Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 307036, 19 pages
doi:10.1155/2012/307036
Research Article
The Real and Complex Hermitian
Solutions to a System of Quaternion Matrix
Equations with Applications
Shao-Wen Yu
Department of Mathematics, East China University of Science and Technology, Shanghai 200237, China
Correspondence should be addressed to Shao-Wen Yu, yushaowen@ecust.edu.cn
Received 9 October 2011; Accepted 17 November 2011
Academic Editor: Qing-Wen Wang
Copyright q 2012 Shao-Wen Yu. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We establish necessary and sufficient conditions for the existence of and the expressions for
the general real and complex Hermitian solutions to the classical system of quaternion matrix
equations A1 X C1 , XB1 C2 , and A3 XA∗3 C3 . Moreover, formulas of the maximal and minimal
ranks of four real matrices X1 , X2 , X3 , and X4 in solution X X1 X2 i X3 j X4 k to the system
mentioned above are derived. As applications, we give necessary and sufficient conditions for the
quaternion matrix equations A1 X C1 , XB1 C2 , A3 XA∗3 C3 , and A4 XA∗4 C4 to have real and
complex Hermitian solutions.
1. Introduction
Throughout this paper, we denote the real number field by R; the complex 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 k2 ijk −1, a0 , a1 , a2 , a3 ∈ R
1.1
by Hm×n ; the identity matrix with the appropriate size by I; the transpose, the conjugate
transpose, the column right space, the row left space of a matrix A over H by AT , A∗ , RA,
NA, respectively; the dimension of RA by dim RA. By 1, for a quaternion matrix
A, dim RA dim NA. dim RA is called the rank of a quaternion matrix A and denoted
by rA. The Moore-Penrose inverse of matrix A over H by A† which satisfies four Penrose
equations AA† A A, A† AA† A† , AA† ∗ AA† , and A† A∗ A† A. In this case, A† is
unique and A† ∗ A∗ † . Moreover, RA and LA stand for the two projectors LA I − A† A,
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International Journal of Mathematics and Mathematical Sciences
and RA I − AA† induced by A. Clearly, RA and LA are idempotent and satisfies RA ∗ RA , LA ∗ LA , RA LA∗ , and RA∗ LA .
Hermitian solutions to some matrix equations were investigated by many authors. In
1976, Khatri and Mitra 2 gave necessary and sufficient conditions for the existence of the
Hermitian solutions to the matrix equations AX B, AXB C and
A1 X C1 ,
XB2 C2 ,
1.2
over the complex field C, and presented explicit expressions for the general Hermitian
solutions to them by generalized inverses when the solvability conditions were satisfied.
Matrix equation that has symmetric patterns with Hermitian solutions appears in some
application areas, such as vibration theory, statistics, and optimal control theory 3–7. Groß
in 8, and Liu et al. in 9 gave the solvability conditions for Hermitian solution and its
expressions of
AXA∗ B
1.3
over C in terms of generalized inverses, respectively. In 10, Tian and Liu established the
solvability conditions for
A3 XA∗3 C3 ,
A4 XA∗4 C4
1.4
to have a common Hermitian solution over C by the ranks of coefficient matrices. In 11,
Tian derived the general common Hermitian solution of 1.4. Wang and Wu in 12 gave
some necessary and sufficient conditions for the existence of the common Hermitian solution
to equations
A1 X C1 ,
A1 X C1 ,
XB2 C2 ,
XB2 C2 ,
A3 XA∗3 C3 ,
A3 XA∗3 C3 ,
A4 XA∗4 C4 ,
1.5
1.6
for operators between Hilbert C∗ -modules by generalized inverses and range inclusion of
matrices.
As is known to us, extremal ranks of some matrix expressions can be used to
characterize nonsingularity, rank invariance, range inclusion of the corresponding matrix
expressions, as well as solvability conditions of matrix equations 4, 7, 9–24. Real matrices
and its extremal ranks in solutions to some complex matrix equation have been investigated
by Tian and Liu 9, 13–15. Tian 13 gave the maximal and minimal ranks of two real
matrices X0 and X1 in solution X X0 iX1 to AXB C over C with its applications. Liu
et al. 9 derived the maximal and minimal ranks of the two real matrices X0 and X1 in a
Hermitian solution X X0 iX1 of 1.3, where B∗ B. In order to investigate the real and
complex solutions to quaternion matrix equations, Wang and his partners have been studying
the real matrices in solutions to some quaternion matrix equations such as AXB C,
A1 XB1 C1 ,
A2 XB2 C2 ,
AXA∗ BXB∗ C,
1.7
International Journal of Mathematics and Mathematical Sciences
3
recently 24–27. To our knowledge, the necessary and sufficient conditions for 1.5 over
H to have the real and complex Hermitian solutions have not been given so far. Motivated
by the work mentioned above, we in this paper investigate the real and complex Hermitian
solutions to system 1.5 over H and its applications.
This paper is organized as follows. In Section 2, we first derive formulas of extremal
ranks of four real matrices X1 , X2 , X3 , and X4 in quaternion solution X X1 X2 i X3 j X4 k to 1.5 over H, then give necessary and sufficient conditions for 1.5 over H to have
real and complex solutions as well as the expressions of the real and complex solutions. As
applications, we in Section 3 establish necessary and sufficient conditions for 1.6 over H to
have real and complex solutions.
2. The Real and Complex Hermitian Solutions to System 1.5 Over H
In this section, we first give a solvability condition and an expression of the general Hermitian
solution to 1.5 over H, then consider the maximal and minimal ranks of four real matrices
X1 , X2 , X3 , and X4 in solution X X1 X2 i X3 j X4 k to 1.5 over H, last, investigate the
real and complex Hermitian solutions to 1.5 over H.
For an arbitrary matrix Mt Mt1 Mt2 i Mt3 j Mt4 k ∈ Hm×n where Mt1 , Mt2 , Mt3 ,
and Mt4 are real matrices, we define a map φ· from Hm×n to R4m×4n by
⎡
Mt1
Mt2
Mt3
Mt4
⎤
⎥
⎢
⎢−Mt2 Mt1 Mt4 −Mt3 ⎥
⎥
⎢
φMt ⎢
⎥.
⎢−Mt3 −Mt4 Mt1 Mt2 ⎥
⎦
⎣
−Mt4 Mt3 −Mt2 Mt1
2.1
By 2.1, it is easy to verify that φ· satisfies the following properties.
a M N ⇔ φM φN.
b φkM lN kφM lφN, φMN φMφN, k, l ∈ R.
c φM∗ φT M, φM† φ† M.
−1
−1
d φM Tm
φMTn R−1
m φMRn Sm φMSn , where t m, n,
⎡
0 −It
⎢
⎢ It
⎢
Tt ⎢
⎢0
⎣
0
0
0
0
0
0
⎤
⎥
0⎥
⎥
⎥,
0 It ⎥
⎦
−It 0
0
⎡
0 0 −It
⎢
⎢0 0
⎢
Rt ⎢
⎢It 0
⎣
0 It
0
⎤
⎥
0 −It ⎥
⎥
⎥,
0 0⎥
⎦
0 0
⎡
0
0
0 −It
⎢
⎢ 0 0 It
⎢
St ⎢
⎢ 0 −It 0
⎣
It
0
0
⎤
⎥
0 ⎥
⎥
⎥.
0 ⎥
⎦
0
2.2
e rφM 4rM.
f M∗ M ⇔ φT M φM, M∗ −M ⇔ φT M −φM.
The following lemmas provide us with some useful results over C, which can be generalized
to H.
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International Journal of Mathematics and Mathematical Sciences
Lemma 2.1 see 2, Lemma 2.1. Let A ∈ Hm×n , B B∗ ∈ Hm×m be known, X ∈ Hn×n unknown;
then the system 1.3 has a Hermitian solution if and only if
AA† B B.
2.3
In that case, the general Hermitian solution of 1.3 can be expressed as
∗
X A† B A† LA V V ∗ LA ,
2.4
where V is arbitrary matrix over H with compatible size.
Lemma 2.2 see 12, Corollary 3.4. Let A1 , C1 ∈ Hm×n ; B1 , C2 ∈ Hn×s ; A3 ∈ Hr×n ; C3 ∈ Hr×r
†
be known, X ∈ Hn×n unknown, and F B1∗ LA1 , M SLF , S A3 LA1 , D C2∗ − B1∗ A1 C1 , J †
A1 C1 F † D, G C3 −A3 J LA1 L∗F J ∗ A∗3 , C3 C3∗ ; then the system 1.5 have a Hermitian solution
if and only if
A1 C2 C1 B1 ,
RA1 C1 0,
A1 C1∗ C1 A∗1 ,
RF D 0,
B1∗ C2 C2∗ B1 ,
RM G 0.
2.5
In that case, the general Hermitian solution of 1.5 can be expressed as
∗
X J LA1 LF J ∗ LA1 LF M† G M† LF LA1 LA1 LF LM V LF LA1 LA1 LF V ∗ LM LF LA1 ,
2.6
where V is arbitrary matrix over H with compatible size.
Lemma 2.3 see 21, 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 rCLA r A
C − rA.
B A
b r B ALC r 0 C − rC.
C 0
c r RCB A r A
B − rB.
A B 0 A BLD
d r RE C 0 r C 0 E − rD − rE.
0 D 0
Lemma 2.3 plays an important role in simplifying ranks of various block matrices.
Lemma 2.4 see 11, Theorem 4.1, Corollary 4.2. Let A ±A∗ ∈ Hm×m , B ∈ Hm×n , and C ∈
Hp×m be given; then
A C∗
A B
∗
max r A − BXC ∓ BXC min r A B C∗ , r
, r
,
X∈Hn×p
C 0
B∗ 0
min
r A − BXC ∓ BXC∗ 2r A B C∗ max {s1 , s2 },
n×p
X∈H
2.7
International Journal of Mathematics and Mathematical Sciences
5
where
s1 r
A B
− 2r
B∗ 0
s2 r
A C∗
C 0
A B C∗
,
B∗ 0 0
− 2r
A B C∗
C 0 0
2.8
.
If RB ⊆ RC∗ ,
A B
∗
max r A − BXC − BXC min r A C∗ , r
,
X∈Hn×p
B∗ 0
2.9
A B
A B
∗
∗
min r A − BXC − BXC 2r A C r
− 2r
.
X∈Hn×p
C 0
B∗ 0
2.10
Lemma 2.5 see 28, Theorem 3.1. Let A ∈ Hm×n , B1 ∈ Hm×p1 , B3 ∈ Hm×p3 , B4 ∈ Hm×p4 , C2 ∈
Hq2 ×n , C3 ∈ Hq3 ×n , and C4 ∈ Hq4 ×n be given. Then the matrix equation
B1 X1 X2 C2 B3 X3 C3 B4 X4 C4 A
2.11
is consistent if and only if
⎡
A B1
⎡
⎤
0 B1
⎤
⎢
⎢
⎥
⎥
⎢C2 0 ⎥
⎢C2 0 ⎥
⎢
⎢
⎥
⎥
r⎢
⎥ r⎢
⎥,
⎢C3 0 ⎥
⎢C3 0 ⎥
⎣
⎣
⎦
⎦
C4 0
C4 0
r
⎡
⎤
⎤
⎡
A B 1 B3 B4
C2 0
0
0
r
0 B1 B3 B4
C2 0
0
0
,
2.12
A B1 B3
0 B1 B 3
⎢
⎥
⎢
⎥
⎥
⎢
⎥
r⎢
⎣C2 0 0 ⎦ r ⎣C2 0 0 ⎦,
C4 0 0
C4 0 0
⎡
⎤
⎡
⎤
A B 1 B4
0 B1 B 4
⎢
⎥
⎢
⎥
⎥
⎢
⎥
r⎢
⎣C2 0 0 ⎦ r ⎣C2 0 0 ⎦.
C3 0 0
C3 0 0
Theorem 2.6. System 1.5 has a Hermitian solution over H if and only if the system of matrix
equations
φA1 Yij 4 × 4 φC1 ,
Yij
4×4
φB1 φC2 ,
φA3 Yij 4 × 4 φT A3 φC3 ,
i, j 1, 2, 3, 4,
2.13
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International Journal of Mathematics and Mathematical Sciences
has a symmetric solution over R. In that case, the general Hermitian solution of 1.5 over H can be
written as
X X1 X2 i X3 j X4 k
1
1
T
T
i
Y12 − Y12
Y34 − Y34
Y11 Y22 Y33 Y44 4
4
2.14
1
1
T
T
T
T
Y13 − Y13
Y14 − Y14
k,
Y24
− Y24 j Y23 − Y23
4
4
T
where Ytt YttT ; t 1, 2, 3, 4; Y1jT Yj1 ; j 2, 3, 4; Y2jT Yj2 ; j 3, 4; Y34
Y43 are the general
solutions of 2.13 over R. Written in an explicit form, X1 , X2 , X3 , and X4 in 2.14 are
X1 1
1
1
1
P1 φX0 P1T P2 φX0 P2T P3 φX0 P3T P4 φX0 P4T
4
4
4
4
⎡
⎤
LφF LφA1 P1T
⎢
⎥
⎢LφF LφA1 P T ⎥
2⎥
⎢
P1 , P2 , P3 , P4 LφA1 LφF LφM V ⎢
⎥
⎢LφF LφA P T ⎥
1
3⎦
⎣
LφF LφA1 P4T
2.15
⎡
⎤
LφM LφF LφA1 P1T
⎢
⎥
⎢L
L
L
PT⎥
T ⎢ φM φF φA1 2 ⎥
P1 , P2 , P3 , P4 LφA1 LφF V ⎢
⎥,
⎢LφM LφF LφA P T ⎥
1
3⎦
⎣
LφM LφF LφA1 P4T
X2 1
1
1
1
P1 φX0 P2T − P2 φX0 P1T P3 φX0 P4T − P4 φX0 P3T
4
4
4
4
⎡
⎤
LφF LφA1 P2T
⎢
⎥
⎢LφF LφA1 P T ⎥
⎢
1⎥
P1 , −P2 , P3 , −P4 LφA1 LφF LφM V ⎢
⎥
⎢LφF LφA P T ⎥
1
4⎦
⎣
LφF LφA1 P3T
⎡
LφM LφF LφA1 P1T
⎤
⎢
⎥
⎢−LφM LφF LφA1 P T ⎥
2⎥
P2 , P1 , P4 , P3 LφA1 LφF V ⎢
⎥,
⎢ LφM LφF LφA P T ⎥
1
3
⎣
⎦
−LφM LφF LφA1 P4T
T⎢
2.16
International Journal of Mathematics and Mathematical Sciences
X3 7
1
1
1
1
P1 φX0 P3T − P3 φX0 P1T P4 φX0 P2T − P2 φX0 P4T
4
4
4
4
⎡
⎤
LφF LφA1 P3T
⎢
⎥
⎢LφF LφA1 P T ⎥
⎢
1⎥
P1 , −P3 , P4 , −P2 LφA1 LφF LφM V ⎢
⎥
⎢LφF LφA P T ⎥
1
2⎦
⎣
LφF LφA1 P4T
⎡
LφM LφF LφA1 P1T
2.17
⎤
⎢
⎥
⎢−LφM LφF LφA1 P T ⎥
3⎥
P3 , P1 , P2 , P4 LφA1 LφF V ⎢
⎥,
⎢ LφM LφF LφA P T ⎥
1
4 ⎦
⎣
−LφM LφF LφA1 P2T
T⎢
X4 1
1
1
1
P1 φX0 P4T − P4 φX0 P1T P2 φX0 P3T − P3 φX0 P2T
4
4
4
4
⎡
⎤
LφF LφA1 P4T
⎢
⎥
⎢LφF LφA1 P T ⎥
⎢
1⎥
P1 , −P4 , P2 , −P3 LφA1 LφF LφM V ⎢
⎥
⎢LφF LφA P T ⎥
1
3⎦
⎣
LφF LφA1 P2T
⎡
LφM LφF LφA1 P1T
2.18
⎤
⎢
⎥
⎢−LφM LφF LφA1 P T ⎥
4⎥
P4 , P1 , P3 , P2 LφA1 LφF V ⎢
⎥,
⎢ LφM LφF LφA P T ⎥
1
2 ⎦
⎣
−LφM LφF LφA1 P3T
T⎢
where
P1 In , 0, 0, 0,
P2 0, In , 0, 0,
P3 0, 0, In , 0,
P4 0, 0, 0, In ,
2.19
φX0 is a particular symmetric solution to 2.13, and V is arbitrary real matrices with compatible
sizes.
Proof. Suppose that 1.5 has a Hermitian solution X over H. Applying properties a and b
of φ· to 1.5 yields
φA1 φX φC1 ,
φXφB2 φC2 ,
φA3 φXφT A3 φC3 ,
2.20
implying that φX is a real symmetric solution to 2.13.
Conversely, suppose that 2.13 has a real symmetric solution
Y Y T Yij 4 × 4 ,
i, j 1, 2, 3, 4.
2.21
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International Journal of Mathematics and Mathematical Sciences
That is,
φA1 Y φC1 ,
Y φB2 φC2 ,
φA3 Y φT A3 φC3 ,
2.22
then by property d of φ·,
−1
−1
Tm
φA1 Tn Y Tm
φC1 Tn ,
Y Tn−1 φB2 Ts Tn−1 φC2 Ts ,
Tr−1 φA3 Tn Y Tn−1 φT A3 Tr Tr−1 φC3 Tr ,
−1
R−1
m φA1 Rn Y Rm φC1 Rn ,
−1
Y R−1
n φB2 Rs Rn φC2 Rs ,
T
R−1
r φA3 Rn Y Rn φ A3 Rr
−1
S−1
m φA1 Sn Y Sm φC1 Sn ,
R−1
r φC3 Rr ,
2.23
−1
Y S−1
n φB2 Ss Sn φC2 Ss ,
−1
−1 T
S−1
r φA3 Sn Y Sn φ A3 Sr Sr φC3 Sr .
Hence,
φA1 Tn Y Tn−1 φC1 ,
Tn Y Tn−1 φB2 φC2 ,
φA3 Tn Y Tn−1 φT A3 φC3 ,
φA1 Rn Y R−1
n φC1 ,
Rn Y R−1
n φB2 φC2 ,
T
φA3 Rn Y R−1
n φ A3 φC3 ,
φA1 Sn Y S−1
n φC1 ,
Sn Y S−1
n φB2 φC2 ,
T
φA3 Sn Y S−1
n φ A3 φC3 ,
2.24
−1
implying that Tn Y Tn−1 , Rn Y R−1
n , and Sn Y Sn are also symmetric solutions of 2.13. Thus,
1
−1
S
Y
S
Y Tn Y Tn−1 Rn Y R−1
n
n
n
4
is a symmetric solution of 2.13, where
−1
S
Y Tn Y Tn−1 Rn Y R−1
Y
S
Yij
n
n
n
4×4
,
i 1, 2, 3, 4,
Y
11 Y11 Y22 Y33 Y44 ,
T
T
Y
12 Y12 − Y12 Y34 − Y34 ,
T
T
Y
13 Y13 − Y13 Y24 − Y24 ,
T
T
Y
14 Y14 − Y14 Y23 − Y23 ,
T
T
Y
21 Y12 − Y12 Y34 − Y34 ,
Y
22 Y11 Y22 Y33 Y44 ,
2.25
International Journal of Mathematics and Mathematical Sciences
T
T
Y
23 Y14 − Y14 Y23 − Y23 ,
T
T
Y
24 Y13 − Y13 Y24 − Y24 ,
T
T
Y
31 Y13 − Y13 Y24 − Y24 ,
T
T
Y
32 Y14 − Y14 Y23 − Y23 ,
Y
33 Y11 Y22 Y33 Y44 ,
T
T
Y
34 Y12 − Y12 Y34 − Y34 ,
T
T
Y
41 Y14 − Y14 Y23 − Y23 ,
T
T
Y
42 Y13 − Y13 Y24 − Y24 ,
T
T
Y
43 Y12 − Y12 Y34 − Y34 ,
Y
44 Y11 Y22 Y33 Y44 .
9
2.26
Let
1 Y11 Y22 Y33 Y44 1 Y12 − Y T Y34 − Y T i
X
12
34
4
4
1
1
T
T
T
T
Y13 − Y13
Y14 − Y14
k.
Y24
− Y24 j Y23 − Y23
4
4
2.27
1
−1
φ X
Y Tn Y Tn−1 Rn Y R−1
n Sn Y Sn .
4
2.28
Then by 2.1,
is a Hermitian solution of 1.5. Observe that
Hence, by the property a, we know that X
Yij , i, j 1, 2, 3, 4, in 2.13 can be written as
Yij Pi Y PjT .
2.29
From Lemma 2.2, the general Hermitian solution to 2.13 can be written as
Y φX0 4LφA1 LφF LφM V LφA1 LφF 4LφF LφA1 V T LφM LφF LφA1 ,
2.30
where V ∈ R is arbitrary. Hence,
Yij Pi φX0 PjT 4Pi LφA1 LφF LφM V LφA1 LφF PjT
4Pi LφF LφA1 V T LφM LφF LφA1 PjT ,
2.31
where i, j 1, 2, 3, 4, substituting them into 2.14, yields the four real matrices X1 , X2 , X3 , and
X4 in 2.15–2.18.
Now we consider the maximal and minimal ranks of four real matrices X1 , X2 , X3 , and
X4 in solution X X1 X2 i X3 j X4 k to 1.5 over H.
10
International Journal of Mathematics and Mathematical Sciences
Theorem 2.7. Suppose that system 1.5 over H has a Hermitian solution, and A1 A11 A12 i A13 j A14 k, C1 C11 C12 i C13 j C14 k ∈ Hm×n B1 B11 B12 i B13 j B14 k, C2 C21 C22 i C23 j C24 k ∈ Hn×s , A3 A31 A32 i A33 j A34 k ∈ Hr×n , C3 C31 C32 i C33 j C34 k ∈ Hr×r
S1 X1 ∈ R
|
n×n
S2 X2 ∈ R
p×q
|
S3 X3 ∈ R
p×q
|
S4 L21
X4 ∈ R
p×q
⎡ ⎤
C21
⎢ ⎥
⎢C22 ⎥
⎢ ⎥
⎢ ⎥,
⎢C23 ⎥
⎣ ⎦
C24
⎡
M11
N11
A1 X C1 , XB1 C2 , A3 XA∗3 C3
A13
A11
A12
A1 X C1 , XB1 C2 , A3 XA∗3 C3
A1 X C1 , XB1 C2 , A3 XA∗3 C3
,
,
,
X X1 X2 i X3 j X4 k
C11
⎢
⎥
⎢−C12 ⎥
⎢
⎥
⎢
⎥,
⎢−C13 ⎥
⎣
⎦
−C14
A14
A14
B14
⎡
⎤
M31
M12
⎤
⎢
⎥
⎥
N13 ⎢
⎣−B12 B11 B14 −B13 ⎦,
−B14 B13 −B12 B11
M14
A33
A34
⎤
⎤
A11
A13
A14
A11
A12
A13
B11
B12
B13 B14
2.32
⎥
⎢
⎥
⎢−A
⎢ 12 A14 −A13 ⎥
⎥,
⎢
⎥
⎢
⎢−A13 A11 A12 ⎥
⎦
⎣
−A14 −A12 A11
⎡
⎤
A32
⎥
⎢
⎥
⎢A
⎢ 31 A34 −A33 ⎥
⎥,
⎢
⎥
⎢
⎢−A34 A31 A32 ⎥
⎦
⎣
A33 −A32 A31
⎡
⎤
⎡
⎤
−B12 B11 B14 −B13
⎢
⎥
⎥
⎢
⎣−B13 −B14 B11 B12 ⎦,
−B14 B13 −B12 B11
B11 B12 B13
X X1 X2 i X3 j X4 k
⎢
⎥
⎢−A12 A11 −A13 ⎥
⎢
⎥
⎢
⎥,
⎢−A13 −A14 A12 ⎥
⎣
⎦
−A14 A13 A11
⎡
,
X X1 X2 i X3 j X4 k
⎡
L11
X X1 X2 i X3 j X4 k
⎢
⎥
⎢ A11 A14 −A13 ⎥
⎢
⎥
⎢
⎥,
⎢−A14 A11 A12 ⎥
⎣
⎦
A13 −A12 A11
⎡
M13
A12
|
A1 X C1 , XB1 C2 , A3 XA∗3 C3
⎤
⎢
⎥
⎢−A12 A11 A14 ⎥
⎢
⎥
⎢
⎥,
⎢−A13 −A14 A11 ⎥
⎣
⎦
−A14 A13 −A12
⎡
⎤
⎢
⎥
⎥
N12 ⎢
⎣−B13 −B14 B11 B12 ⎦,
−B14 B13 −B12 B11
⎡
B11
B12 B13 B14
⎤
⎢
⎥
⎥
N14 ⎢
⎣−B12 B11 B14 −B13 ⎦.
−B13 −B14 B11 B12
International Journal of Mathematics and Mathematical Sciences
11
Then the maximal and minimal ranks of Xi , i 1, 2, 3, 4, in Hermitian solution X X1 X2 i X3 j X4 k to 1.5 are given by
max rXi min{t1i , t},
2.33
min rXi 2t1i t − 2t2i ,
2.34
Xi ∈Si
Xi ∈Si
where
t1i r
⎡
T
M31
T
L21 N1i
− 4r
L11 M1i
B1∗
A1
n,
⎤
T
M11
⎡ ⎤
A3
⎢
⎥
T
⎢M31
⎥
φC3 φA3 φC2 φA3 φ C1 ⎥
⎢ ∗⎥
⎢
⎢
t r⎢ T
⎥ − 8r ⎣ B1 ⎥
⎦ 2n,
⎢ N φT C2 φT A3 φT C2 φB1 φT C2 φT A1 ⎥
⎣ 11
⎦
A1
M11 φC1 φT A3 φC1 φB1 φC1 φT A1 ⎡
⎤
T
0
N1i
M1i
⎡ ⎤
A3
⎢
⎥
∗
⎢M31 φA3 φC2 φA3 φT C1 ⎥
B1
⎢ ∗⎥
⎢
⎥
⎢
⎥
2n.
t2i r ⎢ T
⎥ − 4r ⎣ B1 ⎦ − 4r
⎢ N φT C2 φB1 φT C2 φT A1 ⎥
A1
⎣ 11
⎦
A1
M11 φC1 φB1 φC1 φT A1 0
N11
2.35
Proof. We only prove the case that i 1. Similarly, we can get the results that i 2, 3, 4. Let
1
1
1
1
P1 φX0 P1T P2 φX0 P2T P3 φX0 P3T P4 φX0 P4T A,
4
4
4
4
P1 , P2 , P3 , P4 LφA1 LφF LφM B,
⎡
⎤
LφF LφA1 P1T
⎢
⎥
⎢LφF LφA1 P T ⎥
2⎥
⎢
⎢
⎥ C;
⎢LφF LφA P T ⎥
1
3⎦
⎣
LφF LφA1 P4T
2.36
note that LM is Hermtian; then LφM is symmetric; hence 2.15 can be written as
X1 A BV C BV C∗ .
2.37
Note that A A∗ and RB ⊆ RC∗ ; applying 2.9 and 2.10 to 2.37 yields
∗
max rX1 min rA, C , r
X1 ∈S1
∗
min rX1 2rA, C r
X1 ∈S1
A B
B∗ 0
A B
B∗ 0
− 2r
2.38
,
A B
C 0
.
2.39
12
International Journal of Mathematics and Mathematical Sciences
Let
P1 , P2 , P3 , P4 P,
⎡
φAi 0
0
0
⎢
⎢ 0
φAi 0
0
⎢
ai ⎢
⎢ 0
0
φAi 0
⎣
⎤
⎥
⎥
⎥
⎥,
⎥
⎦
i 1, 3,
0
0
φAi ⎡
⎤
φB1 0
0
0
⎢
⎥
⎢ 0
φB1 0
0 ⎥
⎢
⎥
b1 ⎢
⎥.
⎢ 0
0
φB1 0 ⎥
⎣
⎦
0
0
0
φB1 0
2.40
Note that φX0 is a particular solution to 2.13, it is not difficult to find by Lemma 2.3, block
Gaussian elimination, and property e of φ· that
⎤
⎡
A P
⎥
⎢
T⎥
rA, C∗ r ⎢
⎣ 0 b1 ⎦ − 4r φA1 − 4r φF
0 a1
⎡
⎤
0
P
⎢
⎥
∗ ⎢ 1 T
⎥
φ B1
⎢− φ C2 P T bT ⎥
r⎢ 4
1
1 ⎥ − 4r
⎢
⎥
φA1 ⎣ 1
⎦
T
− φC1 P1 a1
4
⎡
⎤
0
P1 , 0, 0, 0
∗ φ B1
⎢ T
⎥
T
T
⎥ − 4r
φ
r⎢
b
C
P
2
1
1
⎣
⎦
φA1 φC1 P1T
a1
∗ ∗ T
φ B1
φ B1
L21 N1i
− 4r
3r
n
r
φA1 φA1 L11 M1i
∗
T
B1
L21 N1i
− 4r
n.
r
L11 M1i
A1
Note that LA RA∗ , then LφA Rφ∗ A ; hence
⎡
A P
⎢ T
⎢P
⎢
A B
⎢
r⎢
r
⎢0
B∗ 0
⎢
⎢0
⎣
0
0
0
0
⎤
⎥
0 aT3 b1 aT1 ⎥
⎥
⎥
a3 0 0 0 ⎥
⎥ − 8r φM − 8r φF − 8r φA1 ⎥
b1T 0 0 0 ⎥
⎦
a1 0 0 0
2.41
International Journal of Mathematics and Mathematical Sciences
⎡
T
M31
0
13
⎤
N11
T
M11
φC1 φB1 φC1 φT A1 ⎢
⎥
⎢M31
φC3 φA3 φC2 φA3 φT C1 ⎥
⎢
⎥
r⎢ T
⎥
⎢ N φT C2 φT A3 φT C2 φB1 φT C2 φT A1 ⎥
⎣ 11
⎦
M11 φC1 φT A3 ⎡
⎤
⎡
⎤
φA3 φA3 ⎢ ∗ ⎥
⎢ ∗ ⎥
⎥
⎢
⎥
− 8r ⎢
⎣φ B1 ⎦ 6r ⎣φ B1 ⎦ 2n
φA1 φA1 ⎡
T
M31
T
M11
⎤
⎡ ⎤
A3
⎥
⎢
T
⎥
⎢M31
φC3 φA3 φC2 φA3 φ C1 ⎥
⎢ ∗⎥
⎢
⎥
r⎢ T
⎥ − 8r ⎢
⎣ B1 ⎦ 2n.
⎢ N φT C2 φT A3 φT C2 φB1 φT C2 φT A1 ⎥
⎦
⎣ 11
A1
M11 φC1 φT A3 φC1 φB1 φC1 φT A1 0
N11
2.42
Similarly, we can obtain
⎡
T
M11
⎤
⎡ ⎤
A3
⎢
⎥
∗
T
⎢M31 φA3 φC2 φA3 φ C1 ⎥
B1
⎢ ∗⎥
⎢
⎥
⎢
⎥
r
r⎢ T
− 4r ⎣ B1 ⎦ − 4r
2n,
⎥
⎢ N φT C2 φB1 φT C2 φT A1 ⎥
C 0
A1
⎣ 11
⎦
A1
M11 φC1 φB1 φC1 φT A1 A B
0
N11
2.43
Substituting 2.41 and 2.43 into 2.38 and 2.39 yields 2.33 and 2.34, that is i 1.
Corollary 2.8. Suppose system 1.5 over H have a Hermitian solution. Then we have the following.
a System 1.5 has a real hermtian solution if and only if
⎡
2r
T
L21 N1i
L11 M1i
T
M31
0
0
T
M11
φC1 φB1 ⎤
φC1 φT A1 ⎤
⎢
⎥
⎢M31
φC3 φA3 φC2 φA3 φT C1 ⎥
⎢
⎥
r⎢ T
⎥
⎢ N φT C2 φT A3 φT C2 φB1 φT C2 φT A1 ⎥
⎣ 11
⎦
M11 φC1 φT A3 ⎡
N11
N1i
T
M1i
2.44
⎢
⎥
⎢M31 φA3 φC2 φA3 φT C1 ⎥
⎢
⎥
2r ⎢ T
⎥
⎢ N φT C2 φB1 φT C2 φT A1 ⎥
⎣ 11
⎦
M11 φC1 φB1 φC1 φT A1 hold when i 2, 3, 4. In that case, the real solution of 1.5 can be expressed as X X1 in 2.15.
b System 1.5 has a complex solution if and only if 2.44 hold when i 3, 4 or i 2, 4 or
i 2, 3. In that case, the complex solutions of 1.5 can be expressed as X X1 X2 i or X X1 X3 j
14
International Journal of Mathematics and Mathematical Sciences
or X X1 X4 k, where X1 , X2 , X3 , and X4 are expressed as 2.15, 2.16, 2.17, and 2.18,
respectively.
Proof. From 2.34 we can get the necessary and sufficient conditions for Xi 0, i 1, 2, 3, 4.
Thus we can get the results of this Corollary.
3. Solvability Conditions for Real and Complex Hermitian Solutions to
1.6 Over H
In this section, using the results of Theorem 2.6, Theorem 2.7, and Corollary 2.8, we give
necessary and sufficient conditions for 1.6 over H to have real and complex Hermitian
solutions.
Theorem 3.1. Let A1 , A3 , B1 , C1 , C2 , and C3 be defined in Lemma 2.2, A4 ∈ Hl×n , C4 ∈ Hl×l , and
suppose that system 1.5 and the matrix equation A4 Y A∗4 C4 over H have Hermitian solutions X
and Y ∈ Hn×n , respectively. Then system 1.6 over H has a real Hermitian solution if and only if
2.44 hold when i 2, 3, 4, and
0
⎢
r⎢
⎣ 0
M41
⎡
⎡
0
⎢
⎢ 0
⎢
⎢
⎢M41
⎢
r⎢
⎢ 0
⎢
⎢
⎢ 0
⎣
0
T
M31
2rM31 ,
M31 φC3 ⎤
T
⎤
⎡
M41
0
0
T
0
0
M41
⎥
T
T
⎦,
⎣
M411
φB1 φ A1 ⎥
⎦ rM41 r
T
T
φB
φ
M
A
1
1
411
φC4 φA4 φC2 φA4 φT C1 r
⎡
0
3.1
3.2
⎤
⎤
⎡
M41 M411
⎢
⎥
⎢M41 M411
⎥
⎥
⎢
φC4 ⎢
⎥
⎥
⎢
rM41 r ⎢ 0 φT B1 ⎥,
r⎢
⎥
⎢ 0 φT B1 φT C2 φT A4 ⎥
⎦
⎣
⎣
⎦
0
φA
1
0 φA1 φC1 φT A4 ⎤
T
0
M41
0
0
0
⎥
⎡
⎤
T
⎥
M41 M411
0
M411
φT A3 φB1 φT A1 ⎥
⎢
⎥
⎥
⎢ 0 φA3 ⎥
⎥
M411 φC4 0
0
0
⎢
⎥
⎥
⎥ 2r ⎢
⎥,
⎢ 0 φT B1 ⎥
φA3 0
φC3 φA3 φC2 φA3 φT C1 ⎥
⎥
⎣
⎦
⎥
T
T
T
T
T
T
⎥
φ B1 0 φA1 0
φ C2 φ A3 φ C2 φB1 φ C2 φ A1 ⎦
T
T
φA1 0
φC1 φ A3 φC1 φB1 φC1 φ A1 ⎡
0
0
0
0
T
M41
T
M41
0
0
⎤
⎢
⎥
⎡
⎤
T
⎢ 0
M41 M411
0
M411
φB1 φT A1 ⎥
⎢
⎥
⎢
⎢
⎥
⎥
⎥ 2r ⎢ 0 φT B1 ⎥,
r⎢
0
0
⎢M41 M411 φC4 ⎥
⎣
⎦
⎢
⎥
T
T
T
⎢ 0 φT B1 ⎥
0
φA
0
φ
φ
B
φC
B
φ
C
1
2
1
1 ⎦
1
⎣
0 φA1 0
φC1 φB1 φC1 φT A1 3.3
International Journal of Mathematics and Mathematical Sciences
15
where
⎡
M41
A42
A43
A44
⎡
⎤
⎢
⎥
⎢ A41 A44 −A43 ⎥
⎢
⎥
⎢
⎥,
⎢−A44 A41 A42 ⎥
⎣
⎦
A43 −A42 A41
M411
⎤
A21 0 0 0
⎢
⎥
⎢−A22 0 0 0⎥
⎢
⎥
⎢
⎥.
⎢−A23 0 0 0⎥
⎣
⎦
3.4
−A24 0 0 0
Proof. From Corollary 2.8, system 1.5 over H has a real Hermitian solution if and only if
2.44 hold when i 2, 3, 4. By 2.15, the real Hermitian solutions of 1.5 over H can be
expressed as
X1 1
1
1
1
P1 φX0 P1T P2 φX0 P2T P3 φX0 P3T P4 φX0 P4T
4
4
4
4
⎡
⎤
LφF LφA1 P1T
⎢
⎥
⎢LφF LφA1 P T ⎥
2⎥
⎢
P1 , P2 , P3 , P4 LφA1 LφF LφM V ⎢
⎥
⎢LφF LφA P T ⎥
1
3⎦
⎣
LφF LφA1 P4T
⎡
⎤
LφM LφF LφA1 P1T
⎢
⎥
⎢L
L
L
PT⎥
T ⎢ φM φF φA1 2 ⎥
P1 , P2 , P3 , P4 LφA1 LφF V ⎢
⎥,
⎢LφM LφF LφA P T ⎥
1
3⎦
⎣
LφM LφF LφA1 P4T
3.5
where V is arbitrary matrices with compatible sizes.
Let A1 , C1 0; B1 , C2 0; A3 A4 ; C3 C4 in Corollary 2.8 and 2.15. It is easy to
verify that the matrix equation A4 Y A∗4 C4 over H has a real Hermitian solution if and only
if 3.1 hold and the real Hermitian solution can be expressed as
Y1 1
1
1
1
P1 φY0 P1T P2 φY0 P2T P3 φY0 P3T P4 φY0 P4T
4
4
4
4
⎡
⎤
LφA1 P1T
⎢
⎥
⎢L
PT⎥
T ⎢ φA1 2 ⎥
P1 , P2 , P3 , P4 LφA4 U U ⎢
⎥,
⎢LφA P T ⎥
1
3
⎣
⎦
LφA1 P4T
3.6
16
International Journal of Mathematics and Mathematical Sciences
where φY0 is a particular solution to φA4 Yij 4 × 4 φT A4 φC4 and U is arbitrary
matrices with compatible sizes. The expression of Y1 can also be obtained from Lemma 2.1.
Let
P1 , P2 , P3 , P4 P,
G 1
1
1
1
P1 φX0 P1T P2 φX0 P2T P3 φX0 P3T P4 φX0 P4T
4
4
4
4
1
1
1
1
T
T
T
− P1 φY0 P1 − P2 φY0 P2 − P3 φY0 P3 − P4 φY0 P4T .
4
4
4
4
3.7
Equating X1 and Y1 , we obtain the following equation:
⎡
⎤
LφF LφA1 P1T
⎢
⎥
⎢LφF LφA1 P T ⎥
2⎥
⎢
X1 − Y1 G P LφA1 LφF LφM V ⎢
⎥
⎢LφF LφA P T ⎥
1
3⎦
⎣
LφF LφA1 P4T
⎡
⎤
⎡
⎤
LφM LφF LφA1 P1T
LφA4 P1T
⎢
⎥
⎢
⎥
⎢L
⎢L
L
L
PT⎥
PT⎥
T ⎢ φM φF φA1 2 ⎥
T ⎢ φA4 2 ⎥
P LφA1 LφF V ⎢
⎥ − P LφA4 U − U ⎢
⎥.
⎢LφM LφF LφA P T ⎥
⎢LφA P T ⎥
1
4
3⎦
3⎦
⎣
⎣
LφM LφF LφA1 P4T
LφA4 P4T
3.8
It is obvious that system 1.5 and the matrix equation A4 Y A∗4 C4 over H have common real
Hermitian solution if and only if min rX1 − Y1 0, that is, X1 − Y1 0. Hence, we have the
matrix equation
⎡
⎡
⎤
⎤
LφF LφA1 P1T
LφA4 P1T
⎢
⎢
⎥
⎥
⎢LφF LφA1 P T ⎥
⎢L
PT⎥
2⎥
⎢
T ⎢ φA4 2 ⎥
G P LφA4 U U ⎢
⎥ − P LφA1 LφF LφM V ⎢
⎥
⎢LφF LφA P T ⎥
⎢LφA P T ⎥
4
1
3⎦
3⎦
⎣
⎣
LφA4 P4T
LφF LφA1 P4T
⎡
⎤
LφM LφF LφA1 P1T
⎢
⎥
⎢L
L
L
PT⎥
T ⎢ φM φF φA1 2 ⎥
− P LφA1 LφF V ⎢
⎥.
⎢LφM LφF LφA P T ⎥
1
3
⎣
⎦
LφM LφF LφA1 P4T
3.9
International Journal of Mathematics and Mathematical Sciences
17
We know by Lemma 2.5 that 3.9 is solvable if and only if the following four rank equalities
hold
⎡
⎢
r⎢
⎣
G
P LφA4 RφA4 P T
0
⎤
⎡
0
P LφA4 ⎤
⎥
⎢
⎥
⎥ r ⎢ RφA P T
0 ⎥
4
⎦
⎣
⎦,
RφF RφA1 P T
RφF RφA1 P T
0
0
0
P LφA4 P LφA1 LφF
G
P LφA4 P LφA1 LφF
r
,
r
RφA4 P T
0
0
RφA4 P T
0
0
⎡
⎤
G
P LφA4 P LφA1 LφF LφM
⎢
⎥
⎥
r⎢
RφA4 P T
0
0
⎣
⎦
T
RφM RφF RφA1 P
0
0
⎡
⎤
0
P LφA4 P LφA1 LφF LφM
⎢
⎥
⎥,
RφA4 P T
0
0
r⎢
⎣
⎦
T
RφM RφF RφA1 P
0
0
⎡
⎤
⎡
⎤
G
P LφA4 P LφA1 LφF
0
P LφA4 P LφA1 LφF
⎢
⎥
⎢
⎥
T
⎥ r ⎢ RφA P T
⎥.
0
0
0
0
r⎢
4
⎣ RφA4 P
⎦
⎣
⎦
T
T
RφF RφA1 P
RφF RφA1 P
0
0
0
0
3.10
Under the conditions that the system 1.5 and the matrix equation A4 Y A∗4 C4 over H have
Hermitian solutions, it is not difficult to show by Lemma 2.3 and block Gaussian elimination
that 3.10 are equivalent to the four rank equalities 3.2 and 3.3, respectively. Note that
the processes are too much tedious; we omit them here. Obviously, the system 1.5 and the
matrix equation A4 Y A∗4 C4 over H have a common real Hermitian solution if and only if
3.2 and 3.3 hold. Thus, the system 1.6 over H has a real Hermitian solution if and only
if 2.44 hold when i 2, 3, 4, and 3.1–3.3 hold.
Similarly, from Corollary 2.8, we know that the system 1.5 over H has a complex
Hermitian solution if and only if 2.44 hold when i 3, 4, i 2, 4, or; i 2, 3; its complex
Hermitian solutions can be expressed as X X1 X2 i, X X1 X3 j, or X X1 X4 k. It is also
easy to derive the necessary and sufficient condition for the matrix equation A4 Y A∗4 C4 over
H to have a complex Hermitian solution; its complex Hermitian solution can be expressed as
Y Y1 Y2 i, Y Y1 Y3 j, or Y Y1 Y4 k. By equating X1 and Y1 , X2 and Y2 , X3 , and Y3 , X4
and Y4 , respectively, we can derive the necessary and sufficient conditions for the system 1.6
over H to have a complex Hermitian solution.
Acknowledgment
This research was supported by the Excellent Young Teacher Grant of East China University
of Science and Technology yk0157124 and the Natural Science Foundation of China
11001079.
18
International Journal of Mathematics and Mathematical Sciences
References
1 T. W. Hungerford, Algebra, vol. 73 of Graduate Texts in Mathematics, Springer, New York, NY, USA,
1980.
2 C. G. Khatri and S. K. Mitra, “Hermitian and nonnegative definite solutions of linear matrix equations,” SIAM Journal on Applied Mathematics, vol. 31, no. 4, pp. 579–585, 1976.
3 D. Hua and P. Lancaster, “Linear matrix equations from an inverse problem of vibration theory,”
Linear Algebra and Its Applications, vol. 246, pp. 31–47, 1996.
4 Y. H. Liu and Y. G. Tian, “More on extremal ranks of the matrix expressions A − BX ± X ∗ B ∗ with
statistical applications,” Numerical Linear Algebra with Applications, vol. 15, no. 4, pp. 307–325, 2008.
5 M. Wei and Q. Wang, “On rank-constrained Hermitian nonnegative-definite least squares solutions
to the matrix equation AXA∗ B,” International Journal of Computer Mathematics, vol. 84, no. 6, pp.
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6 X. Zhang and M.-Y. Cheng, “The rank-constrained Hermitian nonnegative-definite and positive-definite solutions to the matrix equation AXA∗ B,” Linear Algebra and Its Applications, vol. 370, pp.
163–174, 2003.
7 Y. H. Liu and Y. G. Tian, “Max-min problems on the ranks and inertias of the matrix expressions
A − BXC ± BXC∗ with applications,” Journal of Optimization Theory and Applications, vol. 148, no. 3,
pp. 593–622, 2011.
8 J. Groß, “A note on the general Hermitian solution to AXA∗ B,” Bulletin of the Malaysian Mathematical Society. Second Series, vol. 21, no. 2, pp. 57–62, 1998.
9 Y. H. Liu, Y. G. Tian, and Y. Takane, “Ranks of Hermitian and skew-Hermitian solutions to the matrix
equation AXA∗ B,” Linear Algebra and Its Applications, vol. 431, no. 12, pp. 2359–2372, 2009.
10 Y. G. Tian and Y. H. Liu, “Extremal ranks of some symmetric matrix expressions with applications,”
SIAM Journal on Matrix Analysis and Applications, vol. 28, no. 3, pp. 890–905, 2006.
11 Y. G. Tian, “Maximization and minimization of the rank and inertia of the Hermitian matrix expression A − BX − BX∗ with applications,” Linear Algebra and Its Applications, vol. 434, no. 10, pp. 2109–
2139, 2011.
12 Q. W. Wang and Z.-C. Wu, “Common Hermitian solutions to some operator equations on Hilbert
C∗ -modules,” Linear Algebra and its Applications, vol. 432, no. 12, pp. 3159–3171, 2010.
13 Y. G. Tian, “Ranks of solutions of the matrix equation AXB C,” Linear and Multilinear Algebra, vol.
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14 Y. H. Liu, “Ranks of solutions of the linear matrix equation AX Y B C,” Computers & Mathematics
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15 Y. H. Liu, “Ranks of least squares solutions of the matrix equation AXB C,” Computers & Mathematics with Applications, vol. 55, no. 6, pp. 1270–1278, 2008.
16 Y. G. Tian, “Upper and lower bounds for ranks of matrix expressions using generalized inverses,”
Linear Algebra and Its Applications, vol. 355, pp. 187–214, 2002.
17 Y. G. Tian and S. Cheng, “The maximal and minimal ranks of A − BXC with applications,” New York
Journal of Mathematics, vol. 9, pp. 345–362, 2003.
18 Y. G. Tian, “The maximal and minimal ranks of some expressions of generalized inverses of matrices,”
Southeast Asian Bulletin of Mathematics, vol. 25, no. 4, pp. 745–755, 2002.
19 Y. G. Tian, “The minimal rank of the matrix expression A − BX − Y C,” Missouri Journal of Mathematical
Sciences, vol. 14, no. 1, pp. 40–48, 2002.
20 Q. W. Wang, Z.-C. Wu, and C.-Y. Lin, “Extremal ranks of a quaternion matrix expression subject
to consistent systems of quaternion matrix equations with applications,” Applied Mathematics and
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International Journal of Mathematics and Mathematical Sciences
19
25 Q. W. Wang, S. W. Yu, and W. Xie, “Extreme ranks of real matrices in solution of the quaternion matrix
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27 Q. W. Wang, H.-S. Zhang, and S.-W. Yu, “On solutions to the quaternion matrix equation AXB CY D E,” Electronic Journal of Linear Algebra, vol. 17, pp. 343–358, 2008.
28 Y. G. Tian, “The solvability of two linear matrix equations,” Linear and Multilinear Algebra, vol. 48, no.
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