vol17_pp343-358. 179KB Jun 04 2011 12:06:00 AM
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 17, pp. 343-358, August 2008
ELA
http://math.technion.ac.il/iic/ela
ON SOLUTIONS TO THE QUATERNION MATRIX EQUATION
AXB + CY D = E ∗
QING-WEN WANG† , HUA-SHENG ZHANG† , AND SHAO-WEN YU†
Abstract. Expressions, as well as necessary and sufficient conditions are given for the existence
of the real and pure imaginary solutions to the consistent quaternion matrix equation AXB +CY D =
E. Formulas are established for the extreme ranks of real matrices Xi , Yi , i = 1, · · · , 4, in a solution
pair X = X1 + X2 i + X3 j + X4 k and Y = Y1 + Y2 i + Y3 j + Y4 k to this equation. Moreover, necessary
and sufficient conditions are derived for all solution pairs X and Y of this equation to be real or
pure imaginary, respectively. Some known results can be regarded as special cases of the results in
this paper.
Key words. Quaternion matrix equation, Extreme rank, Generalized inverse.
AMS subject classifications. 15A03, 15A09, 15A24, 15A33.
1. Introduction. Throughout this paper, we denote the real number field by
R, and 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 H m×n . The symbols I, AT , R (A) , N (A) , dim R (A) stand for the identity matrix
of the appropriate size, the transpose, the column right space, the row left space of a
matrix A over H, and the dimension of R (A) , respectively. By [6], for a quaternion
matrix A, dim R (A) = dim N (A) , which is called the rank of A and denoted by r(A).
A generalized inverse of a matrix A which satisfies AA− A = A is denoted by A− .
Moreover, RA and LA stand for the two projectors LA = I − A− A, RA = I − AA−
induced by A. For an arbitrary quaternion matrix M = M1 + M2 i + M3 j + M4 k, we
define a map φ(·) from H m×n to R4m×4n by
(1.1)
M4
−M 3
φ(M ) =
M2
M1
M3
M4
M1
−M 2
∗ Received
M2
M1
−M 4
M3
M1
−M 2
.
−M 3
−M 4
by the editors August 2, 2007. Accepted for publication July 21, 2008. Handling
Editor: Michael Neumann.
† Department
of Mathematics, Shanghai University, Shanghai 200444, P.R. China
(wqw858@yahoo.com.cn). The first author was supported by the Natural Science Foundation
of China (60672160), Shanghai Pujiang Program (06PJ14039), and by the Shanghai Leading
Academic Discipline Project (J50101).
343
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 17, pp. 343-358, August 2008
ELA
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344
Qing-Wen Wang, Hua-Sheng Zhang, and Shao-Wen Yu
By (1.1), it is easy to verify that φ(·) satisfies the following properties:
(a) M = N ⇐⇒ φ(M ) = φ(N ).
(b) φ(M + N ) = φ(M ) + φ(N ), φ(M N ) = φ(M )φ(N ), φ(kM ) = kφ(M ), k ∈ R.
−1
−1
−1
(c) φ(M ) = −Tm
φ(M )Tn = −Rm
φ(M )Rn = Sm
φ(M )Sn , where t = m, n,
0
0
0 −I t
0
0
0 −I t
It
0
0
0
0 −I 2t
0
It
0
, St= 0
.
Rt =
, T t=
0
0
0
It
I2t
0
0 −I t 0
0
0
0
−I t 0
0
0
0
It
(d) r [φ(M )] = 4r(M ).
Tian [17] in 2003 gave the maximal and minimal ranks of two real matrices X0
and X1 in complex solution X = X0 + iX1 to the classical linear matrix equation
(1.2)
AXB = C.
Liu [11] in 2006 investigated the extreme ranks of solution pairs X and Y, and the
extreme ranks of four real matrices X0 , X1 , Y0 and Y1 in a pair of a complex solutions
X = X0 + iX1 and Y = Y0 + iY1 to the generalized Sylvester matrix equation
(1.3)
AX + Y B = C,
which is widely studied (see, e.g., [4], [5], [7], [8], [11], [13], [22]-[24], [33], [34]). As an
extension of (1.2) and (1.3), the matrix equation
(1.4)
AXB + CY D = E
has been well-studied in matrix theory (see, e.g., [2], [9], [10], [12], [14], [18]-[21],
[35]). For instance, Baksalary and Kala [2] gave necessary and sufficient conditions
¨ uler
for the existence and an representation of the general solution to (1.4). Ozg¨
[12] investigated (1.4) over a principal ideal domain. Huang and Zeng [9] considered
(1.4) over a simple Artinian ring. Wang [20]−[21] investigated (1.4) over an arbitrary
division ring and on any regular ring with identity in 1996 and 2004, respectively.
Tian [19] in 2006 established formulas for extremal ranks of the solution of (1.4) over
the complex number field. Note that, to our knowledge, the real and pure imaginary
solutions to (1.4) over the quaternion algebra H have not been investigated so far in
the literature. Motivated by the work mentioned above, and keeping the applications
and interests of quaternion matrices in view (e.g., [1], [3], [25]-[32], [36]-[37]), in this
paper we consider the real and pure imaginary solutions to (1.4) over H. We first
derive the formulas for extremal ranks of real matrices Xi , Yi , i = 1, · · · , 4, in a solution
pair X = X1 + X2 i + X3 j + X4 k and Y = Y1 + Y2 i + Y3 j + Y4 k to (1.4), then use the
results to derive necessary and sufficient conditions for (1.4) over H to have a real
solution and a pure imaginary solution, respectively. Finally, we establish necessary
and sufficient conditions for all solutions (X, Y ) to (1.4) over H to be real or pure
imaginary.
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 17, pp. 343-358, August 2008
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345
On Solutions to the Quaternion Matrix Equation AXB + CY D = E
2. Main results. In this section, we consider (1.4) over H, 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, D = D1 +D2 i+
D3 j + D4 k, E = E1 + E2 i + E3 j + E4 k are known and X = X1 + X2 i + X3 j + X4 k ∈
H n×s , Y = Y1 + Y2 i + Y3 j + Y4 k ∈ H l×k unknown; here Ai , Bi , Ci , Di , Ei , Xi and
Yi , i = 1, · · · , 4, are real matrices with suitable sizes.
The following lemmas are due to Tian (see [15], [16] and [18]) that can be generalized to H.
Lemma 2.1. Let A ∈ H m×k , B ∈ H l×n , C ∈ H m×i , D ∈ H j×n , E
p (X, Y, Z) = E − AXB − CY − ZD. Then
E
E A C
(2.1)
max r [p (X, Y, Z)] = min m, n, r
,r
B
X,Y,Z
D 0 0
D
(2.2)
E C
E A C
min r [p (X, Y, Z)] = r B 0 − r B 0 0
X,Y,Z
D 0
D 0 0
E A C
− r(C) − r(D).
+r
D 0 0
∈ H m×n and
C
0 ;
0
Lemma 2.2. Let A ∈ H m×n , B ∈ H m×k and C ∈ H l×n . Then
C
B A
C 0
r (B, ALC ) = r
− r(B).
=r
− r(C); r
RB A
0 C
A B
Lemma 2.3. Suppose that matrix equation (1.4) is consistent over H. Then its
general solutions can be expressed as
(2.3)
(2.4)
0 + S
1 + LA V1 + V2 RB ,
1 LG U RH T
X=X
2 + LC W1 + W2 RD
2 LG U RH T
0 + S
Y =Y
0 are a pair of special solutions of (1.4), S
1 = (Ip , 0) , S
2 = (0, Is ) ,
0 and Y
where X
T
T
T
T T
T1 = (Iq , 0) , T2 = (0, It ) , G = (A, C) , H = B , −D
, the quaternion matrices
U, V1 , V2 , W1 and W2 are arbitrary with suitable sizes.
0 , Y
0 , S
1 , S
2 , T
1 , T
2 , G and H are deIn the following theorems and corollaries, X
fined as in Lemma 2.3.
Theorem 2.4. Matrix equation (1.4) is consistent over H if and only if the
matrix equation
(2.5)
φ(A) (Xij )4×4 φ(B) + φ(C) (Yij )4×4 φ(D) = φ(E), i, j = 1, 2, 3, 4,
Electronic Journal of Linear Algebra ISSN 1081-3810
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Qing-Wen Wang, Hua-Sheng Zhang, and Shao-Wen Yu
is consistent over R. In that case, the general solution of (1.4) over H can be written
as
(2.6)
(2.7)
X = X1 + X2 i + X3 j + X4 k
1
1
= (X11 + X22 − X33 − X44 ) + (X12 − X21 + X43 − X34 )i
4
4
1
1
+ (X13 + X31 − X24 − X42 )j + (X41 + X14 + X32 + X23 )k,
4
4
Y = Y1 + Y2 i + Y3 j + Y4 k
1
1
= (Y11 + Y22 − Y33 − Y44 ) + (Y12 − Y21 + Y43 − Y34 )i
4
4
1
1
+ (Y31 + Y13 − Y24 − Y42 )j + (Y41 + Y14 + Y32 + Y23 )k.
4
4
Written in an explicit form, Xi and Yi , i = 1, · · · , 4, in (2.6) are as follows.
(2.8)
1
1
1
1
X1 = P1 φ(X 0 )Q1 + P2 φ(X 0 )Q2 − P3 φ(X 0 )Q3 − P4 φ(X 0 )Q4
4
4
4
4
L 1 Q1
L 1 Q2
1
+ (P1 R1 , P 2 R1 , −P 3 R1 , −P 4 R1 ) U
L 1 Q3
4
L 1 Q4
Rφ(B) Q1
Rφ(B) Q2 ,
+ P1 Lφ(A) , P 2 Lφ(A) , −P 3 Lφ(A) , −P 4 Lφ(A) V +U
−Rφ(B) Q3
−Rφ(B) Q4
(2.9)
1
1
1
1
P1 φ(X0 )Q2 − P2 φ(X0 )Q1 + P4 φ(X0 )Q3 − P3 φ(X0 )Q4
4
4
4
4
L 1 Q2
L 1 Q1
1
+ (P1 R1 , −P2 R1 , −P3 R1 , P4 R1 ) U
L 1 Q4
4
X2 =
L 1 Q3
−Rφ(B) Q2
Rφ(B) Q1 ,
+ −P2 Lφ(A) , P1 Lφ(A) , P4 Lφ(A) , −P3 Lφ(A) V + U
Rφ(B) Q4
−Rφ(B) Q3
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
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On Solutions to the Quaternion Matrix Equation AXB + CY D = E
(2.10)
(2.11)
(2.12)
1
1
1
1
P1 φ(X0 )Q3 − P2 φ(X0 )Q4 + P3 φ(X0 )Q1 − P4 φ(X0 )Q2
4
4
4
4
L 1 Q3
L 1 Q4
1
+ (P1 R1 , −P2 R1 , P3 R1 , −P4 R1 ) U
L 1 Q1
4
L 1 Q2
Rφ(B) Q3
−Rφ(B) Q2
+ P3 Lφ(A) , −P2 Lφ(A) , P1 Lφ(A) , −P4 Lφ(A) V + U
Rφ(B) Q1
−Rφ(B) Q4
X3 =
,
1
1
1
1
P1 φ(X0 )Q4 + P2 φ(X0 )Q3 + P3 φ(X0 )Q2 + P4 φ(X0 )Q1
4
4
4
4
L 1 Q4
L 1 Q3
1
+ (P1 R1 , P2 R1 , P3 R1 , P4 R1 ) U
L 1 Q2
4
L 1 Q1
Rφ(B) Q4
Rφ(B) Q3 ,
+ P4 Lφ(A) , P3 Lφ(A) , P2 Lφ(A) , P1 Lφ(A) V + U
Rφ(B) Q2
Rφ(B) Q1
X4 =
1
1
1
1
S1 φ(Y0 )T1 + S2 φ(Y0 )T2 − S3 φ(Y0 )T3 − S4 φ(Y0 )T4
4
4
4
4
L2 T1
L2 T2
1
+ (S1 R2 , S2 R2 , −S3 R2 , −S4 R2 ) U
L2 T3
4
Y1 =
L2 T4
Rφ(B) T1
+ W Rφ(B) T2 ,
+ S1 Lφ(A) , S2 Lφ(A) , −S3 Lφ(A) , −S4 Lφ(A) U
−Rφ(B) T3
−Rφ(B) T4
Electronic Journal of Linear Algebra ISSN 1081-3810
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(2.13)
(2.14)
(2.15)
where
Qing-Wen Wang, Hua-Sheng Zhang, and Shao-Wen Yu
1
1
1
1
S1 φ(Y0 )T2 − S2 φ(Y0 )T1 − S3 φ(Y0 )T4 + S4 φ(Y0 )T3
4
4
4
4
L2 T2
L2 T1
1
+ (S1 R2 , −S2 R2 , −S3 R2 , S4 R2 ) U
L2 T4
4
L2 T3
−Rφ(B) T2
Rφ(B) T1
+W
+ −S2 Lφ(A) , S1 Lφ(A) , S4 Lφ(A) , −S3 Lφ(A) U
Rφ(B) T4
−Rφ(B) T3
Y2 =
,
1
1
1
1
S1 φ(Y0 )T3 − S2 φ(Y0 )T4 + S3 φ(Y0 )T1 − S4 φ(Y0 )T2
4
4
4
4
L2 T3
L2 T4
1
+ (S1 R2 , −S2 R2 , S3 R2 , −S4 R2 ) U
L2 T1
4
L2 T2
Rφ(B) T3
−Rφ(B) T4
+W
+ S3 Lφ(A) , −S 4 Lφ(A) , S 1 Lφ(A) , −S 2 Lφ(A) U
Rφ(B) T1
−Rφ(B) T2
Y3 =
1
1
1
1
S1 φ(Y0 )T4 + S2 φ(Y0 )T3 + S3 φ(Y0 )T2 + S4 φ(Y0 )T1
4
4
4
4
L2 T4
L2 T3
1
+ (S1 R2 , S2 R2 , S3 R2 , S4 R2 ) U
L2 T2
4
L2 T1
Rφ(B) T4
Rφ(B) T3
+W
+ S4 Lφ(A) , S3 Lφ(A) , S2 Lφ(A) , S1 Lφ(A) U
Rφ(B) T2
Rφ(B) T1
,
Y4 =
,
P1 = (Ip , 0, 0, 0) , P2 = (0, Ip , 0, 0) , P3 = (0, 0, Ip , 0) , P4 = (0, 0, 0, Ip ) ,
S1 = (Im , 0, 0, 0) , S2 = (0, Im , 0, 0) , S3 = (0, 0, Im , 0) , S4 = (0, 0, 0, Im ) ,
Q1 = (In , 0, 0, 0)T , Q2 = (0, In , 0, 0)T , Q3 = (0, 0, In , 0)T , Q4 = (0, 0, 0, In )T ,
T
T
T
T
T1 = (Iq , 0, 0, 0) , T2 = (0, Iq , 0, 0) , T3 = (0, 0, Iq , 0) , T4 = (0, 0, 0, Iq ) ,
1 )L
R1 = φ(S
φ(G) , L1 = Rφ(H) φ(T1 ), R2 = φ(S2 )Lφ(G) , L2 = Rφ(H) φ(T2 ),
Electronic Journal of Linear Algebra ISSN 1081-3810
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On Solutions to the Quaternion Matrix Equation AXB + CY D = E
349
, U
ˆ , V and W are arbitrary real matrices with compatible sizes.
and U, U
Proof. Suppose that (1.4) has a solution pair X, Y over H. Applying properties
(a) and (b) of φ(·) above to (1.4) yields
φ(A)φ(X)φ(B) + φ(C)φ(Y )φ(D) = φ(E),
which implies that φ(X), φ(Y ) is a solution pair to (2.5). Conversely, suppose that
(2.5) has a solution pair
ˆ = (Xij )
ˆ
X
4×4 , Y = (Yij )4×4 , i, j = 1, 2, 3, 4.
i.e.
ˆ
φ(A)Xφ(B)
+ φ(C)Yˆ φ(D) = φ(E).
Then, applying property (c) of φ(·) above, it yields
−1
−1
−1
ˆ q−1 φ(B)Tn + Tm
Tm
φ(E)Tn ,
φ(C)Tp Yˆ Tq−1 φ(D)Tn = −Tm
φ(A)Tp XT
−1
−1
−1
−1
−1
ˆ
ˆ
Rm φ(A)Rp XRq φ(B)Rn + Rm φ(C)Rp Y Rq φ(D)Rn = −Rm φ(E)Rn ,
ˆ −1 φ(B)Sn + S −1 φ(C)Sp Yˆ S −1 φ(D)Sn = S −1 φ(E)Sn .
S −1 φ(A)Sp XS
m
q
m
q
m
Hence,
ˆ q−1 φ(B) + φ(C)Tm Yˆ Tq−1 φ(D) = φ(E),
φ(A)Tp XT
ˆ −1 φ(B) + φ(C)Rm Yˆ R−1 φ(D) = φ(E),
φ(A)Rp XR
n
q
ˆ n−1 φ(B) + φ(C)Sm Yˆ Sq−1 φ(D) = φ(E),
φ(A)Sp XS
ˆ −1 , Sm Yˆ S −1 are
ˆ −1 , Rm Yˆ R−1 and Sp XS
ˆ −1 , Tm Yˆ T −1 , Rp XR
which implies that Tp XT
q
n
q
n
q
n
also solutions of (2.5). Thus,
1 ˆ
ˆ −1 ), 1 (Yˆ − Tm Yˆ T −1 − Rm Yˆ R−1 + Sm Yˆ S −1 )
ˆ −1 + Sp XS
ˆ −1 − Rp XR
(X − Tp XT
n
q
n
q
n
q
4
4
is a solution pair of (2.5), where
ˆ − Tp XT
ˆ n−1 = X
ˆ n−1 + Sp XS
ˆ n−1 − Rp XR
X
, i, j = 1, 2, 3, 4
ij
4×4
and
X
11
X
13
X
21
X
23
X
31
X
33
X
41
X
43
= X11 + X22 − X33 − X44 , X
12
= X13 + X31 − X24 − X42 , X
14
= X21 − X12 + X34 − X43 , X
22
= X41 + X14 + X32 + X23 , X
24
= X13 + X31 − X24 − X42 , X
32
= X33 + X44 − X11 − X22 , X
34
= X41 + X14 + X32 + X23 , X
42
= X12 − X21 + X43 − X34 , X
44
= X12 − X21 + X43 − X34 ,
= X41 + X14 + X32 + X23 ,
= X11 + X22 − X33 − X44 ,
= X24 + X42 − X13 − X31 ,
= X41 + X14 + X32 + X23 ,
= X21 − X12 + X34 − X43 ,
= X24 + X42 − X13 − X31 ,
= X33 + X44 − X11 − X22 .
Electronic Journal of Linear Algebra ISSN 1081-3810
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350
Qing-Wen Wang, Hua-Sheng Zhang, and Shao-Wen Yu
Yˆ − Tm Yˆ Tq−1 − Rm Yˆ Rq−1 + Sm Yˆ Sq−1 has a form similar to
here for simplicity.
Let
X
ij
4×4
. We omit it
= 1 (X11 + X22 − X33 − X44 ) + 1 (X12 − X21 + X43 − X34 )i
X
4
4
1
1
+ (X13 + X31 − X24 − X42 )j + (X41 + X14 + X32 + X23 )k,
4
4
1
1
Y = (Y11 + Y22 − Y33 − Y44 ) + (Y12 − Y21 + Y43 − Y34 )i
4
4
1
1
+ (Y31 + Y13 − Y24 − Y42 )j + (Y41 + Y14 + Y32 + Y23 )k.
4
4
Then, by (1.1),
1 ˆ
ˆ n−1 ),
ˆ n−1 + Sp XS
ˆ n−1 − Rp XR
(X − Tp XT
4
1
φ(Y ) = (Yˆ − Tm Yˆ Tq−1 − Rm Yˆ Rq−1 + Sm Yˆ Sq−1 )
4
=
φ(X)
Y is a
is a solution pair of (2.5). Hence, by property (a) of φ(·), we know that X,
solution pair of (1.4). The above discussion shows that the two matrix equations
(1.4) and (2.5) have the same solvability condition and their solutions satisfy (2.6)
and (2.7). Observe that Xij and Yij , i, j = 1, 2, 3, 4 in (2.5) can be written as
j , Yij = Si Y Tj .
Xij = Pi XQ
From Lemma 2.3, the general solution to (2.5) can be written as
= φ(X0 ) + φ(S
1 )Lφ(G) U Rφ(H) φ(T
1 ) + 4Lφ(A) V + 4U
T Rφ(B) ,
X
Hence,
+ 4W T Rφ(D) .
2 )Lφ(G) U Rφ(H) φ(T
2 ) + 4Lφ(C) U
Y = φ(Y0 ) + φ(S
1 )Lφ(G) U Rφ(H) φ(T
1 )Qj + 4Pi Lφ(A) Vj + 4Ui Rφ(B) Qj ,
Xij = Pi φ(X0 )Qj + Pi φ(S
ˆj + 4Wi Rφ(D) Tj ,
2 )Lφ(G) U Rφ(H) φ(T
2 )Tj + 4Si Lφ(C) U
Yij = Si φ(Y0 )Tj + Si φ(S
ˆ1 , · · · , U
ˆ4 ∈ R4p×q , W1 , · · · , W4 ∈
where U, U1 , · · · , U4 ∈ Rp×4q , V1 , · · · , V4 ∈ R4p×q , U
p×4q
R
are arbitrary, and
T = UT , UT , UT , UT ,
V = (V1 , V2 , V3 , V4 ) , U
1
2
3
4
= U
ˆ1 , U
ˆ2 , U
ˆ3 , U
ˆ4 , W T = W1T , W2T , W3T , W4T .
U
Putting them into (2.6), (2.7) yields the eight real matrices X1 , · · · , X4 and Y1 , · · · , Y4 ,
in (2.8)-(2.15).
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 17, pp. 343-358, August 2008
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351
On Solutions to the Quaternion Matrix Equation AXB + CY D = E
Now we consider the extreme ranks of real matrices X1 , · · · , X4 and Y1 , · · · , Y4
in the solution (X, Y ) to (1.4) over H.
Theorem 2.5. Suppose that (1.4) over H is consistent, and for i, j = 1, 2, 3, 4,
Ji = Xi ∈ Rn×s |A(X1 + X2 i + X3 j + X4 k)B + C (Y1 + Y2 i + Y3 j + Y4 k) D = E ,
Kj = Yj ∈ Rl×k |A(X1 + X2 i + X3 j + X4 k)B + C (Y1 + Y2 i + Y3 j + Y4 k) D = E .
Then we have the following:
(a) The maximal and minimal ranks of Xi in the solution X = X1 + X2 i + X3j + X4 k
to (1.4) are given by
i
0
0
B
max r(Xi ) = min p, q, p + q + r
− 4(r (B) − r (A, C)),
i φ(C) φ(E)
Xi ∈Ji
A
i
0
B
B
p + q + r 0 φ(D) − 4r(A) − 4r
,
D
i φ(E)
A
min r(Xi ) = r
Xi ∈Ji
where
1
A
3
A
1
B
3
B
A2
−A1
=
A4
A3
−A1
−A2
=
−A3
A4
−B2
= −B3
B4
−B1
=
−B2
B4
0
i
A
0
φ(C)
A3
−A4
−A1
−A2
A2
−A1
A4
A3
−B1
−B4
B3
B2
−B1
B3
i
B
φ(E)
0
+r 0
i
A
i
0
B
φ(D) −r 0
i
A
φ(E)
−A4
−A1
−A2
−A3
,A
2 =
−A3
A2
−A1
A4
−A4
−A1
−A3
−A2
, A4 =
−A3
A2
−A1
A4
−B4 −B3
2 =
−B1 B2 , B
−B2 −B1
B3 −B4
4 =
−B4 −B3 , B
−B2 −B1
A3
−A4
−A1
−A2
A2
−A1
A4
A3
−B1
−B3
B4
−B1
−B2
−B3
0
0
φ(C)
−A4
−A3
,
A2
−A1
A3
−A4
,
−A1
−A2
B2
−B4
B3
B3
−B1
−B2
B2
−B1
B4
B3
−B4
−B1
i
B
φ(D)
φ(E)
−B4
B2 ,
−B1
−B4
−B3 .
B2
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(b) The maximal and minimal ranks of Yj in the solution Y = Y1 + Y2 i + Y3 j + Y4 k
to (1.4) are given by
j
0
0
D
max r(Yj ) = min s, t, s + t + r
− 4r(B) − 4r (C, A)
j φ(A) φ(E)
Yj ∈Kj
C
j
0
D
B
s + t + r 0 φ(B) − 4r(C) − 4r
,
D
j φ(E)
C
min r(Yj ) = r
Yj ∈Kj
where
1
C
3
C
1
D
3
D
C2
−C1
=
C4
C3
−C1
−C2
=
−C3
C4
−D2
= −D3
D4
−D1
= −D2
D4
0
j
C
0
φ(A)
C3
−C4
−C1
−C2
C2
−C1
C4
C3
−D1
D4
D3
D2
−D1
D3
j
D
φ(E)
0
+r 0
j
C
j
0
D
φ(B) −r 0
j
C
φ(E)
−C4
−C1 C3
−C2 −C4
−C3
2 =
,C
−C3 −C1
C2
−C1
C4 −C2
−C4
−C1 C2
−C2 −C1
−C3
,C
4 =
−C3 C4
C2
−C1
C4
C3
−D4 −D3
−D1
2 = −D3
−D1 D2 , D
−D2 −D1
D4
D3 −D4
−D1
4 = −D2
−D4 −D3 , D
−D2 −D1
−D3
0
0
φ(A)
j
D
φ(B)
φ(E)
−C4
−C3
,
C2
−C1
C3
−C4
,
−C1
−C2
D2
D4
D3
D2
−D1
D4
−D4
D2 ,
−D1
D3 −D4
−D4 −D3 .
−D1 D2
D3
−D1
−D2
Proof. We only derive the maximal and minimal ranks of the matrix X1 ; the
other r (Xi ) , r(Yj ) can be established similarly. Applying (2.1) and (2.2) to (2.8), we
get the following
max r(X1 ) = min {p, q, r(M1 ), r(M2 )} ,
X1 ∈J1
min r(X1 ) = r(M1 ) + r(M2 ) − r(M3 ) − r (P ) − r (Q) ,
X1 ∈J1
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353
where
0 P P
P
X
P
M1 =
0 , M3 = Q 0 0 ,
0
0 0
Q
0
0 = 1 P1 φ(X0 )Q1 + 1 P2 φ(X0 )Q2 − 1 P3 φ(X0 )Q3 − 1 P4 φ(X0 )Q4 ,
X
4
4
4
4
P = P1 Lφ(A) , P2 Lφ(A) , −P3 Lφ(A) , −P4 Lφ(A) ,
1 )Lφ(G) , P2 φ(S
1 )Lφ(G) , −P3 φ(S
1 )Lφ(G) , −P4 φ(S
1 )Lφ(G) ,
P = P1 φ(S
1 )Q1
Rφ(H) φ(T
Rφ(B) Q1
1 )Q2
Rφ(B) Q2
Rφ(H) φ(T
,Q
=
Q=
.
−Rφ(B) Q3
1 )Q3
Rφ(H) φ(T
1 )Q4
−Rφ(B) Q4
Rφ(H) φ(T
0
X
Q
P
0
0
X
, M2 = Q
Q
By Lemma 2.2, block Gaussian elimination and AX0 B + CY0 D = E, we have that
r (LA ) = p − r (A) , r (RB ) = q − r (B) ,
0
P
0
P
X
0
0
Ψ (B)
Q
r(M1 ) =
− 4r (φ(A)) − 4r (φ(B)) − 4r (φ(G))
0 Ψ (A)
0
0
0
0
Ψ (G)
0
#
$
1 )
0
0
φ(B
=r
+ p + q − 4r (B) − 4r (G)
1 ) φ(C) φ(E)
φ(A
where
1 ), P2 φ(S
1 ), −P3 φ(S
1 ), −P4 φ(S
1 ) ,
P = (P1 , P2 , −P3 , −P4 ) , P = P1 φ(S
Q1
φ(Z)
0
0
0
φ(Z)
0
0
= Q2 , Ψ (Z) = 0
, Z = A, B, G.
Q
−Q3
0
0
φ(Z)
0
−Q4
0
0
0
φ(Z)
In the same manner, we can
0
r(M2 ) = r 0
1
A
0
r(M3 ) = r 0
1
A
simplify r(M2 ) and r(M3 ) as follows.
1
B
φ(D) + p + q − 4r(A) − 4r (H) ,
φ(E)
1
0
B
0
φ(D) + p + q − 4r (G) − 4r (H) .
φ(C) φ(E)
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Thus, we have the results for the extreme ranks of the matrix X1 in (a). Similarly,
applying (2.1) and (2.2) to (2.9), (2.10), (2.11), (2.12), (2.13), (2.14), (2.15) yields the
other results in (a), (b).
i , B
i , C
j and D
j (i, j = 1, 2, 3, 4) are defined as in
In the following corollaries, A
Theorem 2.5.
Corollary 2.6. Suppose that (1.4) over H is consistent. Then
(a) (1.4) has a real solution for X if and only if
i
i
0
B
0
0
B
0
0
Bi
+ r 0 φ(D) = r 0
r
0
φ(D) , i = 2, 3, 4.
i φ(C) φ(E)
A
i φ(E)
i φ(C) φ(E)
A
A
In that case, the real solution X can be expressed as X = X1 in (2.8).
(b) All the solutions of (1.4) for X are real if and only if
i
0
0
B
p+q+r
= 4r (B) + 4r (A, C)
i φ(C) φ(E)
A
or
0
p+q+r 0
i
A
i
B
B
, i = 2, 3, 4.
φ(D) = 4r(A) + 4r
D
φ(E)
In that case, the real solutions X can be expressed as X = X1 in (2.8).
(c) (1.4) has a pure imaginary solution for X if and only if
1
1
0
0
B
0
B
0
0
B1
r
+ r 0 φ(D) = r 0
0
φ(D) .
A1 φ(C) φ(E)
1 φ(E)
1 φ(C) φ(E)
A
A
In that case, the pure imaginary solution can be expressed as X = X2 i + X3 j + X4 k
where X2 , X3 and X4 are expressed as (2.9), (2.10) and (2.11), respectively.
(d) All the solutions of (1.4) for X are pure imaginary if and only if
1
0
0
B
p+q+r
= 4r (B) + 4r (A, C)
1 φ(C) φ(E)
A
or
0
p+q +r 0
1
A
1
B
B
.
φ(D) = 4r(A) + 4r
D
φ(E)
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355
In that case, the pure imaginary solutions can be expressed as X = X2 i + X3 j + X4 k,
where X2 , X3 and X4 are expressed as (2.9), (2.10) and (2.11), respectively.
Using the same method, we can get the corresponding results on Y.
We now consider all solution pairs X and Y to (1.4) over H to be real or pure
imaginary, respectively.
Theorem 2.7. Suppose that (1.4) over H is consistent, q (X1 , Y1 ) = E −AX1 B−
CY1 D, and J1 , K1 as in Theorem 2.2 are two independent sets. Then
(2.16)
&
%
B
.
max r [q (X1 , Y1 )] = min r (A) + r (C) − r (A, C) , r (B) + r (D) − r
X1 ∈J1 ,
D
Y ∈K
1
1
In particular:
(a) The solutions (X1 , Y1 ) of (1.4) over H are independent, that is, for any X1 ∈ J1
and Y1 ∈ K1 the solutions X1 and Y1 satisfy (1.4) over H if and only if
(2.17)
r (A, C) = r (A) + r (C) or r
B
D
= r (B) + r (D) .
(b) Under (2.17), the general solution of (1.4) over H can be written as the two
independent forms
1 + LA V1 + V2 RB ,
1 LG U1 RH T
X = X0 + S
(2.18)
2 + LC W1 + W2 RD ,
2 LG U2 RH T
Y = Y0 + S
(2.19)
where (X0 , Y0 ) is a particular real solution of (1.4), U1 , U2 , V1 , V2 , W1 and W2 are
arbitrary.
Proof. Writing (2.3) and (2.4) as two independent matrix expressions, that is,
replacing U in (2.3) and (2.4) by U1 and U2 respectively, then taking them into
E − AX1 B − CY1 D yields
1 B − C S
2 D
1 LG U1 RH T
2 LG U2 RH T
q (X1 , Y1 ) = E − AX0 B − CY0 D − AS
1 B,
1 LG (−U1 + U2 ) RH T
= AS
where U1 and U2 are arbitrary. Then by (2.1), it follows that
1 B
1 LG (−U1 + U2 ) RH T
max r [q (X1 , Y1 )] = max r AS
X1 ∈J1 ,
Y1 ∈K1
U1 ,U2
(
'
1 B
1 LG , r RH T
= min r AS
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Qing-Wen Wang, Hua-Sheng Zhang, and Shao-Wen Yu
where
1 LG = r
r AS
1
AS
G
− r (G) = r (A) + r (C) − r (G) ,
1 B = r T
1 B, H − r (H) = r (B) + r (D) − r (H) .
r RH T
Therefore, we have (2.16). The result in (2.17) follows directly from (2.16) and the
solutions in (2.18) and (2.19) follow from (2.3) and (2.4).
Corollary 2.8. Suppose that (1.4) over H is consistent, and (2.17) holds. Then
(a) All solution pairs X and Y to (1.4) over H are real if and only if
i
0
0
B
p+q+r
= 4r (B) + 4r (A, C)
i φ(C) φ(E)
A
or
and
0
p+q +r 0
i
A
s+t+r
or
0
s+t+r 0
bj
C
0
j
C
i
B
B
φ(D) = 4r(A) + 4r
D
φ(E)
0
φ(A)
j
D
φ(E)
= 4r(B) + 4r (A, C)
bj
D
B
φ(B)
= 4r(C) + 4r
, i, j = 2, 3, 4.
D
φ(E)
(b) All solution pairs X and Y to (1.4) over H are imaginary if and only if
$
#
1
0
0
B
p+q+r
1 φ(C) φ(E) = 4r (B) + 4r (A, C)
A
or
and
0
p+q+r 0
1
A
s+t+r
0
C1
1
B
B
φ(D) = 4r(A) + 4r
D
φ(E)
0
φ(A)
1
D
φ(E)
= 4r(B) + 4r (C, A)
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On Solutions to the Quaternion Matrix Equation AXB + CY D = E
or
0
s+t+r 0
1
C
357
1
D
B
.
φ(B) = 4r(C) + 4r
D
φ(E)
Remark 2.9. The main results of [11], [17] and [19] can be regarded as special
cases of results in this paper.
Acknowledgment. The authors would like to thank Professor Michael Neumann
and a referee very much for their valuable suggestions and comments, which resulted
in great improvement of the original manuscript.
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A publication of the International Linear Algebra Society
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ON SOLUTIONS TO THE QUATERNION MATRIX EQUATION
AXB + CY D = E ∗
QING-WEN WANG† , HUA-SHENG ZHANG† , AND SHAO-WEN YU†
Abstract. Expressions, as well as necessary and sufficient conditions are given for the existence
of the real and pure imaginary solutions to the consistent quaternion matrix equation AXB +CY D =
E. Formulas are established for the extreme ranks of real matrices Xi , Yi , i = 1, · · · , 4, in a solution
pair X = X1 + X2 i + X3 j + X4 k and Y = Y1 + Y2 i + Y3 j + Y4 k to this equation. Moreover, necessary
and sufficient conditions are derived for all solution pairs X and Y of this equation to be real or
pure imaginary, respectively. Some known results can be regarded as special cases of the results in
this paper.
Key words. Quaternion matrix equation, Extreme rank, Generalized inverse.
AMS subject classifications. 15A03, 15A09, 15A24, 15A33.
1. Introduction. Throughout this paper, we denote the real number field by
R, and 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 H m×n . The symbols I, AT , R (A) , N (A) , dim R (A) stand for the identity matrix
of the appropriate size, the transpose, the column right space, the row left space of a
matrix A over H, and the dimension of R (A) , respectively. By [6], for a quaternion
matrix A, dim R (A) = dim N (A) , which is called the rank of A and denoted by r(A).
A generalized inverse of a matrix A which satisfies AA− A = A is denoted by A− .
Moreover, RA and LA stand for the two projectors LA = I − A− A, RA = I − AA−
induced by A. For an arbitrary quaternion matrix M = M1 + M2 i + M3 j + M4 k, we
define a map φ(·) from H m×n to R4m×4n by
(1.1)
M4
−M 3
φ(M ) =
M2
M1
M3
M4
M1
−M 2
∗ Received
M2
M1
−M 4
M3
M1
−M 2
.
−M 3
−M 4
by the editors August 2, 2007. Accepted for publication July 21, 2008. Handling
Editor: Michael Neumann.
† Department
of Mathematics, Shanghai University, Shanghai 200444, P.R. China
(wqw858@yahoo.com.cn). The first author was supported by the Natural Science Foundation
of China (60672160), Shanghai Pujiang Program (06PJ14039), and by the Shanghai Leading
Academic Discipline Project (J50101).
343
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344
Qing-Wen Wang, Hua-Sheng Zhang, and Shao-Wen Yu
By (1.1), it is easy to verify that φ(·) satisfies the following properties:
(a) M = N ⇐⇒ φ(M ) = φ(N ).
(b) φ(M + N ) = φ(M ) + φ(N ), φ(M N ) = φ(M )φ(N ), φ(kM ) = kφ(M ), k ∈ R.
−1
−1
−1
(c) φ(M ) = −Tm
φ(M )Tn = −Rm
φ(M )Rn = Sm
φ(M )Sn , where t = m, n,
0
0
0 −I t
0
0
0 −I t
It
0
0
0
0 −I 2t
0
It
0
, St= 0
.
Rt =
, T t=
0
0
0
It
I2t
0
0 −I t 0
0
0
0
−I t 0
0
0
0
It
(d) r [φ(M )] = 4r(M ).
Tian [17] in 2003 gave the maximal and minimal ranks of two real matrices X0
and X1 in complex solution X = X0 + iX1 to the classical linear matrix equation
(1.2)
AXB = C.
Liu [11] in 2006 investigated the extreme ranks of solution pairs X and Y, and the
extreme ranks of four real matrices X0 , X1 , Y0 and Y1 in a pair of a complex solutions
X = X0 + iX1 and Y = Y0 + iY1 to the generalized Sylvester matrix equation
(1.3)
AX + Y B = C,
which is widely studied (see, e.g., [4], [5], [7], [8], [11], [13], [22]-[24], [33], [34]). As an
extension of (1.2) and (1.3), the matrix equation
(1.4)
AXB + CY D = E
has been well-studied in matrix theory (see, e.g., [2], [9], [10], [12], [14], [18]-[21],
[35]). For instance, Baksalary and Kala [2] gave necessary and sufficient conditions
¨ uler
for the existence and an representation of the general solution to (1.4). Ozg¨
[12] investigated (1.4) over a principal ideal domain. Huang and Zeng [9] considered
(1.4) over a simple Artinian ring. Wang [20]−[21] investigated (1.4) over an arbitrary
division ring and on any regular ring with identity in 1996 and 2004, respectively.
Tian [19] in 2006 established formulas for extremal ranks of the solution of (1.4) over
the complex number field. Note that, to our knowledge, the real and pure imaginary
solutions to (1.4) over the quaternion algebra H have not been investigated so far in
the literature. Motivated by the work mentioned above, and keeping the applications
and interests of quaternion matrices in view (e.g., [1], [3], [25]-[32], [36]-[37]), in this
paper we consider the real and pure imaginary solutions to (1.4) over H. We first
derive the formulas for extremal ranks of real matrices Xi , Yi , i = 1, · · · , 4, in a solution
pair X = X1 + X2 i + X3 j + X4 k and Y = Y1 + Y2 i + Y3 j + Y4 k to (1.4), then use the
results to derive necessary and sufficient conditions for (1.4) over H to have a real
solution and a pure imaginary solution, respectively. Finally, we establish necessary
and sufficient conditions for all solutions (X, Y ) to (1.4) over H to be real or pure
imaginary.
Electronic Journal of Linear Algebra ISSN 1081-3810
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On Solutions to the Quaternion Matrix Equation AXB + CY D = E
2. Main results. In this section, we consider (1.4) over H, 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, D = D1 +D2 i+
D3 j + D4 k, E = E1 + E2 i + E3 j + E4 k are known and X = X1 + X2 i + X3 j + X4 k ∈
H n×s , Y = Y1 + Y2 i + Y3 j + Y4 k ∈ H l×k unknown; here Ai , Bi , Ci , Di , Ei , Xi and
Yi , i = 1, · · · , 4, are real matrices with suitable sizes.
The following lemmas are due to Tian (see [15], [16] and [18]) that can be generalized to H.
Lemma 2.1. Let A ∈ H m×k , B ∈ H l×n , C ∈ H m×i , D ∈ H j×n , E
p (X, Y, Z) = E − AXB − CY − ZD. Then
E
E A C
(2.1)
max r [p (X, Y, Z)] = min m, n, r
,r
B
X,Y,Z
D 0 0
D
(2.2)
E C
E A C
min r [p (X, Y, Z)] = r B 0 − r B 0 0
X,Y,Z
D 0
D 0 0
E A C
− r(C) − r(D).
+r
D 0 0
∈ H m×n and
C
0 ;
0
Lemma 2.2. Let A ∈ H m×n , B ∈ H m×k and C ∈ H l×n . Then
C
B A
C 0
r (B, ALC ) = r
− r(B).
=r
− r(C); r
RB A
0 C
A B
Lemma 2.3. Suppose that matrix equation (1.4) is consistent over H. Then its
general solutions can be expressed as
(2.3)
(2.4)
0 + S
1 + LA V1 + V2 RB ,
1 LG U RH T
X=X
2 + LC W1 + W2 RD
2 LG U RH T
0 + S
Y =Y
0 are a pair of special solutions of (1.4), S
1 = (Ip , 0) , S
2 = (0, Is ) ,
0 and Y
where X
T
T
T
T T
T1 = (Iq , 0) , T2 = (0, It ) , G = (A, C) , H = B , −D
, the quaternion matrices
U, V1 , V2 , W1 and W2 are arbitrary with suitable sizes.
0 , Y
0 , S
1 , S
2 , T
1 , T
2 , G and H are deIn the following theorems and corollaries, X
fined as in Lemma 2.3.
Theorem 2.4. Matrix equation (1.4) is consistent over H if and only if the
matrix equation
(2.5)
φ(A) (Xij )4×4 φ(B) + φ(C) (Yij )4×4 φ(D) = φ(E), i, j = 1, 2, 3, 4,
Electronic Journal of Linear Algebra ISSN 1081-3810
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Qing-Wen Wang, Hua-Sheng Zhang, and Shao-Wen Yu
is consistent over R. In that case, the general solution of (1.4) over H can be written
as
(2.6)
(2.7)
X = X1 + X2 i + X3 j + X4 k
1
1
= (X11 + X22 − X33 − X44 ) + (X12 − X21 + X43 − X34 )i
4
4
1
1
+ (X13 + X31 − X24 − X42 )j + (X41 + X14 + X32 + X23 )k,
4
4
Y = Y1 + Y2 i + Y3 j + Y4 k
1
1
= (Y11 + Y22 − Y33 − Y44 ) + (Y12 − Y21 + Y43 − Y34 )i
4
4
1
1
+ (Y31 + Y13 − Y24 − Y42 )j + (Y41 + Y14 + Y32 + Y23 )k.
4
4
Written in an explicit form, Xi and Yi , i = 1, · · · , 4, in (2.6) are as follows.
(2.8)
1
1
1
1
X1 = P1 φ(X 0 )Q1 + P2 φ(X 0 )Q2 − P3 φ(X 0 )Q3 − P4 φ(X 0 )Q4
4
4
4
4
L 1 Q1
L 1 Q2
1
+ (P1 R1 , P 2 R1 , −P 3 R1 , −P 4 R1 ) U
L 1 Q3
4
L 1 Q4
Rφ(B) Q1
Rφ(B) Q2 ,
+ P1 Lφ(A) , P 2 Lφ(A) , −P 3 Lφ(A) , −P 4 Lφ(A) V +U
−Rφ(B) Q3
−Rφ(B) Q4
(2.9)
1
1
1
1
P1 φ(X0 )Q2 − P2 φ(X0 )Q1 + P4 φ(X0 )Q3 − P3 φ(X0 )Q4
4
4
4
4
L 1 Q2
L 1 Q1
1
+ (P1 R1 , −P2 R1 , −P3 R1 , P4 R1 ) U
L 1 Q4
4
X2 =
L 1 Q3
−Rφ(B) Q2
Rφ(B) Q1 ,
+ −P2 Lφ(A) , P1 Lφ(A) , P4 Lφ(A) , −P3 Lφ(A) V + U
Rφ(B) Q4
−Rφ(B) Q3
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On Solutions to the Quaternion Matrix Equation AXB + CY D = E
(2.10)
(2.11)
(2.12)
1
1
1
1
P1 φ(X0 )Q3 − P2 φ(X0 )Q4 + P3 φ(X0 )Q1 − P4 φ(X0 )Q2
4
4
4
4
L 1 Q3
L 1 Q4
1
+ (P1 R1 , −P2 R1 , P3 R1 , −P4 R1 ) U
L 1 Q1
4
L 1 Q2
Rφ(B) Q3
−Rφ(B) Q2
+ P3 Lφ(A) , −P2 Lφ(A) , P1 Lφ(A) , −P4 Lφ(A) V + U
Rφ(B) Q1
−Rφ(B) Q4
X3 =
,
1
1
1
1
P1 φ(X0 )Q4 + P2 φ(X0 )Q3 + P3 φ(X0 )Q2 + P4 φ(X0 )Q1
4
4
4
4
L 1 Q4
L 1 Q3
1
+ (P1 R1 , P2 R1 , P3 R1 , P4 R1 ) U
L 1 Q2
4
L 1 Q1
Rφ(B) Q4
Rφ(B) Q3 ,
+ P4 Lφ(A) , P3 Lφ(A) , P2 Lφ(A) , P1 Lφ(A) V + U
Rφ(B) Q2
Rφ(B) Q1
X4 =
1
1
1
1
S1 φ(Y0 )T1 + S2 φ(Y0 )T2 − S3 φ(Y0 )T3 − S4 φ(Y0 )T4
4
4
4
4
L2 T1
L2 T2
1
+ (S1 R2 , S2 R2 , −S3 R2 , −S4 R2 ) U
L2 T3
4
Y1 =
L2 T4
Rφ(B) T1
+ W Rφ(B) T2 ,
+ S1 Lφ(A) , S2 Lφ(A) , −S3 Lφ(A) , −S4 Lφ(A) U
−Rφ(B) T3
−Rφ(B) T4
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(2.13)
(2.14)
(2.15)
where
Qing-Wen Wang, Hua-Sheng Zhang, and Shao-Wen Yu
1
1
1
1
S1 φ(Y0 )T2 − S2 φ(Y0 )T1 − S3 φ(Y0 )T4 + S4 φ(Y0 )T3
4
4
4
4
L2 T2
L2 T1
1
+ (S1 R2 , −S2 R2 , −S3 R2 , S4 R2 ) U
L2 T4
4
L2 T3
−Rφ(B) T2
Rφ(B) T1
+W
+ −S2 Lφ(A) , S1 Lφ(A) , S4 Lφ(A) , −S3 Lφ(A) U
Rφ(B) T4
−Rφ(B) T3
Y2 =
,
1
1
1
1
S1 φ(Y0 )T3 − S2 φ(Y0 )T4 + S3 φ(Y0 )T1 − S4 φ(Y0 )T2
4
4
4
4
L2 T3
L2 T4
1
+ (S1 R2 , −S2 R2 , S3 R2 , −S4 R2 ) U
L2 T1
4
L2 T2
Rφ(B) T3
−Rφ(B) T4
+W
+ S3 Lφ(A) , −S 4 Lφ(A) , S 1 Lφ(A) , −S 2 Lφ(A) U
Rφ(B) T1
−Rφ(B) T2
Y3 =
1
1
1
1
S1 φ(Y0 )T4 + S2 φ(Y0 )T3 + S3 φ(Y0 )T2 + S4 φ(Y0 )T1
4
4
4
4
L2 T4
L2 T3
1
+ (S1 R2 , S2 R2 , S3 R2 , S4 R2 ) U
L2 T2
4
L2 T1
Rφ(B) T4
Rφ(B) T3
+W
+ S4 Lφ(A) , S3 Lφ(A) , S2 Lφ(A) , S1 Lφ(A) U
Rφ(B) T2
Rφ(B) T1
,
Y4 =
,
P1 = (Ip , 0, 0, 0) , P2 = (0, Ip , 0, 0) , P3 = (0, 0, Ip , 0) , P4 = (0, 0, 0, Ip ) ,
S1 = (Im , 0, 0, 0) , S2 = (0, Im , 0, 0) , S3 = (0, 0, Im , 0) , S4 = (0, 0, 0, Im ) ,
Q1 = (In , 0, 0, 0)T , Q2 = (0, In , 0, 0)T , Q3 = (0, 0, In , 0)T , Q4 = (0, 0, 0, In )T ,
T
T
T
T
T1 = (Iq , 0, 0, 0) , T2 = (0, Iq , 0, 0) , T3 = (0, 0, Iq , 0) , T4 = (0, 0, 0, Iq ) ,
1 )L
R1 = φ(S
φ(G) , L1 = Rφ(H) φ(T1 ), R2 = φ(S2 )Lφ(G) , L2 = Rφ(H) φ(T2 ),
Electronic Journal of Linear Algebra ISSN 1081-3810
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On Solutions to the Quaternion Matrix Equation AXB + CY D = E
349
, U
ˆ , V and W are arbitrary real matrices with compatible sizes.
and U, U
Proof. Suppose that (1.4) has a solution pair X, Y over H. Applying properties
(a) and (b) of φ(·) above to (1.4) yields
φ(A)φ(X)φ(B) + φ(C)φ(Y )φ(D) = φ(E),
which implies that φ(X), φ(Y ) is a solution pair to (2.5). Conversely, suppose that
(2.5) has a solution pair
ˆ = (Xij )
ˆ
X
4×4 , Y = (Yij )4×4 , i, j = 1, 2, 3, 4.
i.e.
ˆ
φ(A)Xφ(B)
+ φ(C)Yˆ φ(D) = φ(E).
Then, applying property (c) of φ(·) above, it yields
−1
−1
−1
ˆ q−1 φ(B)Tn + Tm
Tm
φ(E)Tn ,
φ(C)Tp Yˆ Tq−1 φ(D)Tn = −Tm
φ(A)Tp XT
−1
−1
−1
−1
−1
ˆ
ˆ
Rm φ(A)Rp XRq φ(B)Rn + Rm φ(C)Rp Y Rq φ(D)Rn = −Rm φ(E)Rn ,
ˆ −1 φ(B)Sn + S −1 φ(C)Sp Yˆ S −1 φ(D)Sn = S −1 φ(E)Sn .
S −1 φ(A)Sp XS
m
q
m
q
m
Hence,
ˆ q−1 φ(B) + φ(C)Tm Yˆ Tq−1 φ(D) = φ(E),
φ(A)Tp XT
ˆ −1 φ(B) + φ(C)Rm Yˆ R−1 φ(D) = φ(E),
φ(A)Rp XR
n
q
ˆ n−1 φ(B) + φ(C)Sm Yˆ Sq−1 φ(D) = φ(E),
φ(A)Sp XS
ˆ −1 , Sm Yˆ S −1 are
ˆ −1 , Rm Yˆ R−1 and Sp XS
ˆ −1 , Tm Yˆ T −1 , Rp XR
which implies that Tp XT
q
n
q
n
q
n
also solutions of (2.5). Thus,
1 ˆ
ˆ −1 ), 1 (Yˆ − Tm Yˆ T −1 − Rm Yˆ R−1 + Sm Yˆ S −1 )
ˆ −1 + Sp XS
ˆ −1 − Rp XR
(X − Tp XT
n
q
n
q
n
q
4
4
is a solution pair of (2.5), where
ˆ − Tp XT
ˆ n−1 = X
ˆ n−1 + Sp XS
ˆ n−1 − Rp XR
X
, i, j = 1, 2, 3, 4
ij
4×4
and
X
11
X
13
X
21
X
23
X
31
X
33
X
41
X
43
= X11 + X22 − X33 − X44 , X
12
= X13 + X31 − X24 − X42 , X
14
= X21 − X12 + X34 − X43 , X
22
= X41 + X14 + X32 + X23 , X
24
= X13 + X31 − X24 − X42 , X
32
= X33 + X44 − X11 − X22 , X
34
= X41 + X14 + X32 + X23 , X
42
= X12 − X21 + X43 − X34 , X
44
= X12 − X21 + X43 − X34 ,
= X41 + X14 + X32 + X23 ,
= X11 + X22 − X33 − X44 ,
= X24 + X42 − X13 − X31 ,
= X41 + X14 + X32 + X23 ,
= X21 − X12 + X34 − X43 ,
= X24 + X42 − X13 − X31 ,
= X33 + X44 − X11 − X22 .
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Qing-Wen Wang, Hua-Sheng Zhang, and Shao-Wen Yu
Yˆ − Tm Yˆ Tq−1 − Rm Yˆ Rq−1 + Sm Yˆ Sq−1 has a form similar to
here for simplicity.
Let
X
ij
4×4
. We omit it
= 1 (X11 + X22 − X33 − X44 ) + 1 (X12 − X21 + X43 − X34 )i
X
4
4
1
1
+ (X13 + X31 − X24 − X42 )j + (X41 + X14 + X32 + X23 )k,
4
4
1
1
Y = (Y11 + Y22 − Y33 − Y44 ) + (Y12 − Y21 + Y43 − Y34 )i
4
4
1
1
+ (Y31 + Y13 − Y24 − Y42 )j + (Y41 + Y14 + Y32 + Y23 )k.
4
4
Then, by (1.1),
1 ˆ
ˆ n−1 ),
ˆ n−1 + Sp XS
ˆ n−1 − Rp XR
(X − Tp XT
4
1
φ(Y ) = (Yˆ − Tm Yˆ Tq−1 − Rm Yˆ Rq−1 + Sm Yˆ Sq−1 )
4
=
φ(X)
Y is a
is a solution pair of (2.5). Hence, by property (a) of φ(·), we know that X,
solution pair of (1.4). The above discussion shows that the two matrix equations
(1.4) and (2.5) have the same solvability condition and their solutions satisfy (2.6)
and (2.7). Observe that Xij and Yij , i, j = 1, 2, 3, 4 in (2.5) can be written as
j , Yij = Si Y Tj .
Xij = Pi XQ
From Lemma 2.3, the general solution to (2.5) can be written as
= φ(X0 ) + φ(S
1 )Lφ(G) U Rφ(H) φ(T
1 ) + 4Lφ(A) V + 4U
T Rφ(B) ,
X
Hence,
+ 4W T Rφ(D) .
2 )Lφ(G) U Rφ(H) φ(T
2 ) + 4Lφ(C) U
Y = φ(Y0 ) + φ(S
1 )Lφ(G) U Rφ(H) φ(T
1 )Qj + 4Pi Lφ(A) Vj + 4Ui Rφ(B) Qj ,
Xij = Pi φ(X0 )Qj + Pi φ(S
ˆj + 4Wi Rφ(D) Tj ,
2 )Lφ(G) U Rφ(H) φ(T
2 )Tj + 4Si Lφ(C) U
Yij = Si φ(Y0 )Tj + Si φ(S
ˆ1 , · · · , U
ˆ4 ∈ R4p×q , W1 , · · · , W4 ∈
where U, U1 , · · · , U4 ∈ Rp×4q , V1 , · · · , V4 ∈ R4p×q , U
p×4q
R
are arbitrary, and
T = UT , UT , UT , UT ,
V = (V1 , V2 , V3 , V4 ) , U
1
2
3
4
= U
ˆ1 , U
ˆ2 , U
ˆ3 , U
ˆ4 , W T = W1T , W2T , W3T , W4T .
U
Putting them into (2.6), (2.7) yields the eight real matrices X1 , · · · , X4 and Y1 , · · · , Y4 ,
in (2.8)-(2.15).
Electronic Journal of Linear Algebra ISSN 1081-3810
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On Solutions to the Quaternion Matrix Equation AXB + CY D = E
Now we consider the extreme ranks of real matrices X1 , · · · , X4 and Y1 , · · · , Y4
in the solution (X, Y ) to (1.4) over H.
Theorem 2.5. Suppose that (1.4) over H is consistent, and for i, j = 1, 2, 3, 4,
Ji = Xi ∈ Rn×s |A(X1 + X2 i + X3 j + X4 k)B + C (Y1 + Y2 i + Y3 j + Y4 k) D = E ,
Kj = Yj ∈ Rl×k |A(X1 + X2 i + X3 j + X4 k)B + C (Y1 + Y2 i + Y3 j + Y4 k) D = E .
Then we have the following:
(a) The maximal and minimal ranks of Xi in the solution X = X1 + X2 i + X3j + X4 k
to (1.4) are given by
i
0
0
B
max r(Xi ) = min p, q, p + q + r
− 4(r (B) − r (A, C)),
i φ(C) φ(E)
Xi ∈Ji
A
i
0
B
B
p + q + r 0 φ(D) − 4r(A) − 4r
,
D
i φ(E)
A
min r(Xi ) = r
Xi ∈Ji
where
1
A
3
A
1
B
3
B
A2
−A1
=
A4
A3
−A1
−A2
=
−A3
A4
−B2
= −B3
B4
−B1
=
−B2
B4
0
i
A
0
φ(C)
A3
−A4
−A1
−A2
A2
−A1
A4
A3
−B1
−B4
B3
B2
−B1
B3
i
B
φ(E)
0
+r 0
i
A
i
0
B
φ(D) −r 0
i
A
φ(E)
−A4
−A1
−A2
−A3
,A
2 =
−A3
A2
−A1
A4
−A4
−A1
−A3
−A2
, A4 =
−A3
A2
−A1
A4
−B4 −B3
2 =
−B1 B2 , B
−B2 −B1
B3 −B4
4 =
−B4 −B3 , B
−B2 −B1
A3
−A4
−A1
−A2
A2
−A1
A4
A3
−B1
−B3
B4
−B1
−B2
−B3
0
0
φ(C)
−A4
−A3
,
A2
−A1
A3
−A4
,
−A1
−A2
B2
−B4
B3
B3
−B1
−B2
B2
−B1
B4
B3
−B4
−B1
i
B
φ(D)
φ(E)
−B4
B2 ,
−B1
−B4
−B3 .
B2
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Qing-Wen Wang, Hua-Sheng Zhang, and Shao-Wen Yu
(b) The maximal and minimal ranks of Yj in the solution Y = Y1 + Y2 i + Y3 j + Y4 k
to (1.4) are given by
j
0
0
D
max r(Yj ) = min s, t, s + t + r
− 4r(B) − 4r (C, A)
j φ(A) φ(E)
Yj ∈Kj
C
j
0
D
B
s + t + r 0 φ(B) − 4r(C) − 4r
,
D
j φ(E)
C
min r(Yj ) = r
Yj ∈Kj
where
1
C
3
C
1
D
3
D
C2
−C1
=
C4
C3
−C1
−C2
=
−C3
C4
−D2
= −D3
D4
−D1
= −D2
D4
0
j
C
0
φ(A)
C3
−C4
−C1
−C2
C2
−C1
C4
C3
−D1
D4
D3
D2
−D1
D3
j
D
φ(E)
0
+r 0
j
C
j
0
D
φ(B) −r 0
j
C
φ(E)
−C4
−C1 C3
−C2 −C4
−C3
2 =
,C
−C3 −C1
C2
−C1
C4 −C2
−C4
−C1 C2
−C2 −C1
−C3
,C
4 =
−C3 C4
C2
−C1
C4
C3
−D4 −D3
−D1
2 = −D3
−D1 D2 , D
−D2 −D1
D4
D3 −D4
−D1
4 = −D2
−D4 −D3 , D
−D2 −D1
−D3
0
0
φ(A)
j
D
φ(B)
φ(E)
−C4
−C3
,
C2
−C1
C3
−C4
,
−C1
−C2
D2
D4
D3
D2
−D1
D4
−D4
D2 ,
−D1
D3 −D4
−D4 −D3 .
−D1 D2
D3
−D1
−D2
Proof. We only derive the maximal and minimal ranks of the matrix X1 ; the
other r (Xi ) , r(Yj ) can be established similarly. Applying (2.1) and (2.2) to (2.8), we
get the following
max r(X1 ) = min {p, q, r(M1 ), r(M2 )} ,
X1 ∈J1
min r(X1 ) = r(M1 ) + r(M2 ) − r(M3 ) − r (P ) − r (Q) ,
X1 ∈J1
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353
where
0 P P
P
X
P
M1 =
0 , M3 = Q 0 0 ,
0
0 0
Q
0
0 = 1 P1 φ(X0 )Q1 + 1 P2 φ(X0 )Q2 − 1 P3 φ(X0 )Q3 − 1 P4 φ(X0 )Q4 ,
X
4
4
4
4
P = P1 Lφ(A) , P2 Lφ(A) , −P3 Lφ(A) , −P4 Lφ(A) ,
1 )Lφ(G) , P2 φ(S
1 )Lφ(G) , −P3 φ(S
1 )Lφ(G) , −P4 φ(S
1 )Lφ(G) ,
P = P1 φ(S
1 )Q1
Rφ(H) φ(T
Rφ(B) Q1
1 )Q2
Rφ(B) Q2
Rφ(H) φ(T
,Q
=
Q=
.
−Rφ(B) Q3
1 )Q3
Rφ(H) φ(T
1 )Q4
−Rφ(B) Q4
Rφ(H) φ(T
0
X
Q
P
0
0
X
, M2 = Q
Q
By Lemma 2.2, block Gaussian elimination and AX0 B + CY0 D = E, we have that
r (LA ) = p − r (A) , r (RB ) = q − r (B) ,
0
P
0
P
X
0
0
Ψ (B)
Q
r(M1 ) =
− 4r (φ(A)) − 4r (φ(B)) − 4r (φ(G))
0 Ψ (A)
0
0
0
0
Ψ (G)
0
#
$
1 )
0
0
φ(B
=r
+ p + q − 4r (B) − 4r (G)
1 ) φ(C) φ(E)
φ(A
where
1 ), P2 φ(S
1 ), −P3 φ(S
1 ), −P4 φ(S
1 ) ,
P = (P1 , P2 , −P3 , −P4 ) , P = P1 φ(S
Q1
φ(Z)
0
0
0
φ(Z)
0
0
= Q2 , Ψ (Z) = 0
, Z = A, B, G.
Q
−Q3
0
0
φ(Z)
0
−Q4
0
0
0
φ(Z)
In the same manner, we can
0
r(M2 ) = r 0
1
A
0
r(M3 ) = r 0
1
A
simplify r(M2 ) and r(M3 ) as follows.
1
B
φ(D) + p + q − 4r(A) − 4r (H) ,
φ(E)
1
0
B
0
φ(D) + p + q − 4r (G) − 4r (H) .
φ(C) φ(E)
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Qing-Wen Wang, Hua-Sheng Zhang, and Shao-Wen Yu
Thus, we have the results for the extreme ranks of the matrix X1 in (a). Similarly,
applying (2.1) and (2.2) to (2.9), (2.10), (2.11), (2.12), (2.13), (2.14), (2.15) yields the
other results in (a), (b).
i , B
i , C
j and D
j (i, j = 1, 2, 3, 4) are defined as in
In the following corollaries, A
Theorem 2.5.
Corollary 2.6. Suppose that (1.4) over H is consistent. Then
(a) (1.4) has a real solution for X if and only if
i
i
0
B
0
0
B
0
0
Bi
+ r 0 φ(D) = r 0
r
0
φ(D) , i = 2, 3, 4.
i φ(C) φ(E)
A
i φ(E)
i φ(C) φ(E)
A
A
In that case, the real solution X can be expressed as X = X1 in (2.8).
(b) All the solutions of (1.4) for X are real if and only if
i
0
0
B
p+q+r
= 4r (B) + 4r (A, C)
i φ(C) φ(E)
A
or
0
p+q+r 0
i
A
i
B
B
, i = 2, 3, 4.
φ(D) = 4r(A) + 4r
D
φ(E)
In that case, the real solutions X can be expressed as X = X1 in (2.8).
(c) (1.4) has a pure imaginary solution for X if and only if
1
1
0
0
B
0
B
0
0
B1
r
+ r 0 φ(D) = r 0
0
φ(D) .
A1 φ(C) φ(E)
1 φ(E)
1 φ(C) φ(E)
A
A
In that case, the pure imaginary solution can be expressed as X = X2 i + X3 j + X4 k
where X2 , X3 and X4 are expressed as (2.9), (2.10) and (2.11), respectively.
(d) All the solutions of (1.4) for X are pure imaginary if and only if
1
0
0
B
p+q+r
= 4r (B) + 4r (A, C)
1 φ(C) φ(E)
A
or
0
p+q +r 0
1
A
1
B
B
.
φ(D) = 4r(A) + 4r
D
φ(E)
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On Solutions to the Quaternion Matrix Equation AXB + CY D = E
355
In that case, the pure imaginary solutions can be expressed as X = X2 i + X3 j + X4 k,
where X2 , X3 and X4 are expressed as (2.9), (2.10) and (2.11), respectively.
Using the same method, we can get the corresponding results on Y.
We now consider all solution pairs X and Y to (1.4) over H to be real or pure
imaginary, respectively.
Theorem 2.7. Suppose that (1.4) over H is consistent, q (X1 , Y1 ) = E −AX1 B−
CY1 D, and J1 , K1 as in Theorem 2.2 are two independent sets. Then
(2.16)
&
%
B
.
max r [q (X1 , Y1 )] = min r (A) + r (C) − r (A, C) , r (B) + r (D) − r
X1 ∈J1 ,
D
Y ∈K
1
1
In particular:
(a) The solutions (X1 , Y1 ) of (1.4) over H are independent, that is, for any X1 ∈ J1
and Y1 ∈ K1 the solutions X1 and Y1 satisfy (1.4) over H if and only if
(2.17)
r (A, C) = r (A) + r (C) or r
B
D
= r (B) + r (D) .
(b) Under (2.17), the general solution of (1.4) over H can be written as the two
independent forms
1 + LA V1 + V2 RB ,
1 LG U1 RH T
X = X0 + S
(2.18)
2 + LC W1 + W2 RD ,
2 LG U2 RH T
Y = Y0 + S
(2.19)
where (X0 , Y0 ) is a particular real solution of (1.4), U1 , U2 , V1 , V2 , W1 and W2 are
arbitrary.
Proof. Writing (2.3) and (2.4) as two independent matrix expressions, that is,
replacing U in (2.3) and (2.4) by U1 and U2 respectively, then taking them into
E − AX1 B − CY1 D yields
1 B − C S
2 D
1 LG U1 RH T
2 LG U2 RH T
q (X1 , Y1 ) = E − AX0 B − CY0 D − AS
1 B,
1 LG (−U1 + U2 ) RH T
= AS
where U1 and U2 are arbitrary. Then by (2.1), it follows that
1 B
1 LG (−U1 + U2 ) RH T
max r [q (X1 , Y1 )] = max r AS
X1 ∈J1 ,
Y1 ∈K1
U1 ,U2
(
'
1 B
1 LG , r RH T
= min r AS
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Qing-Wen Wang, Hua-Sheng Zhang, and Shao-Wen Yu
where
1 LG = r
r AS
1
AS
G
− r (G) = r (A) + r (C) − r (G) ,
1 B = r T
1 B, H − r (H) = r (B) + r (D) − r (H) .
r RH T
Therefore, we have (2.16). The result in (2.17) follows directly from (2.16) and the
solutions in (2.18) and (2.19) follow from (2.3) and (2.4).
Corollary 2.8. Suppose that (1.4) over H is consistent, and (2.17) holds. Then
(a) All solution pairs X and Y to (1.4) over H are real if and only if
i
0
0
B
p+q+r
= 4r (B) + 4r (A, C)
i φ(C) φ(E)
A
or
and
0
p+q +r 0
i
A
s+t+r
or
0
s+t+r 0
bj
C
0
j
C
i
B
B
φ(D) = 4r(A) + 4r
D
φ(E)
0
φ(A)
j
D
φ(E)
= 4r(B) + 4r (A, C)
bj
D
B
φ(B)
= 4r(C) + 4r
, i, j = 2, 3, 4.
D
φ(E)
(b) All solution pairs X and Y to (1.4) over H are imaginary if and only if
$
#
1
0
0
B
p+q+r
1 φ(C) φ(E) = 4r (B) + 4r (A, C)
A
or
and
0
p+q+r 0
1
A
s+t+r
0
C1
1
B
B
φ(D) = 4r(A) + 4r
D
φ(E)
0
φ(A)
1
D
φ(E)
= 4r(B) + 4r (C, A)
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On Solutions to the Quaternion Matrix Equation AXB + CY D = E
or
0
s+t+r 0
1
C
357
1
D
B
.
φ(B) = 4r(C) + 4r
D
φ(E)
Remark 2.9. The main results of [11], [17] and [19] can be regarded as special
cases of results in this paper.
Acknowledgment. The authors would like to thank Professor Michael Neumann
and a referee very much for their valuable suggestions and comments, which resulted
in great improvement of the original manuscript.
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