
A4
A1 −A2
−A3 A2
A1
.
In that case, the pure imaginary Hermitian solution X can be expressed as
X = X2 i + X3 j + X4 k, where X2 , X3 , and X4 are expressed as (3.6), (3.7),
and (3.8).
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Q.W. Wang and J. Jiang
(f ) All the solutions of (1.1) for X are pure imaginary Hermitian if and only if
#
"
A1
0
0
r
e1 φ(B) φ(C) + 2n = 4r(A) + 4r(G).
A
In that case, the pure imaginary Hermitian solution X can be expressed as
X = X2 i + X3 j + X4 k, where X2 , X3 , and X4 are expressed as (3.6), (3.7),
and (3.8).
Using the same method, we can get the corresponding results on Y.
Remark 3.4. Similarly, we can get the corresponding results on the skewHermitian solution of (1.1).
4. Ranks of Hermitian solution to some special cases of (1.1). In this
section, we consider some special cases of (1.1) over C. When B vanishes, (1.1)
becomes (1.2) where A ∈ Cm×n , C ∈ Cm×m . We get the corresponding results on
(1.2) as follows.
Corollary 4.1. Let A = A1 + A2 i ∈ Cm×n , C = C ∗ = C1 + C2 i ∈ Cm×m be
given. Then
(a) The matrix equation (1.2) has a Hermitian solution if and only if the real
matrix equation
·
¸ ·
¸·
¸
¸·
A1 −A2
C1 −C2
X11 X12
AT2
AT1
(4.1)
=
A2 A1
C2 C1
−AT2 AT1
X21 X22
has a symmetric solution over R. In this case, the general Hermitian solution
of (1.2) over C can be written as
(4.2)
X = X1 + X2 i =
1
1 T
(X11 + X22 ) + (X12
− X12 )i,
2
2
T
T
where Xtt = Xtt
, t = 1, 2; and X12
= X21 are the general solutions of (4.1)
over R. Written in an explicit form, X1 , X2 in (4.2) are
·
¸
1
1
V1
T
T
X1 = P1 φ(X0 )P1 + P2 φ(X0 )P2 + [P1 , P2 ] Lφ(A)
V2
2
2
·
¸T
V1
T
+
LTφ(A) [P1 , P2 ] ,
V2
·
¸
1
1
V1
T
T
X2 = P2 φ(X0 )P1 − P1 φ(X0 )P2 − [−P2 , P1 ] Lφ(A)
V2
2
2
·
¸T
V1
T
+
LTφ(A) [−P2 , P1 ] ,
V2
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Extreme Ranks of (Skew-) Hermitian Solutions
571
where X0 = A† C(A† )∗ , P1 = [In , 0] , P2 = [0, In ] , and V1 and V2 are arbitrary
real matrices with compatible sizes.
(b) Put
©
ª
J1 = X1 ∈ Rn×n | A(X1 + X2 i)A∗ = C ,
©
ª
J2 = X2 ∈ Rn×n | A(X1 + X2 i)A∗ = C .
Then we have the following:
(i) The maximal and minimal ranks of X1 in the Hermitian solution X =
X1 + X2 i to (1.2) are given by
C1 −C2 A1
max r(X1 ) = min n, r C2
C1 A2 + 2n − 4r(A) ,
X1 ∈J1
0
AT2
AT1
·
¸
C1 −C2 A1
A1
min r(X1 ) = r C2
.
C1 A2 − 2r
X1 ∈J1
A2
0
AT2
AT1
(ii) The maximal and minimal ranks of X2 in the Hermitian solution X =
X1 + X2 i to (1.2) are given by
C1 −C2 A1
max r(X2 ) = min n, r C2
C1
A2 + 2n − 4r(A) ,
X2 ∈J2
0
AT2 −AT1
·
¸
C1 −C2 A1
A1
min r(X2 ) = r C2
.
C1
A2 − 2r
X2 ∈J2
A2
0
AT2 −AT1
Remark 4.2. Corollary 4.1 is Theorem 2.2 of [13]. Similarly, Theorem 3.2 of
[13] can be regarded as a special case of (1.1) with skew-Hermitian solutions.
Acknowledgment. The authors would like to thank Professor Michael Neumann
and a referee very much for their valuable suggestions and comments, which resulted
in a great improvement of the original manuscript.
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