1. Introduction
An
n
×
n
matrix
A
is called an arrowhead matrix if it has the following form:
(1)
A
=
[
a
1
b
1
b
2
⋯
b
n
-
1
c
1
a
2
0
⋯
0
c
2
0
a
3
⋯
0
⋮
⋮
⋮
⋱
⋮
c
n
-
1
0
0
⋯
a
n
]
.
If
b
i
=
c
i
,
i
=
1
,
…
,
n
-
1
, then
A
is said to be a symmetric arrowhead matrix. We denote all real-valued symmetric
n
×
n
arrowhead matrices by
S
A
R
n
×
n
. Such matrices arise in the description of radiationless transitions in isolated molecules [1], oscillators vibrationally coupled with a Fermi liquid [2] and quantum optics [3], and so forth. Numerically efficient algorithms for computing eigenvalues and eigenvectors of arrowhead matrices were discussed in [4–8]. The inverse problem of constructing the symmetric arrowhead matrix from spectral data has been investigated by Xu [9], Peng et al. [10], and Borges et al. [11]. In this paper, we will further consider the least-squares solutions of the matrix equations for symmetric arrowhead matrices and associated approximation problems, which can be described as follows.
Problem 1.
Given
A
∈
R
m
×
n
,
B
∈
R
n
×
p
,
C
∈
R
m
×
q
,
D
∈
R
q
×
p
, and
H
∈
R
m
×
p
, find nontrivial real-valued symmetric arrowhead matrices
X
∈
R
n
×
n
and
Y
∈
R
q
×
q
such that
(2)
∥
A
X
B
+
C
Y
D
-
H
∥
=
min
.
Problem 2.
Given real-valued symmetric arrowhead matrices
X
~
∈
R
n
×
n
,
Y
~
∈
R
q
×
q
, find
(
X
^
,
Y
^
)
∈
S
1
such that
(3)
∥
X
^
-
X
~
∥
2
+
∥
Y
^
-
Y
~
∥
2
=
min
(
X
,
Y
)
∈
S
1
(
∥
X
-
X
~
∥
2
+
∥
Y
-
Y
~
∥
2
)
,
where
S
1
is the solution set of Problem 1.
Problem 3.
Given
A
,
C
∈
R
m
×
n
,
B
,
D
∈
R
n
×
p
, and
H
∈
R
m
×
p
, find nontrivial real-valued symmetric arrowhead matrix
X
such that
(4)
∥
A
X
B
+
C
X
D
-
H
∥
=
min
.
Problem 4.
Given a real-valued symmetric arrowhead matrix
X
~
∈
R
n
×
n
, find
X
^
∈
S
3
such that
(5)
∥
X
^
-
X
~
∥
=
min
X
∈
S
3
∥
X
-
X
~
∥
,
where
S
3
is the solution set of Problem 3.
Recently, Li et al. [12] considered the least-squares solutions of the matrix equation
A
X
B
+
C
Y
D
=
E
for symmetric arrowhead matrices. By using Moore-Penrose inverses and the Kronecker product, the minimum-norm and least-squares solution to the matrix equation for symmetric arrowhead matrices was provided. However, we can easily see that the method used in [12] involves complicated computations for Moore-Penrose generalized inverses of partitioned matrices, and the expression of the minimum-norm and least-squares solution was not explicit. Compared with the approach proposed in [12], the method in this paper is more concise and easy to perform.
The paper is organized as follows. In Section 2, using the Kronecker product and stretching function
vec
(
·
)
of matrices, we give an explicit representation of the solution set
S
1
of Problem 1. Furthermore, we show that there exists a unique solution in Problem 2 and present the expression of the unique solution
(
X
^
,
Y
^
)
of Problem 2. In Section 3, we provide an explicit representation of the solution set
S
3
of Problem 3 and present the expression of the unique solution
X
^
of Problem 4. In Section 4, a numerical algorithm to acquire the optimal approximation solution for Problem 2 under the Frobenius norm sense is described and a numerical example is provided. Some concluding remarks are given in Section 5.
Throughout this paper, we denote the real
m
×
n
matrix space by
R
m
×
n
and the transpose and the Moore-Penrose generalized inverse of a real matrix
A
by
A
⊤
and
A
+
, respectively.
I
n
represents the identity matrix of size
n
. For
A
,
B
∈
R
m
×
n
, an inner product in
R
m
×
n
is defined by
(
A
,
B
)
=
trace
(
B
⊤
A
)
; then
R
m
×
n
is a Hilbert space. The matrix norm
∥
·
∥
induced by the inner product is the Frobenius norm. Given two matrices
A
=
[
a
i
j
]
∈
R
m
×
n
and
B
=
[
b
i
j
]
∈
R
p
×
q
, the Kronecker product of
A
and
B
is defined by
A
⊗
B
=
[
a
i
j
B
]
∈
R
m
p
×
n
q
. Also, for an
m
×
n
matrix
A
=
[
a
1
,
a
2
,
…
,
a
n
]
, where
a
i
,
i
=
1
,
…
,
n
, is the
i
th column vector of
A
, the stretching function
vec
(
A
)
is defined by
vec
(
A
)
=
[
a
1
⊤
,
a
2
⊤
,
…
,
a
n
⊤
]
⊤
.
2. The Solutions of Problems 1 and 2
To begin with, we introduce two lemmas.
Lemma 5 (see [13]).
If
L
∈
R
m
×
q
,
b
∈
R
m
, then the general solution of
∥
L
y
-
b
∥
=
min
can be expressed as
y
=
L
+
b
+
(
I
q
-
L
+
L
)
z
, where
z
∈
R
q
is an arbitrary vector.
Lemma 6 (see [14]).
Let
D
∈
R
m
×
n
,
H
∈
R
n
×
l
,
J
∈
R
l
×
s
. Then
(6)
vec
(
D
H
J
)
=
(
J
⊤
⊗
D
)
vec
(
H
)
.
Let
d
1
=
2
n
-
1
and
d
2
=
2
q
-
1
. It is easily seen that
dim
(
S
A
R
n
×
n
)
=
d
1
and
dim
(
S
A
R
q
×
q
)
=
d
2
. Define
(7)
Z
i
j
=
{
2
2
(
e
i
(
n
)
(
e
j
(
n
)
)
⊤
+
e
j
(
n
)
(
e
i
(
n
)
)
⊤
)
,
i
=
1
;
j
=
2
,
…
,
n
,
e
i
(
n
)
(
e
i
(
n
)
)
⊤
,
i
=
j
=
1
,
…
,
n
,
(8)
W
k
l
=
{
2
2
(
e
k
(
q
)
(
e
l
(
q
)
)
⊤
+
e
l
(
p
)
(
e
k
(
q
)
)
⊤
)
,
k
=
1
;
l
=
2
,
…
,
q
,
e
k
(
q
)
(
e
k
(
q
)
)
⊤
,
k
=
l
=
1
,
…
,
q
,
where
e
i
(
n
)
is the
i
th column vector of the identity matrix
I
n
. It is easy to verify that
{
Z
i
j
}
and
{
W
k
l
}
form orthonormal bases of the subspaces
S
A
R
n
×
n
and
S
A
R
q
×
q
, respectively. That is,
(9)
(
Z
i
j
,
Z
k
l
)
=
{
0
,
i
≠
k
or
j
≠
l
,
1
,
i
=
k
,
j
=
l
,
(
W
i
j
,
W
k
l
)
=
{
0
,
i
≠
k
or
j
≠
l
,
1
,
i
=
k
,
j
=
l
.
Now, if
X
∈
S
A
R
n
×
n
and
Y
∈
S
A
R
q
×
q
, then
X
and
Y
can be expressed as
(10)
X
=
∑
i
,
j
α
i
j
Z
i
j
,
Y
=
∑
k
,
l
β
k
l
W
k
l
,
where the real numbers
α
i
j
,
i
=
1
,
j
=
2
,
…
,
n
;
i
=
j
=
1
,
…
,
n
, and
β
k
l
,
k
=
1
,
l
=
2
,
…
,
q
;
k
=
l
=
1
,
…
,
q
, are yet to be determined.
It follows from (10) that the relation of (2) can be equivalently written as
(11)
∥
∑
i
,
j
α
i
j
A
Z
i
j
B
+
∑
k
,
l
β
k
l
C
W
k
l
D
-
H
∥
=
min
.
When setting
(12)
α
=
[
α
11
,
…
,
α
n
,
n
,
α
12
,
…
,
α
1
,
n
]
⊤
,
β
=
[
β
11
,
…
,
β
q
,
q
,
β
12
,
…
,
β
1
,
q
]
⊤
,
(13)
G
=
[
vec
(
Z
11
)
,
…
,
vec
(
Z
n
,
n
)
,
vec
(
Z
12
)
,
…
,
vec
(
Z
1
,
n
)
]
∈
R
n
2
×
d
1
(14)
L
=
[
vec
(
W
11
)
,
…
,
vec
(
W
q
,
q
)
,
vec
(
W
12
)
,
…
,
vec
(
W
1
,
q
)
]
∈
R
q
2
×
d
2
,
(15)
M
=
(
B
⊤
⊗
A
)
G
,
N
=
(
D
⊤
⊗
C
)
L
,
h
=
vec
(
H
)
.
By Lemma 6, we see that the relation of (11) is equivalent to
(16)
∥
M
α
+
N
β
-
h
∥
=
min
.
We note that
(17)
∥
M
α
+
N
β
-
h
∥
2
=
∥
M
[
α
+
M
+
(
N
β
-
h
)
]
+
E
M
(
N
β
-
h
)
∥
2
=
∥
M
[
α
+
M
+
(
N
β
-
h
)
]
∥
2
+
∥
E
M
(
N
β
-
h
)
∥
2
=
∥
M
[
α
+
M
+
(
N
β
-
h
)
]
∥
2
+
∥
E
M
N
[
β
-
(
E
M
N
)
+
E
M
h
]
h
h
h
h
h
h
h
-
(
I
m
p
-
E
M
N
(
E
M
N
)
+
)
E
M
h
∥
2
=
∥
M
[
α
+
M
+
(
N
β
-
h
)
]
∥
2
+
∥
E
M
N
[
β
-
(
E
M
N
)
+
E
M
h
]
∥
2
+
∥
(
I
m
p
-
E
M
N
(
E
M
N
)
+
)
E
M
h
∥
2
,
where
E
M
=
I
m
p
-
M
M
+
. It follows from Lemma 5 and (17) that
∥
M
α
+
N
β
-
h
∥
=
min
if and only if
(18)
α
=
-
M
+
N
β
+
M
+
h
+
F
M
v
,
(19)
β
=
(
E
M
N
)
+
E
M
h
+
W
u
,
where
F
M
=
I
d
1
-
M
+
M
,
W
=
I
d
2
-
(
E
M
N
)
+
E
M
N
, and
u
∈
R
d
2
,
v
∈
R
d
1
are arbitrary vectors.
Substituting (19) into (18), we obtain
(20)
α
=
α
~
-
M
+
N
W
u
+
F
M
v
,
where
α
~
=
M
+
h
-
M
+
N
(
E
M
N
)
+
E
M
h
.
In summary of the above discussion, we have proved the following result.
Theorem 7.
Suppose that
A
∈
R
m
×
n
,
B
∈
R
n
×
p
,
C
∈
R
m
×
q
,
D
∈
R
q
×
p
, and
H
∈
R
m
×
p
. Let
{
Z
i
j
}
,
{
W
k
l
}
,
G
,
L
,
M
,
N
,
h
be given as in (7), (8), (13), (14), and (15), respectively. Write
d
1
=
2
n
-
1
,
d
2
=
2
q
-
1
,
E
M
=
I
m
p
-
M
M
+
,
F
M
=
I
d
1
-
M
+
M
,
W
=
I
d
2
-
(
E
M
N
)
+
E
M
N
, and
α
~
=
M
+
h
-
M
+
N
(
E
M
N
)
+
E
M
h
. Then the solution set
S
1
of Problem 1 can be expressed as
(21)
S
1
=
{
=
K
1
(
α
⊗
I
n
)
,
Y
=
K
2
(
β
⊗
I
q
)
(
X
,
Y
)
∈
S
A
R
n
×
n
×
S
A
R
q
×
q
∣
h
X
=
K
1
(
α
⊗
I
n
)
,
Y
=
K
2
(
β
⊗
I
q
)
}
,
where
(22)
K
1
=
[
Z
11
,
…
,
Z
n
,
n
,
Z
12
,
…
,
Z
1
,
n
]
∈
R
n
×
n
d
1
,
(23)
K
2
=
[
W
11
,
…
,
W
q
,
q
,
W
12
,
…
,
W
1
,
q
]
∈
R
q
×
q
d
2
,
α
,
β
are, respectively, given by (20) and (19) with
u
∈
R
d
2
,
v
∈
R
d
1
being arbitrary vectors.
From (17), we can easily obtain the following corollary.
Corollary 8.
Under the same assumptions as in Theorem 7, the matrix equation
(24)
A
X
B
+
C
Y
D
=
H
has a solution if and only if
(25)
E
M
N
(
E
M
N
)
+
E
M
N
=
E
M
h
.
In this case, the solution set
S
1
of (24) is given by (21).
It follows from Theorem 7 that the solution set
S
1
is always nonempty. It is easy to verify that
S
1
is a closed convex subset of
S
A
R
n
×
n
×
S
A
R
q
×
q
. From the best approximation theorem [15], we know there exists a unique solution
(
X
^
,
Y
^
)
in
S
1
such that (3) holds.
We now focus our attention on seeking the unique solution
(
X
^
,
Y
^
)
in
S
1
. For the real-valued symmetric arrowhead matrices
X
~
and
Y
~
, it is easily seen that
X
~
,
Y
~
can be expressed as the linear combinations of the orthonormal bases
{
Z
i
j
}
and
{
W
i
j
}
; that is,
(26)
X
~
=
∑
i
,
j
γ
i
j
Z
i
j
,
Y
~
=
∑
k
,
l
δ
k
l
W
k
l
,
where
γ
i
j
,
i
=
1
,
j
=
2
,
…
,
n
;
i
=
j
=
1
,
…
,
n
, and
δ
k
l
,
k
=
1
,
l
=
2
,
…
,
q
;
k
=
l
=
1
,
…
,
q
, are uniquely determined by the elements of
X
~
and
Y
~
. Let
(27)
γ
=
[
γ
11
,
…
,
γ
n
,
n
,
γ
12
,
…
,
γ
1
,
n
]
⊤
,
δ
=
[
δ
11
,
…
,
δ
q
,
q
,
δ
12
,
…
,
δ
1
,
q
]
⊤
.
Then, for any pair of matrices
(
X
,
Y
)
∈
S
1
in (21), by the relations of (9) and (26), we see that
(28)
f
=
∥
X
-
X
~
∥
2
+
∥
Y
-
Y
~
∥
2
=
∥
∑
i
,
j
(
α
i
j
-
γ
i
j
)
Z
i
j
∥
2
+
∥
∑
k
,
l
(
β
k
l
-
δ
k
l
)
W
k
l
∥
2
=
(
∑
i
,
j
(
α
i
j
-
γ
i
j
)
Z
i
j
,
∑
i
,
j
(
α
i
j
-
γ
i
j
)
Z
i
j
)
+
(
∑
k
,
l
(
β
k
l
-
δ
k
l
)
W
k
l
,
∑
k
,
l
(
β
k
l
-
δ
k
l
)
W
k
l
)
=
∑
i
,
j
(
α
i
j
-
γ
i
j
)
(
Z
i
j
,
∑
i
,
j
(
α
i
j
-
γ
i
j
)
Z
i
j
)
+
∑
k
,
l
(
β
k
l
-
δ
k
l
)
(
W
k
l
,
∑
k
,
l
(
β
k
l
-
δ
k
l
)
W
k
l
)
=
∑
i
,
j
(
α
i
j
-
γ
i
j
)
2
+
∑
k
,
l
(
β
k
l
-
δ
k
l
)
2
=
∥
α
-
γ
∥
2
+
∥
β
-
δ
∥
2
.
Substituting (19) and (20) into the function of
f
, we have
(29)
f
=
∥
α
~
-
M
+
N
W
u
+
F
M
v
-
γ
∥
2
+
∥
(
E
M
N
)
+
E
M
h
+
W
u
-
δ
∥
2
=
u
⊤
W
N
⊤
(
M
M
⊤
)
+
N
W
u
+
2
(
γ
-
α
~
)
⊤
M
+
N
W
u
-
2
(
γ
-
α
~
)
⊤
F
M
v
+
v
⊤
F
M
v
+
(
γ
-
α
~
)
⊤
(
γ
-
α
~
)
+
u
⊤
W
u
-
2
(
δ
-
(
E
M
N
)
+
E
M
h
)
⊤
W
u
+
(
δ
-
(
E
M
N
)
+
E
M
h
)
⊤
(
δ
-
(
E
M
N
)
+
E
M
h
)
.
Therefore,
(30)
∂
f
∂
u
=
2
W
N
⊤
(
M
M
⊤
)
+
N
W
u
+
2
W
N
⊤
(
M
+
)
⊤
(
γ
-
α
~
)
+
2
W
u
-
2
W
(
δ
-
(
E
M
N
)
+
E
M
h
)
,
∂
f
∂
v
=
2
F
M
v
-
2
F
M
(
γ
-
α
~
)
.
Clearly,
∥
X
-
X
~
∥
2
+
∥
Y
-
Y
~
∥
2
=
min
if and only if
(31)
∂
f
∂
u
=
0
,
∂
f
∂
v
=
0
which yields
(32)
W
u
=
(
I
d
2
+
W
N
⊤
(
M
M
⊤
)
+
N
W
)
-
1
×
W
(
δ
-
(
E
M
N
)
+
E
M
h
-
N
⊤
(
M
+
)
⊤
(
γ
-
α
~
)
)
,
F
M
v
=
F
M
(
γ
-
α
~
)
.
Upon substituting (32) into (19) and (20), we obtain
(33)
α
^
=
-
M
+
N
W
(
I
d
2
+
W
N
⊤
(
M
M
⊤
)
+
N
W
)
-
1
×
W
(
δ
-
(
E
M
N
)
+
E
M
h
-
N
⊤
(
M
+
)
⊤
(
γ
-
α
~
)
)
+
α
~
+
F
M
γ
,
(34)
β
^
=
(
E
M
N
)
+
E
M
h
+
(
I
d
2
+
W
N
⊤
(
M
M
⊤
)
+
N
W
)
-
1
×
W
(
δ
-
(
E
M
N
)
+
E
M
h
-
N
⊤
(
M
+
)
⊤
(
γ
-
α
~
)
)
.
By now, we have proved the following result.
Theorem 9.
Let the real-valued symmetric arrowhead matrices
X
~
and
Y
~
be given. Then Problem 2 has a unique solution and the unique solution of Problem 2 can be expressed as
(35)
X
^
=
K
1
(
α
^
⊗
I
n
)
,
Y
^
=
K
2
(
β
^
⊗
I
q
)
,
where
α
^
,
β
^
are given by (33) and (34), respectively.
3. The Solutions of Problems 3 and 4
It follows from (10) that the minimization problem of (4) can be equivalently written as
(36)
∥
∑
i
,
j
α
i
j
A
Z
i
j
B
+
∑
i
,
j
α
i
j
C
Z
i
j
D
-
H
∥
=
min
.
Using Lemma 6, we see that the relation of (36) is equivalent to
(37)
∥
M
α
+
Q
α
-
h
∥
=
min
,
where
Q
=
(
D
⊤
⊗
C
)
G
. It follows from Lemma 5 that the general solution of
∥
M
α
+
Q
α
-
h
∥
=
min
with respect to
α
can be expressed as
(38)
α
=
U
+
h
+
(
I
d
1
-
U
+
U
)
z
,
where
U
=
M
+
Q
and
z
∈
R
d
1
is an arbitrary vector.
To summarize, we have obtained the following result.
Theorem 10.
Suppose that
A
∈
R
m
×
n
,
B
∈
R
n
×
p
,
C
∈
R
m
×
q
,
D
∈
R
q
×
p
, and
H
∈
R
m
×
p
. Let
{
Z
i
j
}
,
G
,
M
,
h
be given as in (7), (13), and (15), respectively. Write
Q
=
(
D
⊤
⊗
C
)
G
,
d
1
=
2
n
-
1
, and
U
=
M
+
Q
. Then the solution set
S
3
of Problem 3 can be expressed as
(39)
S
3
=
{
X
∈
S
A
R
n
×
n
∣
X
=
K
1
(
α
⊗
I
n
)
}
,
where
K
1
and
α
are given by (22) and (38) with
z
∈
R
d
1
being arbitrary vectors.
Similarly, for the real-valued symmetric arrowhead matrix
X
~
, it is easily seen that
X
~
can be expressed as the linear combination of the orthonormal basis
{
Z
i
j
}
; that is,
X
~
=
∑
i
,
j
γ
i
j
Z
i
j
, where
γ
i
j
,
i
=
1
,
j
=
2
,
…
,
n
;
i
=
j
=
1
,
…
,
n
, are uniquely determined by the elements of
X
~
. Then, for any matrix
X
∈
S
3
in (39), by the relation of (9), we have
(40)
∥
X
-
X
~
∥
2
=
∥
∑
i
,
j
(
α
i
j
-
γ
i
j
)
Z
i
j
∥
2
=
(
∑
i
,
j
(
α
i
j
-
δ
i
j
)
Z
i
j
,
∑
i
,
j
(
α
i
j
-
γ
i
j
)
Z
i
j
)
=
∑
i
,
j
(
α
i
j
-
γ
i
j
)
(
Z
i
j
,
∑
i
,
j
(
α
i
j
-
γ
i
j
)
Z
i
j
)
=
∑
i
,
j
(
α
i
j
-
γ
i
j
)
2
=
∥
α
-
γ
∥
2
=
∥
J
z
-
(
γ
-
U
+
h
)
∥
2
,
where
J
=
I
d
1
-
U
+
U
,
γ
=
[
γ
11
,
…
,
γ
n
,
n
,
γ
12
,
…
,
γ
1
,
n
]
⊤
.
In order to solve Problem 4, we need the following lemma [16].
Lemma 11.
Suppose that
P
∈
R
q
×
m
,
Δ
∈
R
q
×
q
, and
Γ
∈
R
m
×
m
where
Δ
2
=
Δ
=
Δ
⊤
and
Γ
2
=
Γ
=
Γ
⊤
.Then
(41)
∥
P
-
Δ
D
Γ
∥
=
min
E
∈
R
q
×
m
∥
P
-
Δ
E
Γ
∥
if and only if
Δ
(
P
-
D
)
Γ
=
0
, in which case,
∥
P
-
Δ
D
Γ
∥
=
∥
P
-
Δ
P
Γ
∥
.
It follows from Lemma 11 and
J
2
=
J
=
J
⊤
that
(42)
∥
X
-
X
~
∥
=
∥
J
z
-
(
γ
-
U
+
h
)
∥
=
min
if and only if
J
(
γ
-
U
+
h
-
z
)
=
0
; that is,
(43)
J
z
=
J
(
γ
-
U
+
h
)
.
Substituting (43) into (38), we obtain
(44)
α
^
=
U
+
h
+
J
(
γ
-
U
+
h
)
.
By now, we have proved the following result.
Theorem 12.
Let the real-valued symmetric arrowhead matrix
X
~
be given. Then Problem 4 has a unique solution and the unique solution of Problem 4 can be expressed as
(45)
X
^
=
K
1
(
α
^
⊗
I
n
)
,
where
J
=
I
d
1
-
U
+
U
,
γ
=
[
γ
11
,
…
,
γ
n
,
n
,
γ
12
,
…
,
γ
1
,
n
]
⊤
, and
α
^
is given by (44).