# (Anti-)Hermitian Generalized (Anti-)Hamiltonian Solution to a System of Matrix Equations

Authors: Juan Yu; Qing-Wen Wang; Chang-Zhou Dong
Journal: Mathematical Problems in Engineering
 (2014)

---

## Abstract

We mainly solve three problems. Firstly, by the decomposition of the (anti-)Hermitian generalized (anti-)Hamiltonian matrices, the necessary and sufficient conditions for the existence of and the expression for the (anti-)Hermitian generalized (anti-)Hamiltonian solutions to the system of matrix equationsA
X
=
B
,
X
C
=
D are derived, respectively. Secondly, the optimal approximation solution min
⁡
X
∈
K
⁡
∥
X
^
-
X
∥ is obtained, where K is the (anti-)Hermitian generalized (anti-)Hamiltonian solution set of the above system and X
^ is the given matrix. Thirdly, the least squares (anti-)Hermitian generalized (anti-)Hamiltonian solutions are considered. In addition, algorithms about computing the least squares (anti-)Hermitian generalized (anti-)Hamiltonian solution and the corresponding numerical examples are presented.


---

## Body

## 1. Introduction

Throughout this paper, the set of allm
×
n complex matrices, the set of all n
×
n Hermitian matrices, the set of all n
×
n anti-Hermitian matrices, the set of all n
×
n unitary matrices, and the set of all n
×
n antisymmetric orthogonal matrices are denoted, respectively, by ℂ
m
×
n, H
ℂ
n
×
n
,
A
H
ℂ
n
×
n
,
U
ℂ
n
×
n, and A
S
O
ℝ
n
×
n. The symbol I
n represents an identity matrix of order n and r
(
A
)
,
A
†, and A
*, respectively, stand for the rank, the Moore-Penrose inverse, and the conjugate transpose of matrix A. For two matrices A
,
B
∈
ℂ
m
×
n, the inner product is defined by 〈
A
,
B
〉
=
tr
⁡
(
B
*
A
). Obviously, ℂ
m
×
n is a complete inner product space. The norm ∥
·
∥, induced by the inner product, is called the Frobenius norm. A
*
B stands for the Hadamard product of two matrices A and B. For A
∈
ℂ
m
×
n, two matrices L
A  and   R
A, respectively, represent two orthogonal projectors L
A
=
I
n
-
A
†
A and R
A
=
I
m
-
A
A
†, both of which satisfy
(1)
L
A
=
(
L
A
)
2
=
(
L
A
)
*
=
(
L
A
)
†
,
R
A
=
(
R
A
)
2
=
(
R
A
)
*
=
(
R
A
)
†
.The Hamiltonian matrices defined as in [1] are very important in engineering (see [2] and the references therein). Moreover, using Hamiltonian matrices to solve algebraic matrix Riccati equation is a very effective method in optimal control theory [3–5]. As the extension of the Hamiltonian matrices, the following four definitions, which can also be found in [1, 6, 7], are given. Without special statement, we in this paper always assume that J
∈
A
S
O
ℝ
2
k
×
2
k satisfies
(2)
J
T
=
-
J
,
J
T
J
=
J
J
T
=
I
n
.Definition 1.
A matrixX
∈
H
H
ℂ
2
k
×
2
k is said to be a Hermitian generalized Hamiltonian matrix if X
=
X
* and J
X
J
=
X
*.Definition 2.
A matrixX
∈
H
A
H
ℂ
2
k
×
2
k is said to be a Hermitian generalized anti-Hamiltonian matrix if X
=
X
* and J
X
J
=
-
X
*.Definition 3.
A matrixX
∈
A
H
A
H
ℂ
2
k
×
2
k is said to be an anti-Hermitian generalized anti-Hamiltonian matrix if X
=
-
X
* and J
X
J
=
-
X
*.Definition 4.
A matrixX
∈
A
H
H
ℂ
2
k
×
2
k is said to be an anti-Hermitian generalized Hamiltonian matrix if X
=
-
X
* and J
X
J
=
X
*.The well-known system of matrix equations(3)
A
X
=
B
,
X
C
=
D
,
with unknown matrix X, has attracted much attention and has been widely and deeply studied by many authors. For example, Khatri and Mitra [8] in 1976 established the Hermitian and nonnegative definite solution to the system (3). Mitra [9] in 1984 gave the system (3) the minimal rank solution over the complex field ℂ. Wang in [10] and Wang et al. [11], respectively, investigated the bisymmetric and centrosymmetric solutions over the quaternion algebra and obtained the bisymmetric nonnegative definite solutions with extremal ranks and inertias to the system (3). Xu in [12] studied the common Hermitian and positive solutions to the adjointable operator equations (3). Yuan in [13] presented the least squares solutions to the system (3). Some other results concerning the system (3) can be found in [14–23].As special cases of the system (3), the classical matrix equations A
X
=
B and X
C
=
D have also been investigated (see, e.g., [1, 2, 5–7, 24–31]). For instance, Dai [24], by means of the singular value decomposition, derived the symmetric solution to equation A
X
=
B. Guan and Jiang [6], using the decomposition of the anti-Hermitian generalized anti-Hamiltonian matrices, derived the least squares solution to equation A
X
=
B. Zhang et al. in [29] and [1], respectively, obtained the general expression of the least squares Hermitian generalized Hamiltonian solutions to equation X
C
=
D and got the unite optimal approximation solution in the least squares solutions set and gave the solvable conditions and the general representation of the Hermitian generalized Hamiltonian solutions to equation A
X
=
B, by using the singular value decomposition and the properties of Hermitian generalized Hamiltonian matrices.As far as we know, there has been little information on studying the (anti-)Hermitian generalized (anti-)Hamiltonian solution to the system (3) over ℂ
2
k
×
2
k. So, motived by the work mentioned above, especially the work in [6, 7, 26, 29, 30], we, in this paper, are mainly concerned with the following three problems.Problem 5.
GivenA
,
B
∈
ℂ
m
×
2
k,C
,
D
∈
ℂ
2
k
×
q, find X
∈
H
H
ℂ
2
k
×
2
k
(
H
A
H
ℂ
2
k
×
2
k
,
A
H
H
ℂ
2
k
×
2
k, or A
H
A
H
ℂ
2
k
×
2
k
) such that the system (3) holds.Problem 6.
GivenX
^
∈
ℂ
2
k
×
2
k, find X
~
∈
K such that
(4)
∥
X
^
-
X
~
∥
=
min
⁡
X
∈
K
∥
X
^
-
X
∥
,
where K is the solution set of Problem 5.Problem 7.
LetA
,
B
∈
ℂ
m
×
2
k,C
,
D
∈
ℂ
2
k
×
q. Find X
∈
H
H
ℂ
2
k
×
2
k
(
H
A
H
ℂ
2
k
×
2
k
,
A
H
H
ℂ
2
k
×
2
k, or A
H
A
H
ℂ
2
k
×
2
k
) such that
(5)
min
⁡
X
=
∥
A
X
-
B
∥
2
+
∥
X
C
-
D
∥
2
.The remainder of this paper is arranged as follows. In Section2, some lemmas will be introduced, which will be useful for us to obtain the solutions to Problems 5–7. In Section 3, by applying the decomposition of the (anti-)Hermitian generalized (anti-)Hamiltonian matrices, the solvability condition and the explicit expression of the solution to Problem 5 will be derived. In Section 4, the optimal approximation solution to Problem 6 will be established. In Section 5, the solution to Problem 7 will be investigated and meanwhile the minimum norm of the solution will be obtained. In Section 6, algorithms and numerical examples about computing the solution to Problem 7 will be provided. Finally, in Section 7, some conclusions will be made.


## 2. Preliminaries

In this section, we focus on introducing some lemmas, which will play key roles in solving Problems5–7.Taking into account Definitions1–4 and the eigenvalue decomposition of the matrix J
∈
A
S
O
ℝ
2
k
×
2
k, it is not difficult to conclude that the following decompositions of the (anti-)Hermitian generalized (anti-)Hamiltonian matrices hold, some of which can also be seen in [6, 26, 29, 30].Lemma 8.
Let the eigenvalue decomposition of matrixJ
∈
A
S
O
ℝ
2
k
×
2
k be
(6)
J
=
P
(
i
I
k
0
0
-
i
I
k
)
P
*
,
where P
∈
U
ℂ
2
k
×
2
k. Then X
∈
H
H
ℂ
2
k
×
2
k if and only if X can be expressed as
(7)
X
=
P
(
0
X
12
X
12
*
0
)
P
*
,
where X
12
∈
ℂ
k
×
k are arbitrary.Lemma 9.
Let the eigenvalue decomposition of matrixJ
∈
A
S
O
ℝ
2
k
×
2
k be (6). Then X
∈
A
H
A
H
ℂ
2
k
×
2
k if and only if X can be expressed as
(8)
X
=
P
(
0
X
12
-
X
12
*
0
)
P
*
,
where X
12
∈
ℂ
k
×
k is arbitrary.Lemma 10.
Let the eigenvalue decomposition of matrixJ
∈
A
S
O
ℝ
2
k
×
2
k be (6). Then X
∈
H
A
H
ℂ
2
k
×
2
k if and only if X can be expressed as
(9)
X
=
P
(
X
11
0
0
X
22
)
P
*
,
where X
11
,
X
22
∈
H
ℂ
k
×
k are arbitrary.Lemma 11.
Let the eigenvalue decomposition of matrixJ
∈
A
S
O
ℝ
2
k
×
2
k be (6). Then X
∈
A
H
H
ℂ
2
k
×
2
k if and only if X can be expressed as
(10)
X
=
P
(
X
11
0
0
X
22
)
P
*
,
where X
11
,
X
22
∈
A
H
ℂ
k
×
k are arbitrary.Lemma 12 (see [20]).
GivenA
∈
ℂ
m
×
n
,
B
∈
ℂ
p
×
l
,
C
∈
ℂ
m
×
p, and D
∈
ℂ
n
×
l, then the system of matrix equations
(11)
A
X
=
C
,
X
B
=
D
has a solution X
∈
ℂ
n
×
p if and only if
(12)
A
A
†
C
=
C
,
D
B
†
B
=
D
,
A
D
=
C
B
,
in which case the general solutions can be expressed as
(13)
X
=
A
†
C
+
D
B
†
-
A
†
A
D
B
†
+
(
I
-
A
†
A
)
W
(
I
-
B
B
†
)
,
where W
∈
ℂ
n
×
p is arbitrary.By applying the singular value decomposition, similar to the proof of Theorem  1 in [24], the following lemma can be shown.Lemma 13.
AssumeE
,
F
∈
ℂ
m
×
n. Let the singular value decomposition of E be
(14)
E
=
U
(
Σ
0
0
0
)
V
*
,
where
(15)
U
∈
U
ℂ
m
×
m
,
V
∈
U
ℂ
n
×
n
,
Σ
=
diag
⁡
(
α
1
,
…
,
α
r
)
,
α
i
>
0
,
i
=
1
,
…
,
r
;
r
=
r
(
E
)
.
Partition
(16)
V
X
V
*
=
(
X
11
X
12
X
12
*
X
22
)
,
U
*
F
V
=
(
F
11
F
12
F
21
F
22
)
,
where
(17)
X
11
∈
H
ℂ
r
×
r
,
X
22
∈
H
ℂ
(
n
-
r
)
×
(
n
-
r
)
,
F
11
∈
ℂ
r
×
r
,
F
22
∈
ℂ
(
m
-
r
)
×
(
n
-
r
)
.
Then the matrix equation
(18)
E
X
=
F
has Hermitian solutions if and only if
(19)
E
E
†
F
=
F
,
E
F
*
=
F
E
*
,
F
21
=
0
,
F
22
=
0
,
in which case the Hermitian solution can be expressed as
(20)
X
=
V
(
Σ
-
1
F
11
Σ
-
1
F
12
F
12
*
Σ
-
1
X
22
)
V
*
,
where X
22
∈
H
ℂ
(
n
-
r
)
×
(
n
-
r
) is arbitrary.By the similar way, the following lemma can also be verified.Lemma 14.
AssumeM
,
N
∈
ℂ
m
×
n. Let the singular value decomposition of M be
(21)
M
=
U
(
Π
0
0
0
)
V
*
,
where
(22)
U
∈
U
ℂ
m
×
m
,
V
∈
U
ℂ
n
×
n
,
Π
=
diag
⁡
(
β
1
,
…
,
β
s
)
,
β
i
>
0
,
i
=
1
,
…
,
s
;
s
=
r
(
M
)
.
Partition
(23)
V
X
V
*
=
(
X
11
X
12
-
X
12
*
X
22
)
,
U
*
N
V
=
(
N
11
N
12
N
21
N
22
)
,
where
(24)
X
11
∈
A
H
ℂ
s
×
s
,
X
22
∈
A
H
ℂ
(
n
-
s
)
×
(
n
-
s
)
,
N
11
∈
ℂ
s
×
s
,
N
22
∈
ℂ
(
m
-
s
)
×
(
n
-
s
)
.
Then the matrix equation
(25)
M
X
=
N
has an anti-Hermitian solution if and only if
(26)
M
M
†
N
=
N
,
M
N
*
=
-
N
M
*
,
N
21
=
0
,
N
22
=
0
,
in which case the anti-Hermitian solution can be expressed as
(27)
X
=
V
(
Π
-
1
N
11
Π
-
1
N
12
-
N
12
*
Π
-
1
X
22
)
V
*
,
where X
22
∈
A
H
ℂ
(
n
-
s
)
×
(
n
-
s
) is arbitrary.Lemma 15 (see [31]).
GivenA
′
,
B
′
∈
ℂ
k
×
(
m
+
q
), C
′
,
D
′
∈
ℂ
k
×
(
m
+
q
), suppose that the matrices A
′ and C
′, respectively, have the following singular value decompositions:
(28)
A
′
=
P
1
(
Γ
0
0
0
)
Q
1
*
,
C
′
=
U
1
(
Λ
0
0
0
)
V
1
*
,
where
(29)
P
1
=
(
P
11
P
12
)
∈
U
ℂ
k
×
k
,
P
11
∈
ℂ
k
×
t
1
;
Q
1
=
(
Q
11
Q
12
)
∈
U
ℂ
(
m
+
q
)
×
(
m
+
q
)
,
Q
11
∈
ℂ
(
m
+
q
)
×
t
1
;
U
1
=
(
U
11
U
12
)
∈
U
ℂ
k
×
k
,
U
11
∈
ℂ
k
×
t
2
;
V
1
=
(
V
11
V
12
)
∈
U
ℂ
(
m
+
q
)
×
(
m
+
q
)
,
V
11
∈
ℂ
(
m
+
q
)
×
t
2
;
Γ
=
diag
⁡
(
δ
1
,
δ
2
,
…
,
δ
t
1
)
,
δ
i
>
0
,
1
≤
i
≤
t
1
;
t
1
=
r
(
A
′
)
;
Λ
=
diag
⁡
(
γ
1
,
γ
2
,
…
,
γ
t
2
)
,
γ
i
>
0
,
1
≤
i
≤
t
2
;
t
2
=
r
(
C
′
)
.
Then the solution set of the problem
(30)
f
(
X
12
)
=
Δ
∥
(
A
′
)
*
X
12
-
(
B
′
)
*
∥
2
+
∥
X
12
C
′
-
D
′
∥
2
=
min
⁡
consists of matrices X
12
∈
ℂ
k
×
k with the following form:
(31)
X
12
=
P
1
(
ϕ
*
(
P
11
*
D
′
V
11
Λ
+
Γ
Q
11
*
(
B
′
)
*
U
12
)
Γ
-
1
Q
11
*
(
B
′
)
*
U
12
P
12
*
D
′
V
11
Λ
-
1
X
22
′
)
U
1
*
,
where
(32)
ϕ
=
(
ϕ
i
j
)
,
ϕ
i
j
=
1
δ
i
2
+
γ
j
2
,
1
≤
i
≤
t
1
,
1
≤
j
≤
t
2
,
and X
22
′
∈
ℂ
(
k
-
t
1
)
×
(
k
-
t
2
) is arbitrary.Lemma 16.
GivenE
,
F
∈
ℂ
m
×
n, let the singular value decomposition of E, the partitions of V
X
V
* and U
*
F
V be, respectively, as in (14)–(16). Then the least squares Hermitian solution to the matrix equation (18) can be expressed as
(33)
X
=
V
(
Φ
*
(
Σ
F
11
+
F
11
*
Σ
)
Σ
-
1
F
12
F
12
*
Σ
-
1
X
22
)
V
*
,
where
(34)
Φ
=
(
1
α
i
2
+
α
j
2
)
,
1
≤
i
,
j
≤
r
,
and X
22
∈
H
ℂ
(
n
-
r
)
×
(
n
-
r
) is arbitrary.Proof.
Combining (14)–(16) and the unitary invariance of the Frobenius norm, it is easy to obtain that
(35)
∥
E
X
-
F
∥
2
=
∥
(
Σ
0
0
0
)
V
*
V
(
X
11
X
12
X
12
*
X
22
)
-
U
*
F
V
∥
2
=
∥
(
Σ
X
11
Σ
X
12
0
0
)
-
(
F
11
F
12
F
21
F
22
)
∥
2
=
∥
Σ
X
11
-
F
11
∥
2
+
∥
Σ
X
12
-
F
12
∥
2
+
∥
F
21
∥
2
+
∥
F
22
∥
2
.
Then ∥
E
X
-
F
∥
2 reaches its minimum if and only if
(36)
∥
Σ
X
11
-
F
11
∥
2
,
(37)
∥
Σ
X
12
-
F
12
∥
2
reach their minimum. For X
11
=
(
x
i
j
)
∈
H
ℂ
r
×
r, F
11
=
(
f
i
j
)
∈
ℂ
r
×
r, since x
i
j
=
x
i
j
*
,
1
≤
i
,
j
≤
r, then
(38)
∥
Σ
X
11
-
F
11
∥
2
=
∑
‍
i
=
1
r
∑
‍
j
=
1
r
(
α
i
x
i
j
-
f
i
j
)
2
=
∑
‍
1
≤
i
,
j
≤
r
r
[
(
α
i
2
+
α
j
2
)
|
x
i
j
|
2
aaaaaaaa
+
2
(
α
i
f
i
j
+
α
j
f
i
j
*
)
x
i
j
+
2
|
f
i
j
|
2
]
.
Hence, there exists a unique solution X
11
=
(
x
^
i
j
)
∈
H
ℂ
r
×
r for (36) such that
(39)
x
^
i
j
=
α
i
f
i
j
+
α
j
f
i
j
*
α
i
2
+
α
j
2
,
1
≤
i
,
j
≤
r
.
That is,
(40)
X
11
=
Φ
*
(
Σ
F
11
+
F
11
*
Σ
)
,
where
(41)
Φ
=
(
1
α
i
2
+
α
j
2
)
,
1
≤
i
,
j
≤
r
.
When X
12 can be expressed as
(42)
X
12
=
Σ
-
1
F
12
,
(37) gets its minimum. Therefore, the least squares Hermitian solution to (18) can be described as (33).By the similar way, the following result can be obtained.Lemma 17.
GivenM
,
N
∈
ℂ
m
×
n, let the singular value decomposition of M, the partitions of V
X
V
*, and U
*
N
V be, respectively, as in (21)–(23). Then the least squares anti-Hermitian solution to the matrix equation M
X
=
N can be expressed as
(43)
X
=
V
(
Ψ
*
(
Π
N
11
-
N
11
*
Π
)
Π
-
1
N
12
-
N
12
*
Π
-
1
X
22
)
V
*
,
where
(44)
Ψ
=
(
1
β
i
2
+
β
j
2
)
,
1
≤
i
,
j
≤
s
,
and X
22
∈
A
H
ℂ
(
n
-
s
)
×
(
n
-
s
) is arbitrary.Lemma 18 (see [20]).
GivenF
∈
ℂ
m
×
n, G
∈
ℂ
p
×
q, and L
∈
ℂ
m
×
q, then the matrix equation F
X
G
=
L has a solution if and only if
(45)
F
F
†
L
G
†
G
=
L
,
in which case the general solution is
(46)
X
=
F
†
L
G
†
+
Y
-
F
†
F
Y
G
G
†
,
where Y
∈
ℂ
n
×
p is arbitrary.The following lemma is due to [25, 32] or [29, Lemma  5].Lemma 19.
LetM
,
N
∈
ℂ
m
×
n. Then there exists a unique matrix W
1
∈
ℂ
m
×
n such that
(47)
∥
W
1
-
M
∥
2
+
∥
W
1
-
N
∥
2
=
min
⁡
W
∈
ℂ
m
×
n
(
∥
W
-
M
∥
2
+
∥
W
-
N
∥
2
)
,
where
(48)
W
1
=
M
+
N
2
.


## 3. The Solvability Conditions and the Expression of the Solution to Problem5

In this section, our purpose is to derive the necessary and sufficient conditions of and the explicit expression of the solution to Problem5 by using the results introduced in Section 2.Theorem 20.
GivenA
,
B
∈
ℂ
m
×
2
k
,
C
,
D
∈
ℂ
2
k
×
q, let the decomposition of X
∈
H
H
ℂ
2
k
×
2
k be (7). Partition
(49)
A
P
=
(
A
1
A
2
)
,
A
1
∈
ℂ
m
×
k
,
A
2
∈
ℂ
m
×
k
;
(50)
B
P
=
(
B
1
B
2
)
,
B
1
∈
ℂ
m
×
k
,
B
2
∈
ℂ
m
×
k
;
(51)
P
*
C
=
(
C
1
C
2
)
,
C
1
∈
ℂ
k
×
q
,
C
2
∈
ℂ
k
×
q
;
(52)
P
*
D
=
(
D
1
D
2
)
,
D
1
∈
ℂ
k
×
q
,
D
2
∈
ℂ
k
×
q
;
(53)
A
′
=
(
A
1
C
1
*
)
,
B
′
=
(
B
2
D
2
*
)
,
(54)
C
′
=
(
A
2
*
C
2
)
,
D
′
=
(
B
1
*
D
1
)
.
Then Problem 5 has a solution X
∈
H
H
ℂ
2
k
×
2
k if and only if
(55)
A
′
(
A
′
)
†
B
′
=
B
′
,
D
′
(
C
′
)
†
C
′
=
D
′
,
A
′
D
′
=
B
′
C
′
,
in which case the Hermitian generalized Hamiltonian solution to Problem 5 can be expressed as
(56)
X
=
P
(
0
X
12
X
12
*
0
)
P
*
,
where
(57)
X
12
=
(
A
′
)
†
B
′
+
D
′
(
C
′
)
†
-
(
A
′
)
†
A
′
D
′
(
C
′
)
†
+
L
A
′
W
R
C
′
and W
∈
ℂ
k
×
k is arbitrary.Proof.
It follows from (7) and (49)–(52) that the system (3) can be transformed into the following system of matrix equations:
(58)
A
1
X
12
=
B
2
,
X
12
A
2
*
=
B
1
*
,
C
1
*
X
12
=
D
2
*
,
X
12
C
2
=
D
1
.
Then, combining (53) and (54) yields that
(59)
A
′
X
12
=
B
′
,
X
12
C
′
=
D
′
.
Thus, by Lemma 12, the system (59) has a solution X
12
∈
ℂ
k
×
k if and only if all equalities in (55) hold, in which case the solution can be written as (57). So the solution to system (3) can be expressed as (56).Remark 21.
LetC and D vanish in Theorem 20. Partition
(60)
A
P
=
(
A
1
A
2
)
,
A
1
∈
ℂ
m
×
k
,
A
2
∈
ℂ
m
×
k
;
B
P
=
(
B
1
B
2
)
,
B
1
∈
ℂ
m
×
k
,
B
2
∈
ℂ
m
×
k
.
Then the matrix equation A
X
=
B has Hermitian generalized Hamiltonian solutions if and only if
(61)
A
1
A
1
†
B
2
=
B
2
,
A
2
A
2
†
B
1
=
B
1
,
A
1
B
1
*
=
B
2
A
2
*
,
in which case its solution can be described as
(62)
X
=
P
(
0
X
12
X
12
*
0
)
P
*
,
where
(63)
X
12
=
A
1
†
B
2
+
B
1
*
(
A
2
†
)
*
-
A
1
†
A
1
B
1
*
(
A
2
†
)
*
+
L
A
1
W
L
A
2
and W
∈
ℂ
k
×
k is arbitrary. It is clear that this result is different from Theorem  3.1 given in [1].Similarly, by Lemmas9 and 12, we can get the anti-Hermitian generalized anti-Hamiltonian solution to system (3).Theorem 22.
GivenA
,
B
∈
ℂ
m
×
2
k, C
,
D
∈
ℂ
2
k
×
q, let the decomposition of X
∈
A
H
A
H
ℂ
2
k
×
2
k be (8). A
P
,
B
P
,
P
*
C, and P
*
D, respectively, have the partitions as in (49)–(52). Put
(64)
A
~
=
(
A
1
C
1
*
)
,
B
~
=
(
B
2
-
D
2
*
)
,
C
~
=
(
A
2
*
C
2
)
,
D
~
=
(
-
B
1
*
D
1
)
.
Then Problem 5 has a solution X
∈
A
H
A
H
ℂ
2
k
×
2
k if and only if
(65)
A
~
A
~
†
B
~
=
B
~
,
D
~
C
~
†
C
~
=
D
~
,
A
~
D
~
=
B
~
C
~
,
in which case the anti-Hermitian generalized anti-Hamiltonian solution to Problem 5 can be expressed as
(66)
X
=
P
(
0
X
12
-
X
12
*
0
)
P
*
,
where
(67)
X
12
=
A
~
†
B
~
+
D
~
C
~
†
-
A
~
†
A
~
D
~
C
~
†
+
L
A
~
Z
R
C
~
and Z
∈
ℂ
k
×
k is arbitrary.Now, we investigate the Hermitian generalized anti-Hamiltonian solution to the system (3).Theorem 23.
GivenA
,
B
∈
ℂ
m
×
2
k, C
,
D
∈
ℂ
2
k
×
q, let the decomposition of X
∈
H
A
H
ℂ
2
k
×
2
k be (9). A
P
,
B
P
,
P
*
C, and P
*
D, respectively, have the partitions as in (49)–(52). Denote
(68)
A
¯
=
(
A
1
C
1
*
)
,
B
¯
=
(
B
1
D
1
*
)
,
(69)
C
¯
=
(
A
2
C
2
*
)
,
D
¯
=
(
B
2
D
2
*
)
.
Let the singular value decompositions of A
¯ and C
¯ be, respectively,
(70)
A
¯
=
U
(
Σ
0
0
0
)
V
*
,
(71)
C
¯
=
Q
(
Π
0
0
0
)
R
*
,
where
(72)
U
∈
U
ℂ
(
m
+
q
)
×
k
,
V
∈
U
ℂ
k
×
k
,
Σ
=
diag
⁡
(
α
1
,
…
,
α
r
)
,
α
i
>
0
,
i
=
1
,
…
,
r
;
r
=
r
(
A
¯
)
,
Q
∈
U
ℂ
(
m
+
q
)
×
k
,
R
∈
U
ℂ
k
×
k
,
Π
=
diag
⁡
(
β
1
,
…
,
β
s
)
,
β
j
>
0
,
j
=
1
,
…
,
s
;
s
=
r
(
C
¯
)
.
Set
(73)
V
X
11
V
*
=
(
X
¯
11
X
¯
12
X
¯
12
*
X
¯
22
)
;
(74)
U
*
B
¯
V
=
(
B
¯
11
B
¯
12
B
¯
21
B
¯
22
)
;
(75)
R
X
22
R
*
=
(
X
^
11
X
^
12
X
^
12
*
X
^
22
)
;
(76)
Q
*
D
¯
R
=
(
D
¯
11
D
¯
12
D
¯
21
D
¯
22
)
,
where
(77)
X
¯
11
∈
H
ℂ
r
×
r
,
X
^
11
∈
H
ℂ
s
×
s
,
X
¯
22
∈
H
ℂ
(
k
-
r
)
×
(
k
-
r
)
,
X
^
22
∈
H
ℂ
(
k
-
s
)
×
(
k
-
s
)
,
B
¯
11
∈
ℂ
r
×
r
,
D
¯
11
∈
ℂ
s
×
s
,
B
¯
22
∈
ℂ
(
m
+
q
-
r
)
×
(
k
-
r
)
,
D
¯
22
∈
ℂ
(
m
+
q
-
s
)
×
(
k
-
s
)
.
Then Problem 5 has a solution X
∈
H
A
H
ℂ
2
k
×
2
k if and only if
(78)
A
¯
(
A
¯
)
†
B
¯
=
B
¯
,
A
¯
(
B
¯
)
*
=
B
¯
(
A
¯
)
*
,
B
¯
21
=
0
,
B
¯
22
=
0
,
(79)
C
¯
(
C
¯
)
†
D
¯
=
D
¯
,
C
¯
(
D
¯
)
*
=
D
¯
(
C
¯
)
*
,
D
¯
21
=
0
,
D
¯
22
=
0
,
in which case the Hermitian generalized anti-Hamiltonian solution to Problem 5 can be described as
(80)
X
=
P
(
X
11
0
0
X
22
)
P
*
,
where
(81)
X
11
=
V
(
Σ
-
1
B
¯
11
Σ
-
1
B
¯
12
(
B
¯
12
)
*
Σ
-
1
X
¯
22
)
V
*
,
(82)
X
22
=
R
(
Π
-
1
D
¯
11
Π
-
1
D
¯
12
(
D
¯
12
)
*
Π
-
1
X
^
22
)
R
*
,
and X
¯
22
∈
H
ℂ
(
k
-
r
)
×
(
k
-
r
), X
^
22
∈
H
ℂ
(
k
-
s
)
×
(
k
-
s
) are arbitrary.Proof.
It can be derived from (9), (49)–(52), and (68)-(69) that the system (3) is consistent if and only if the following two equations:
(83)
A
¯
X
11
=
B
¯
,
(84)
C
¯
X
22
=
D
¯
,
are solvable. By (70), (73), and (74), and then combining Lemma 13, we can obtain that there exists Hermitian solution X
11 such that (83) holds if and only if all equalities in (78) hold, in which case the solution can be written as (81). By the similar way, there exists Hermitian solution X
22 such that (84) holds if and only if all equalities in (79) hold, in which case the solution can be described as (82). Therefore, the Hermitian generalized anti-Hamiltonian solution to Problem 5 can be expressed as (80).From Lemmas11 and 14, it is not difficult to obtain the anti-Hermitian generalized Hamiltonian solution to Problem 5, which can be described as follows.Theorem 24.
GivenA
,
B
∈
ℂ
m
×
2
k, C
,
D
∈
ℂ
2
k
×
q, let the decomposition of X
∈
A
H
H
ℂ
2
k
×
2
k be (10). A
P
,
B
P
,
P
*
C, and P
*
D, respectively, have the partitions as in (49)–(52). Denote
(85)
A
^
=
(
A
1
C
1
*
)
,
B
^
=
(
B
1
-
D
1
*
)
,
C
^
=
(
A
2
C
2
*
)
,
D
^
=
(
B
2
-
D
2
*
)
.
Let the singular value decompositions of A
^ and C
^ be, respectively,
(86)
A
^
=
U
(
Σ
0
0
0
)
V
*
,
C
^
=
Q
(
Π
0
0
0
)
R
*
,
where
(87)
U
∈
U
ℂ
(
m
+
q
)
×
k
,
V
∈
U
ℂ
k
×
k
,
Σ
=
diag
⁡
(
α
1
,
…
,
α
r
)
,
α
i
>
0
,
i
=
1
,
…
,
r
;
r
=
r
(
A
^
)
,
Q
∈
U
ℂ
(
m
+
q
)
×
k
,
R
∈
U
ℂ
k
×
k
,
Π
=
diag
⁡
(
β
1
,
…
,
β
s
)
,
β
j
>
0
,
j
=
1
,
…
,
s
;
s
=
r
(
C
^
)
.
Set
(88)
V
X
11
V
*
=
(
X
¯
11
X
¯
12
-
X
¯
12
*
X
¯
22
)
;
U
*
B
^
V
=
(
B
^
11
B
^
12
B
^
21
B
^
22
)
;
R
X
22
R
*
=
(
X
^
11
X
^
12
-
X
^
12
*
X
^
22
)
;
Q
*
D
^
R
=
(
D
^
11
D
^
12
D
^
21
D
^
22
)
,
where
(89)
X
¯
11
∈
H
ℂ
r
×
r
,
X
^
11
∈
H
ℂ
s
×
s
,
X
¯
22
∈
H
ℂ
(
k
-
r
)
×
(
k
-
r
)
,
X
^
22
∈
H
ℂ
(
k
-
s
)
×
(
k
-
s
)
,
B
^
11
∈
ℂ
r
×
r
,
D
^
11
∈
ℂ
s
×
s
,
B
^
22
∈
ℂ
(
m
+
q
-
r
)
×
(
k
-
r
)
,
D
^
22
∈
ℂ
(
m
+
q
-
s
)
×
(
k
-
s
)
.
Then Problem 5 has a solution X
∈
A
H
H
ℂ
2
k
×
2
k if and only if
(90)
A
^
A
^
†
B
^
=
B
^
,
A
^
B
^
*
=
-
B
^
A
^
*
,
B
^
21
=
0
,
B
^
22
=
0
,
C
^
C
^
†
D
^
=
D
^
,
C
^
D
^
*
=
-
D
^
C
^
*
,
D
^
21
=
0
,
D
^
22
=
0
,
in which case the anti-Hermitian generalized Hamiltonian solution to Problem 5 can be described as
(91)
X
=
P
(
X
11
0
0
X
22
)
P
*
,
where
(92)
X
11
=
V
(
Σ
-
1
B
^
11
Σ
-
1
B
^
12
-
B
^
12
*
Σ
-
1
X
¯
22
)
V
*
,
X
22
=
R
(
Π
-
1
D
^
11
Π
-
1
D
^
12
-
D
^
12
*
Π
-
1
X
^
22
)
R
*
,
and X
¯
22
∈
A
H
ℂ
(
k
-
r
)
×
(
k
-
r
), X
^
22
∈
A
H
ℂ
(
k
-
s
)
×
(
k
-
s
) are arbitrary.


## 4. The Expression of the Unique Solution to Problem6

In this section, our aim is to derive the optimal approximation solution to Problem6.Theorem 25.
GivenX
^
∈
ℂ
2
k
×
2
k, under the hypotheses of Theorem 20, let
(93)
P
*
X
^
P
=
(
X
^
11
X
^
12
X
^
21
X
^
22
)
,
X
^
11
∈
ℂ
k
×
k
,
X
^
22
∈
ℂ
k
×
k
.
If Problem 5 has Hermitian generalized Hamiltonian solutions, then Problem 6 has a unique solution X
~
∈
H
H
ℂ
2
k
×
2
k if and only if
(94)
L
A
′
(
X
^
12
+
(
X
^
21
)
*
2
-
X
0
)
R
C
′
=
X
^
12
+
(
X
^
21
)
*
2
-
X
0
,
in which case the unique solution X
~ can be expressed as
(95)
X
~
=
P
(
0
X
0
¯
(
X
0
¯
)
*
0
)
P
*
,
where
(96)
X
0
¯
=
X
^
12
+
(
X
^
21
)
*
2
,
X
0
=
(
A
′
)
†
B
′
+
D
′
(
C
′
)
†
-
(
A
′
)
†
A
′
D
′
(
C
′
)
†
.Proof.
When the Hermitian generalized Hamiltonian solution setK of Problem 5 is nonempty, it is not difficult to verify that K is a closed convex set. Then by [33], Problem 6 has a unique solution X
~
∈
H
H
ℂ
2
k
×
2
k. From Theorem 20, for any X
∈
K, X can be expressed as
(97)
X
=
P
(
0
X
0
X
0
*
0
)
P
*
+
P
(
0
L
A
′
W
R
C
′
R
C
′
W
*
L
A
′
0
)
P
*
,
where
(98)
X
0
=
(
A
′
)
†
B
′
+
D
′
(
C
′
)
†
-
(
A
′
)
†
A
′
D
′
(
C
′
)
†
and W
∈
ℂ
k
×
k is arbitrary. Then it follows from the equalities in (93) and (97) and the unitary invariance of the Frobenius norm that
(99)
∥
X
^
-
X
∥
2
=
∥
P
*
X
^
P
-
P
*
X
P
∥
2
=
∥
(
X
^
11
X
^
12
-
X
0
-
L
A
′
W
R
C
′
X
^
21
-
X
0
*
-
R
C
′
W
*
L
A
′
X
^
22
)
∥
2
=
∥
X
^
11
∥
2
+
∥
X
^
22
∥
2
+
∥
X
^
12
-
X
0
-
L
A
′
W
R
C
′
∥
2
+
∥
X
^
21
-
X
0
*
-
R
C
′
W
*
L
A
′
∥
2
=
∥
X
^
11
∥
2
+
∥
X
^
22
∥
2
+
∥
L
A
′
W
R
C
′
-
(
-
X
0
+
X
^
12
)
∥
2
+
∥
L
A
′
W
R
C
′
-
(
-
X
0
+
X
^
21
*
)
∥
2
.
Thus, Problem 6 has a unique solution X
~
∈
H
H
ℂ
2
k
×
2
k if and only if there exists W such that
(100)
∥
L
A
′
W
R
C
′
-
(
-
X
0
+
X
^
12
)
∥
2
+
∥
L
A
′
W
R
C
′
-
(
-
X
0
+
(
X
^
21
)
*
)
∥
2
reaches its minimum. Therefore, by Lemma 19, (100) arrives at its minimum if and only if there exists W such that the matrix equation
(101)
L
A
′
W
R
C
′
=
-
X
0
+
X
^
12
-
X
0
+
(
X
^
21
)
*
2
=
X
^
12
+
(
X
^
21
)
*
2
-
X
0
holds, which, by Lemma 18, has a solution if and only if (94) holds, in which case the solution can be expressed as
(102)
W
=
L
A
′
(
X
^
12
+
(
X
^
21
)
*
2
-
X
0
)
R
C
′
+
Z
-
L
A
′
Z
R
C
′
,
where Z
∈
ℂ
k
×
k is arbitrary. Inserting (102) into (97), and then combining (94) yields (95).Analogously, the following theorem can be shown.Theorem 26.
GivenX
^
∈
ℂ
2
k
×
2
k, under the hypotheses of Theorem 22, let
(103)
P
*
X
^
P
=
(
X
^
11
X
^
12
X
^
21
X
^
22
)
,
X
^
11
∈
ℂ
k
×
k
,
X
^
22
∈
ℂ
k
×
k
.
If Problem 5 has anti-Hermitian generalized anti-Hamiltonian solutions, then Problem 6 has a unique solution X
~
∈
A
H
A
H
ℂ
2
k
×
2
k if and only if
(104)
L
A
~
(
X
^
12
-
(
X
^
21
)
*
2
-
X
0
)
R
C
~
=
X
^
12
-
(
X
^
21
)
*
2
-
X
0
,
in which case the unique solution X
~ can be expressed as
(105)
X
~
=
P
(
0
X
0
¯
-
(
X
0
¯
)
*
0
)
P
*
,
where
(106)
X
0
¯
=
X
^
12
-
(
X
^
21
)
*
2
,
X
0
=
(
A
~
)
†
B
~
+
D
~
(
C
~
)
†
-
(
A
~
)
†
A
~
D
~
(
C
~
)
†
.Now, we give the unique Hermitian generalized anti-Hamiltonian solution to Problem6.Theorem 27.
GivenX
^
∈
ℂ
2
k
×
2
k, under the hypotheses of Theorem 23, let
(107)
P
*
X
^
+
X
^
*
2
P
=
(
X
^
11
′
X
^
12
′
(
X
^
12
′
)
*
X
^
22
′
)
,
X
^
11
′
∈
H
ℂ
k
×
k
,
X
^
22
′
∈
H
ℂ
k
×
k
;
(108)
V
*
X
^
11
′
V
=
(
X
^
11
∘
X
^
12
∘
(
X
^
12
∘
)
*
X
^
22
∘
)
,
X
^
11
∘
∈
H
ℂ
r
×
r
,
X
^
22
∘
∈
H
ℂ
(
k
-
r
)
×
(
k
-
r
)
;
(109)
R
*
X
^
22
′
R
=
(
X
^
11
′
X
^
12
′′
(
X
^
12
′′
)
*
X
^
22
′′
)
,
X
^
11
′′
∈
H
ℂ
s
×
s
,
X
^
22
′′
∈
H
ℂ
(
k
-
s
)
×
(
k
-
s
)
.
If Problem 5 has Hermitian generalized anti-Hamiltonian solutions, then the unique solution X
~
∈
H
A
H
ℂ
2
k
×
2
k to Problem 6 can be expressed as
(110)
X
~
=
P
(
X
11
∘
0
0
X
22
∘
)
P
*
,
where
(111)
X
11
∘
=
V
(
Σ
-
1
B
¯
11
Σ
-
1
B
¯
12
(
B
¯
12
)
*
Σ
-
1
X
^
22
∘
)
V
*
,
(112)
X
22
∘
=
R
(
Π
-
1
D
¯
11
Π
-
1
D
¯
12
(
D
¯
12
)
*
Π
-
1
X
^
22
′′
)
R
*
.Proof.
When the Hermitian generalized anti-Hamiltonian solution setK of Problem 5 is nonempty, it is easy to prove that K is a closed convex set. Then, Problem 6 has a unique solution X
~
∈
H
A
H
ℂ
2
k
×
2
k by the aid of [33]. For any X
∈
K, due to Theorem 23, X can be expressed as
(113)
X
=
P
(
X
11
0
0
X
22
)
P
*
,
where X
11 and X
22 have the expressions as in (81) and (82). Combining the equalities in (80)–(82) and (107) and the unitary invariance of the Frobenius norm yields that
(114)
∥
X
-
X
^
∥
2
=
∥
X
-
X
^
+
X
^
*
2
∥
2
+
∥
X
^
-
X
^
*
2
∥
2
=
∥
P
(
X
11
0
0
X
22
)
P
*
-
X
^
+
X
^
*
2
∥
2
+
∥
X
^
-
X
^
*
2
∥
2
=
∥
(
X
11
0
0
X
22
)
-
(
X
^
11
′
X
^
12
′
(
X
^
12
′
)
*
X
^
22
′
)
∥
2
+
∥
X
^
-
X
^
*
2
∥
2
=
∥
X
11
-
X
^
11
′
∥
2
+
∥
X
22
-
X
^
22
′
∥
2
+
2
∥
X
^
12
′
∥
2
+
∥
X
^
-
X
^
*
2
∥
2
.
So,
(115)
min
⁡
X
∈
∈
H
A
H
ℂ
2
k
×
2
k
∥
X
-
X
^
∥
2
holds
⟺

min
⁡
X
11
∈
H
ℂ
k
×
k
∥
X
11

-

X
^
11
′
∥
2
holds
and
min
⁡
X
22
∈
H
ℂ
k
×
k
∥
X
22

-

X
^
22
′
∥
2

holds
.
By (81), (108), and the unitary invariance of the Frobenius norm, we obtain
(116)
∥
X
11
-
X
^
11
′
∥
2
=
∥
(
Σ
-
1
B
¯
11
Σ
-
1
B
¯
12
(
B
¯
12
)
*
Σ
-
1
X
¯
22
)
-
(
X
^
11
∘
X
^
12
∘
(
X
^
12
∘
)
*
X
^
22
∘
)
∥
2
=
∥
Σ
-
1
B
¯
11
-
X
^
11
∘
∥
2
+
∥
X
¯
22
-
X
^
22
∘
∥
2
+
2
∥
Σ
-
1
B
¯
12
-
X
^
12
∘
∥
2
.
Then
(117)
min
⁡
X
11
∈
H
ℂ
k
×
k
∥
X
11

-

X
^
11
′
∥
2

holds
⟺

min
⁡
X
¯
22
∈
H
ℂ
(
k
-
r
)
×
(
k
-
r
)
∥
X
¯
22

-

X
^
22
∘
∥
2
holds
.
Therefore, when X
¯
22 can be expressed as
(118)
X
¯
22
=
X
^
22
∘
,
min
⁡
X
11
∈
H
ℂ
k
×
k
⁡
∥
X
11
-
X
^
11
′
∥
2 holds. Then combining (81) yields (111). Similarly, we can derive the expression in (112) by (82) and (109). Thus, (110) is the unique solution to Problem 6.By the method used in Theorem27, the following theorem can also be shown.Theorem 28.
GivenX
^
∈
ℂ
2
k
×
2
k, under the hypotheses of Theorem 24, let
(119)
P
*
X
^
-
X
^
*
2
P
=
(
X
^
11
′
X
^
12
′
-
(
X
^
12
′
)
*
X
^
22
′
)
,
X
^
11
′
∈
A
H
ℂ
k
×
k
,
X
^
22
′
∈
A
H
ℂ
k
×
k
;
V
*
X
^
11
′
V
=
(
X
^
11
∘
X
^
12
∘
-
(
X
^
12
∘
)
*
X
^
22
∘
)
,
X
^
11
∘
∈
A
H
ℂ
r
×
r
,
X
^
22
∘
∈
A
H
ℂ
(
k
-
r
)
×
(
k
-
r
)
;
R
*
X
^
22
′
R
=
(
X
^
11
′′
X
^
12
′′
-
(
X
^
12
′′
)
*
X
^
22
′′
)
,
X
^
11
′′
∈
A
H
ℂ
s
×
s
,
X
^
22
′′
∈
A
H
ℂ
(
k
-
s
)
×
(
k
-
s
)
.
If Problem 5 has anti-Hermitian generalized Hamiltonian solutions, then the unique solution X
~
∈
A
H
H
ℂ
2
k
×
2
k to Problem 6 can be expressed as
(120)
X
~
=
P
(
X
11
∘
0
0
X
22
∘
)
P
*
,
where
(121)
X
11
∘
=
V
(
Σ
-
1
B
^
11
Σ
-
1
B
^
12
-
B
^
12
*
Σ
-
1
X
^
22
∘
)
V
*
,
X
22
∘
=
R
(
Π
-
1
D
^
11
Π
-
1
D
^
12
-
D
^
12
*
Π
-
1
X
^
22
′′
)
R
*
.


## 5. The Expression of the Solution to Problem7

If the solvability conditions of linear matrix equations are not satisfied, the least squares solution is usually considered. So, in this section, the solution to Problem7 is constructed.Theorem 29.
GivenA
,
B
∈
ℂ
m
×
2
k, C
,
D
∈
ℂ
2
k
×
q, let the decomposition of X
∈
H
H
ℂ
2
k
×
2
k be (7). A
P
,
B
P
,
P
*
C
,
P
*
D
,
C
′, and D
′, respectively, have the partitions as in (49)–(52) and (54). Denote
(122)
A
′
=
(
A
1
*
C
1
)
,
B
′
=
(
B
2
*
D
2
)
.
Let the singular value decompositions of A
′ and C
′ be as given in (28). Then the least squares Hermitian generalized Hamiltonian solution to Problem 7 can be described as (7), where X
12 has the expression as in (31).Proof.
Combining (7), (49)–(52), (54), (122), and the unitary invariance of the Frobenius norm yields that
(123)
∥
A
X
-
B
∥
2
+
∥
X
C
-
D
∥
2
=
∥
(
A
′
)
*
X
12
-
(
B
′
)
*
∥
2
+
∥
X
12
C
′
-
D
′
∥
2
.
Therefore, by Lemma 15, if X
12 has the expression as in (31), then (123) reaches its minimum. Then, substituting (31) into (7), we obtain the least squares Hermitian generalized Hamiltonian solution to Problem 7.Corollary 30.
GivenA
,
B
∈
ℂ
m
×
2
k, C
,
D
∈
ℂ
2
k
×
q, under the conditions of Theorem 29, the least squares Hermitian generalized Hamiltonian solution with minimum norm to Problem 7 can be described as (7), where X
12 has the expression as in (31) with X
22
′
=
0.By the same way, we can also derive the least squares anti-Hermitian generalized anti-Hamiltonian solution to Problem7.Theorem 31.
GivenA
,
B
∈
ℂ
m
×
2
k, C
,
D
∈
ℂ
2
k
×
q, let the decomposition of X
∈
A
H
A
H
ℂ
2
k
×
2
k be (8). A
P
,
B
P
,
P
*
C, and P
*
D, respectively, have the partitions as in (49)–(52). Denote
(124)
A
′
=
(
A
1
*
C
1
)
,
B
′
=
(
B
2
*
-
D
2
)
,
C
′
=
(
A
2
*
C
2
)
,
D
′
=
(
-
B
1
*
D
1
)
.
Let the singular value decompositions of A
′ and C
′ be as in (28). Then the least squares anti-Hermitian generalized anti-Hamiltonian solution to Problem 7 can be described as (8), where X
12 has the expression as in (31).Corollary 32.
GivenA
,
B
∈
ℂ
m
×
2
k, C
,
D
∈
ℂ
2
k
×
q, under the conditions of Theorem 31, the least squares anti-Hermitian generalized anti-Hamiltonian solution with minimum norm to Problem 7 can be described as (8), where X
12 has the expression as in (31) with X
22
′
=
0.At present, we give the least squares Hermitian generalized anti-Hamiltonian solution to Problem7.Theorem 33.
AssumeA
,
B
∈
ℂ
m
×
2
k, C
,
D
∈
ℂ
2
k
×
q. Let the decomposition of X
∈
H
A
H
ℂ
2
k
×
2
k be (9). A
P
,
B
P
,
P
*
C, P
*
D, A
¯
,
B
¯
,
C
¯, and D
¯, respectively, have the partitions as in (49)–(52), (68), and (69). Let the singular value decompositions of A
¯ and C
¯ be, respectively, (70) and (71), V
X
11
V
*
,
U
*
B
¯
V, R
X
22
R
*
,
and
Q
*
D
¯
R have the partitions as in (73)–(76). Then the least squares Hermitian generalized anti-Hamiltonian solution to Problem 7 can be expressed as (9) with
(125)
X
11
=
V
(
Φ
1
*
(
Σ
B
¯
11
+
B
¯
11
*
Σ
)
Σ
-
1
B
¯
12
B
¯
12
*
Σ
-
1
X
¯
22
)
V
*
,
(126)
X
22
=
R
(
Φ
2
*
(
Π
D
¯
11
+
D
¯
11
*
Π
)
Π
-
1
D
¯
12
D
¯
12
*
Π
-
1
X
^
22
)
R
*
,
where
(127)
Φ
1
=
(
1
α
i
2
+
α
j
2
)
,
1
≤
i
,
j
≤
r
;
Φ
2
=
(
1
β
i
2
+
β
j
2
)
,
1
≤
i
,
j
≤
s
,
and X
¯
22
∈
H
ℂ
(
k
-
r
)
×
(
k
-
r
), X
^
22
∈
H
ℂ
(
k
-
s
)
×
(
k
-
s
) are arbitrary.Proof.
It follows from (9), (49)–(52), (68), (69), and the unitary invariance of the Frobenius norm that
(128)
∥
A
X
-
B
∥
2
+
∥
X
C
-
D
∥
2
=
∥
A
¯
X
11
-
B
¯
∥
2
+
∥
C
¯
X
22
-
D
¯
∥
2
.
Then
(129)
∥
A
X
-
B
∥
2
+
∥
X
C
-
D
∥
2
gains its minimum value if and only if
(130)
min
⁡
=
∥
A
¯
X
11
-
B
¯
∥
2
holds
,
(131)
min
⁡
=
∥
C
¯
X
22
-
D
¯
∥
2
holds
.
So, by (68), (70), (73), and (74) and then combining Lemma 16, we get that if X
11 has the expression as in (125), then (130) holds. Similarly, if X
22 has the expression as in (126), then (131) holds. Thus, the least squares Hermitian generalized anti-Hamiltonian solution to Problem 7 can be expressed as (9), where X
11 and X
22 have the expressions as in (125) and (126).Corollary 34.
GivenA
,
B
∈
ℂ
m
×
2
k, C
,
D
∈
ℂ
2
k
×
q, under the conditions of Theorem 33, the least squares Hermitian generalized anti-Hamiltonian solution with minimum norm to Problem 7 can be expressed as (9) with X
11 and X
22 having the expressions as in (125) and (126), where X
¯
22
=
0
,
X
^
22
=
0.At last, on the basis of Lemma17, we can obtain the least squares anti-Hermitian generalized Hamiltonian solution to Problem 7, the proof of which is analogous to the proof of Theorem 33.Theorem 35.
GivenA
,
B
∈
ℂ
m
×
2
k, C
,
D
∈
ℂ
2
k
×
q, let the decomposition of X
∈
A
H
H
ℂ
2
k
×
2
k be (10). A
P
,
B
P
,
P
*
C
,
P
*
D
,
A
^
,
B
^
,
C
^, and D
^, respectively, have the partitions as in (49)–(52), (85). Assume that the singular value decompositions of A
^ and C
^ are, respectively, expressed as in (86) and V
X
11
V
*
,
U
*
B
^
V, R
X
22
R
*
,
Q
*
D
^
R have the partitions as in (88). Then the least squares anti-Hermitian generalized Hamiltonian solution to Problem 7 can be expressed as (10) with
(132)
X
11
=
V
(
Ψ
1
*
(
Σ
B
^
11
-
B
^
11
*
Σ
)
Σ
-
1
B
^
12
-
B
^
12
*
Σ
-
1
X
¯
22
)
V
*
,
X
22
=
R
(
Ψ
2
*
(
Π
D
^
11
-
D
^
11
*
Π
)
Π
-
1
D
^
12
-
D
^
12
*
Π
-
1
X
^
22
)
R
*
,
where
(133)
Ψ
1
=
(
1
α
i
2
+
α
j
2
)
,
1
≤
i
,
j
≤
r
;
Ψ
2
=
(
1
β
i
2
+
β
j
2
)
,
1
≤
i
,
j
≤
s
,
and X
¯
22
∈
A
H
ℂ
(
k
-
r
)
×
(
k
-
r
), X
^
22
∈
A
H
ℂ
(
k
-
s
)
×
(
k
-
s
) are arbitrary.Corollary 36.
GivenA
,
B
∈
ℂ
m
×
2
k, C
,
D
∈
ℂ
2
k
×
q, under the conditions of Theorem 35, the least squares anti-Hermitian generalized Hamiltonian solution with minimum norm to Problem 7 can be expressed as (10) with X
11 and X
22 having the expressions as in (132), where X
¯
22
=
0
,
X
^
22
=
0.


## 6. Algorithms and Numerical Examples

In this section, algorithms are given to compute the solution to Problem7, and meanwhile some numerical examples are presented to show that the algorithms provided are feasible. Note that all the tests are performed by MATLAB 7.6.An algorithm is firstly presented to compute the least squares Hermitian generalized Hamiltonian solution to Problem7.Algorithm 37.
Step  1. Input A
,
B
,
C
,
D
,
J.
Step  2. Compute the eigenvalue decomposition of J according to (6).
Step  3. Compute A
P
,
B
P
,
P
*
C
,
P
*
D according to (49)–(52).
Step  4. Compute A
′
,
B
′
,
C
′
,
D
′ according to (53) and (54). If the conditions in (55) hold, then compute the Hermitian generalized Hamiltonian solution to Problem 5 according to (56) and (57). Otherwise, turn to Step  5.
Step  5. Compute A
′
,
B
′
,
C
′
,
D
′ according to (53) and (122).
Step  6. Compute the singular value decompositions of A
′ and C
′ according to (28).
Step  7. Compute X
12 according to (31).
Step  8. Compute X according to (7), and output X.Example 38.
Given(134)
A
=
(
3
+
6
i
2
+
i
7
-
2
i
8
+
3
i
2
-
3
i
5
-
4
i
1
+
4
i
9
+
3
i
)
,
B
=
(
2
-
4
i
3
+
2
i
5
+
i
4
+
i
6
+
i
2
-
5
i
1
+
6
i
5
+
3
i
)
,
C
=
(
4
+
7
i
10
+
3
i
7
+
i
8
+
7
i
3
+
9
i
1
-
6
i
2
-
5
i
5
+
6
i
2
+
7
i
2
3
i
3
+
7
i
)
,
D
=
(
7
+
3
i
5
2
i
5
+
2
i
2
-
3
i
6
-
i
3
+
i
9
-
i
4
4
-
2
i
5
+
2
i
1
+
4
i
)
,
J
=
(
0
0
0
1
0
0
-
1
0
0
1
0
0
-
1
0
0
0
)
,it can be easily verified that the conditions in (55) are not satisfied. Then, according to Algorithm 37, the least squares Hermitian generalized Hamiltonian solution X to Problem 7 can be expressed as
(135)
X
=
(
0.0865
-
i
0.0141
-
0.0359
i
0.0317
-
0.0312
i
0.1174
+
0.0824
i
0.0141
+
0.0359
i
-
0.0148
-
0.0201
-
0.0682
i
0.1189
+
0.0954
i
0.0317
+
0.0302
i
-
0.0201
+
0.0682
i
-
0.0724
0.1114
+
0.0359
i
0.1173
-
0.0824
i
0.1189
-
0.0954
i
0.1114
-
0.0359
i
0.0006
)
,
min
⁡
X
∈
H
H
ℂ
2
k
×
2
k
∥
X
-
X
*
∥
=
0.0000
,
min
⁡
X
∈
H
H
ℂ
2
k
×
2
k
∥
X
*
-
J
X
J
∥
=
0.6000
.Remark 39.
(1) There exists a unique least squares Hermitian generalized Hamiltonian solution to Problem7 if and only if both A
′ and C
′ in Theorem 29 have full row ranks. Example 38 just illustrates it.
(2) Similarly, the algorithm about computing the least squares anti-Hermitian generalized anti-Hamiltonian solution to Problem7 can be shown. We omit it here.
Now, we provide another algorithm to compute the least squares Hermitian generalized anti-Hamiltonian solution to Problem7.Algorithm 40.
Step  1. Input A
,
B
,
C
,
D
,
J.
Step  2. Compute the eigenvalue decomposition of J according to (6).
Step  3. Compute A
P
,
B
P
,
P
*
C
,
P
*
D according to (49)–(52).
Step  4. Compute A
¯
,
B
¯
,
C
¯
,
D
¯ according to (68) and (69).
Step  5. Compute the singular value decompositions of A
¯ and C
¯ according to (70)-(71).
Step  6. Compute the partitions of U
*
B
¯
V
,
Q
*
D
¯
R according to (74) and (76). If the conditions in (78) and (79) are all satisfied, then compute the Hermitian generalized anti-Hamiltonian solution to Problem 5 according to (80)–(82). Otherwise, turn to Step  7.
Step  7. Compute X
11 and X
22 according to (125) and (126).
Step  8. Compute X according to (9), and output X.Example 41.
LetA
,
B
,
C
,
D
,
J be as given in Example 38.
It is not difficult to prove that the conditions in (78) and (79) do not hold. So, according to Algorithm 40, the least squares Hermitian generalized anti-Hamiltonian solution to Problem 7 can be written as
(136)
X
=
(
0.3671
0.1579
-
0.1804
i
0.1822
+
0.1400
i
0.0179
-
0.0012
i
0.1579
+
0.1804
i
0.2324
0.0179
+
0.1149
i
-
0.0662
-
0.1400
i
0.1822
-
0.1400
i
0.0179
-
0.1149
i
0.2143
0.1221
-
0.0849
i
0.0179
+
0.0012
i
-
0.0662
+
0.1400
i
0.1221
+
0.0849
i
0.5764
)
,
min
⁡
X
∈
H
A
H
ℂ
2
k
×
2
k
∥
X
-
X
*
∥
=
0.0000
,
min
⁡
X
∈
H
A
H
ℂ
2
k
×
2
k
∥
X
*
+
J
X
J
∥
=
0.8309
.Remark 42.
(1) There exists a unique least squares Hermitian generalized anti-Hamiltonian solution to Problem7 if and only if both A
¯ and C
¯ in Theorem 33 have full column ranks. Example 41 is just the case.
(2) Similarly, the algorithm about computing the least squares anti-Hermitian generalized Hamiltonian solution to Problem7 can be obtained. We also omit it here.


## 7. Conclusions

In the previous sections, using the decomposition of the (anti-)Hermitian generalized (anti-)Hamiltonian matrices, the necessary and sufficient conditions for the existence of and the expression for the solution to Problem5 have been firstly derived, respectively. Then the solutions to Problems 6 and 7 have been individually given. Finally, algorithms have been given to compute the least squares Hermitian generalized Hamiltonian solution and the least squares Hermitian generalized anti-Hamiltonian solution to Problem 7, and the corresponding examples have also been presented to show that the algorithms are reasonable.



---

*Source: 539215-2014-03-25.xml*