# The

Authors: Yu-Ping Zhang; Chang-Zhou Dong
Journal: Journal of Applied Mathematics
 (2014)

---

## Abstract

We present the
*congruence class of the least-square and the minimum norm least-square solutions to the system of complex matrix equation A
X
=
C
,
X
B
=
D by generalized singular value decomposition and canonical correlation decomposition.


---

## Body

## 1. Introduction

Throughout we denote the complexm
×
n matrix space by C
m
×
n. The symbols I, A
*, and ∥
A
∥ stand for the identity matrix with the appropriate size, the conjugate transpose, and the Frobenius norm of A
∈
C
m
×
n, respectively. Recall that matrices X, Y
∈
C
n
×
n are in the same 
*congruence class if there is a nonsingular P
∈
C
n
×
n such that X
=
P
*
Y
P [1].Investigating the classical system of matrix equations(1)
A
X
=
C
,
X
B
=
D
has attracted many people’s attention and many results have been obtained about system (1) with various constraints, such as Hermitian, positive definite, positive semidefinite, reflexive, and generalized reflexive solutions (see [2–10]). Studying the least-square solutions of the system of matrix equations (1) is also a very active research topic (see [11–16]). It is well known that Hermitian, positive definite and positive semidefinite matrices are the special case of 
*congruence. Therefore investigating the 
*congruence class of a solution of the matrix equation (1) is very meaningful.In 2005, Horn et al. [1] studied the possible 
*congruence class of a square solution when linear matrix equation A
X
=
B is solvable. In 2009, Zheng et al. [17] describe 
*congruence class of least-square and minimum norm least-square solutions of the equation A
X
=
B when it is not solvable and discuss a 
*congruence class of the solutions of the system (1) when it is solvable. To our knowledge, so far there has been little investigation of 
*congruence class of the least-square and minimum norm least-square solutions to (1) when it is not solvable.Motivated by the work mentioned above, we investigate the
*congruence class of the least-square and the minimum norm least-square solutions to the system of complex matrix equation (1) by generalized singular value decomposition (GSVD) and canonical correlation decomposition (CCD).


## 2. The
*Congruence Class of the Solutions to (1)

Lemma 1 (see [4]).
LetA
∈
C
m
×
n and B
∈
C
n
×
p. Then the GSVD of A and B
* can be expressed as
(2)
A
=
U
Σ
A
P
,
B
*
=
V
Σ
B
P
,
where  U
∈
C
m
×
m and V
∈
C
p
×
p are unitary matrices, P
∈
C
n
×
n is nonsingular matrix,
(3)
Σ
A
∈
C
m
×
n
,
Σ
B
∈
C
p
×
n
,
r
=
rank
⁡
(
A
B
*
)
,
Σ
A
=
(
I
A
·
S
A
·
0
O
A
·
)
t
s
r
-
s
-
t
n
-
r
,
Σ
B
=
(
O
B
·
S
B
·
0
I
B
·
)
t
s
r
-
s
-
t
n
-
r
,
I
A and I
B are identity matrices, O
A and O
B are zero matrices, and
(4)
S
A
=
diag
⁡
(
α
1
,
…
,
α
s
)
,
S
B
=
diag
⁡
(
β
1
,
…
,
β
s
)
with 1
>
α
1
≥
⋯
≥
α
s
>
0, 0
<
β
1
≤
⋯
≤
β
s
<
1, and α
i
2
+
β
i
2
=
1
(
i
=
1
,
…
,
s
).For convenience, in the following theorem we denote(5)
P
X
P
*
=
(
X
11
X
12
X
13
X
14
X
21
X
22
X
23
X
24
X
31
X
32
X
33
X
34
X
41
X
42
X
43
X
44
)
,
t
s
r
-
s
-
t
n
-
r
(6)
U
*
C
P
*
=
(
C
11
C
12
C
13
C
14
C
21
C
22
C
23
C
24
C
31
C
32
C
33
C
34
)
,
t
s
r
-
s
-
t
n
-
r
P
D
V
=
(
D
11
D
12
D
13
D
21
D
22
D
23
D
31
D
32
D
33
D
41
D
42
D
43
)
p
-
r
+
t
s
r
-
s
-
t
.Theorem 2.
LetA, C
∈
C
m
×
n, B, D
∈
C
n
×
p, and the GSVD of A and B
* be expressed as (2), and then one has the following. (a)
The system of matrix equation (1) has a solution in C
n
×
n if and only if
(7)
C
3
i
=
0
,
D
i
1
=
0
,
(
i
=
1,2
,
3,4
)
,
C
12
=
D
12
S
B
-
1
,
C
13
=
D
13
,
S
A
-
1
C
22
=
D
22
S
B
-
1
,
S
A
-
1
C
23
=
D
23
.
(b)
In that case, the general solutions of (1) are
(8)
X
=
P
-
1
(
C
11
C
12
C
13
C
14
S
A
-
1
C
21
S
A
-
1
C
22
D
23
S
A
-
1
C
24
X
31
D
32
S
B
-
1
D
33
X
34
X
41
D
42
S
B
-
1
D
43
X
44
)
(
P
-
1
)
*
,where X
31, X
41, X
34, and X
44 are arbitrary. (c)
For arbitraryX
31, X
41, X
34, and X
44, there exists a solution in C
n
×
n of (1) which is *congruent to
(9)
Y
=
(
C
11
C
12
C
13
C
14
S
A
-
1
C
21
S
A
-
1
C
22
D
23
S
A
-
1
C
24
X
31
D
32
S
B
-
1
D
33
X
34
X
41
D
42
S
B
-
1
D
43
X
44
)
.
(d)
There exists a minimum norm solution inC
n
×
n of (1) which is 
*congruent to
(10)
Y
=
(
C
11
C
12
C
13
C
14
S
A
-
1
C
21
S
A
-
1
C
22
D
23
S
A
-
1
C
24
0
D
32
S
B
-
1
D
33
0
0
D
42
S
B
-
1
D
43
0
)
.Proof.
Using the GSVD ofA and B
* given by (2), we get
(11)
A
X
=
C
⟺
U
Σ
A
P
X
=
C
⟺
Σ
A
P
X
P
*
=
U
*
C
P
*
,
X
B
=
D
⟺
X
P
*
Σ
B
*
V
*
=
D
⟺
P
X
P
*
Σ
B
*
=
P
D
V
.
By (2) and (5), Σ
A
P
X
P
* and P
X
P
*
Σ
B
* have the following matrix decomposition:
(12)
Σ
A
P
X
P
*
=
(
X
11
X
12
X
13
X
14
S
A
X
21
S
A
X
22
S
A
X
23
S
A
X
24
0
0
0
0
)
,
P
X
P
*
Σ
B
*
=
(
0
X
12
S
B
X
13
0
X
22
S
B
X
23
0
X
32
S
B
X
33
0
X
42
S
B
X
43
)
,
and we have that system (1) is equivalent to
(13)
(
X
11
X
12
X
13
X
14
S
A
X
21
S
A
X
22
S
A
X
23
S
A
X
24
0
0
0
0
)
=
(
C
11
C
12
C
13
C
14
C
21
C
22
C
23
C
24
C
31
C
32
C
33
C
34
)
,
(
0
X
12
S
B
X
13
0
X
22
S
B
X
23
0
X
32
S
B
X
33
0
X
42
S
B
X
43
)
=
(
D
11
D
12
D
13
D
21
D
22
D
23
D
31
D
32
D
33
D
41
D
42
D
43
)
;
obviously, the system of matrix equation (1) has a solution in C
n
×
n if and only if
(14)
C
3
i
=
0
,
D
i
1
=
0
,
S
A
X
2
i
=
C
2
i
,
X
i
2
S
B
=
D
i
2
,
X
1
i
=
C
1
i
,
X
i
3
=
D
i
3
,
vvvvvvvvvvvvvvvvvvvv
(
i
=
1,2
,
3,4
)
.
Therefore, (1) has a solution in C
n
×
n if and only if (7) holds, and a general form of the solutions can be expressed as (8); for arbitrary X
31, X
41, X
34, and X
44, there exists a solution in C
n
×
n of (1) which is 
*congruent to (9), and the part (d) follows from the definition of Frobenius norm.Remark 3.
In 2009, Zheng et al. [17] discuss a 
*congruence class of the solutions of the system (1) when it is solvable. Our result in Theorem 2 is different with the result mentioned above.


## 3. The
*Congruence Class of the Least-Square Solutions to (1)

Lemma 4 (see [18]).
Let the CCD of matrix pair[
A
,
C
] with A
∈
C
m
×
n, C
∈
C
m
×
k,  rank
A
=
g, and  rank
C
=
h be given as
(15)
A
=
U
(
Σ
A
,
0
)
E
A
-
1
,
C
=
U
(
Σ
C
,
0
)
E
C
-
1
,
where U is a unitary matrix and
(16)
Σ
A
=
(
I
i
Λ
j
0
¯
-
0
_
Δ
j
I
t
)
,
Σ
C
=
(
I
h
0
¯
)
are nonsingular matrices with the same row partitioning, and g
=
i
+
j
+
t,
(17)
Λ
j
=
diag
⁡
(
λ
i
+
1
,
…
,
λ
i
+
j
)
,
1
>
λ
i
+
1
≥
⋯
≥
λ
i
+
j
>
0
,
Δ
j
=
diag
⁡
(
Δ
i
+
1
,
…
,
Δ
i
+
j
)
,
0
>
Δ
i
+
1
≥
⋯
≥
Δ
i
+
j
>
1
,
Λ
j
2
+
Δ
j
2
=
I
j
,
U
=
(
u
1
u
2
u
3
u
4
u
5
u
6
)
i
j
h
-
i
-
j
m
-
h
-
j
-
t
j
t
.Lemma 5 (see [18]).
GivenE, F
∈
C
m
×
n, then there exists a unique matrix S
¯
∈
C
m
×
n such that
(18)
∥
S
-
E
∥
2
+
∥
S
-
F
∥
2
=
min
⁡
,
and S
¯ can be expressed as
(19)
S
¯
=
E
+
F
.Lemma 6 (see [10]).
GivenE, F
∈
C
m
×
n, Ω
1
=
diag
⁡
(
a
1
,
…
,
a
m
), Ω
2
=
diag
⁡
(
b
1
,
…
,
b
n
), a
i
>
0
(
i
=
1
,
…
,
m
), and b
j
>
0
(
j
=
1
,
…
,
n
), then there exists a unique matrix S
¯
∈
C
m
×
n such that
(20)
∥
Ω
1
S
-
E
∥
2
+
∥
S
Ω
2
-
F
∥
2
=
min
⁡
,
and S
¯ can be expressed as
(21)
S
¯
=
Φ
*
(
Ω
1
E
+
F
Ω
2
)
,
where
(22)
Φ
=
(
1
a
i
2
+
b
j
2
)
∈
C
m
×
n
.Using Lemmas5 and 6, we can easily obtain the following.Lemma 7.
GivenE, F, G
∈
C
m
×
n, Ω
1
=
diag
⁡
(
a
1
,
…
,
a
m
), Ω
2
=
diag
⁡
(
b
1
,
…
,
b
m
), I
n
=
diag
⁡
(
i
1
,
…
,
i
n
), a
i
>
0
(
i
=
1
,
…
,
m
), b
j
>
0
(
j
=
1
,
…
,
m
), and i
k
=
1
(
k
=
1
,
…
,
n
), then there exist unique matrices S and W such that
(23)
∥
Ω
1
S
+
Ω
2
W
-
E
∥
2
+
∥
S
-
F
∥
2
+
∥
W
-
G
∥
2
=
min
⁡
,
and S and W can be expressed as
(24)
S
=
F
,
W
=
Φ
*
(
Ω
2
(
Ω
1
F
-
E
)
+
G
)
,
where
(25)
Φ
=
(
1
b
j
2
+
i
k
2
)
∈
C
m
×
n
.Lemma 8.
GivenE, F
∈
C
m
×
n, Ω
1
=
diag
⁡
(
a
1
,
…
,
a
m
), Ω
2
=
diag
⁡
(
b
1
,
…
,
b
n
), a
i
>
0
(
i
=
1
,
…
,
m
), and b
j
>
0
(
j
=
1
,
…
,
n
), then there exist unique matrices S and W such that
(26)
∥
Ω
1
S
+
Ω
2
W
-
E
∥
2
=
min
⁡
,
and S and W can be expressed as
(27)
S
=
0
,
W
=
Ω
2
-
1
E
.LetA, C
∈
C
m
×
n, B, D
∈
C
n
×
l, and rank
A
=
p
≥
rank
B
=
q. According to Lemma 4, there exist a unitary matrix U
∈
C
n
×
n and nonsingular matrices R
A
∈
C
m
×
m and R
B
∈
C
l
×
l, such that the CCD of matrix pair [
A
*
,
B
] is given as
(28)
A
*
=
U
(
Σ
A
,
0
)
R
A
-
1
,
B
=
U
(
Σ
B
,
0
)
R
B
-
1
,
where Σ
A
∈
C
n
×
p, Σ
B
∈
C
n
×
q,
(29)
Σ
A
=
(
I
r
0
0
0
G
s
0
0
0
0
0
0
0
0
S
s
0
0
0
I
t
)
,
Σ
B
=
(
I
r
0
0
0
I
s
0
0
0
I
q
-
r
-
s
0
0
0
0
0
0
0
0
0
)
,
where p
=
r
+
s
+
t,
(30)
G
s
=
diag
⁡
(
g
r
+
1
,
…
,
g
r
+
s
)
,
1
>
g
r
+
1
≥
⋯
≥
g
r
+
s
>
0
,
S
s
=
diag
⁡
(
w
r
+
1
,
…
,
w
r
+
s
)
,
0
>
w
r
+
1
≥
⋯
≥
w
r
+
s
>
1
,
G
s
2
+
S
s
2
=
I
s
,
U
=
(
u
1
u
2
u
3
u
4
u
5
u
6
)
r
s
q
-
r
-
s
n
-
q
-
s
-
t
s
t
.
Without loss of generality, let p
=
q, and then we have the following results.Theorem 9.
LetA, C
∈
C
m
×
n, B, D
∈
C
n
×
l, and the CCD of matrix pair [
A
*
,
B
] be expressed as (28), and then one has the following. (a)
The least-square solutions to the system (1) are(31)
X
=
U
(
C
11
+
D
11
C
12
+
D
12
C
13
+
D
13
C
14
C
15
C
16
D
21
D
22
D
23
0
0
0
D
31
D
32
D
33
X
34
X
35
X
36
D
41
D
42
D
43
X
44
X
45
X
46
Y
51
Y
52
Y
53
S
-
1
C
24
S
-
1
C
25
S
-
1
C
26
D
31
+
D
61
D
32
+
D
62
D
33
+
D
63
C
34
C
35
C
36
)
U
*
,where X
34, X
35, X
36, X
44, X
45, and X
46 are arbitrary, Y
5
i
=
Φ
*
(
S
(
G
D
2
i
-
C
2
i
)
+
D
5
i
),  i
=
1,2
,
3,  Φ
=
(
1
/
(
w
r
+
j
2
+
e
k
2
)
)
∈
C
s
×
s, and e
k
=
1,  j
=
1
,
…
,
s,  k
=
1
,
…
,
s.(b)
For arbitraryX
34, X
35, X
36, X
44, X
45, and X
46, there exists a least-square solution in C
n
×
n of (1) which is *congruent to(32)
Y
=
(
C
11
+
D
11
C
12
+
D
12
C
13
+
D
13
C
14
C
15
C
16
D
21
D
22
D
23
0
0
0
D
31
D
32
D
33
X
34
X
35
X
36
D
41
D
42
D
43
X
44
X
45
X
46
Y
51
Y
52
Y
53
S
-
1
C
24
S
-
1
C
25
S
-
1
C
26
D
31
+
D
61
D
32
+
D
62
D
33
+
D
63
C
34
C
35
C
36
)
,where Y
5
i
=
Φ
*
(
S
(
G
D
2
i
-
C
2
i
)
+
D
5
i
),  i
=
1,2
,
3,  Φ
=
(
1
/
(
w
r
+
j
2
+
e
k
2
)
)
∈
C
s
×
s,  and e
k
=
1,  j
=
1
,
…
,
s,  k
=
1
,
…
,
s.(c)
There exists a minimum norm least-square solution inC
n
×
n of (1) which is *congruent to(33)
Y
=
(
C
11
+
D
11
C
12
+
D
12
C
13
+
D
13
C
14
C
15
C
16
D
21
D
22
D
23
0
0
0
D
31
D
32
D
33
0
0
0
D
41
D
42
D
43
0
0
0
Y
51
Y
52
Y
53
S
-
1
C
24
S
-
1
C
25
S
-
1
C
26
D
31
+
D
61
D
32
+
D
62
D
33
+
D
63
C
34
C
35
C
36
)
,where Y
5
i
=
Φ
*
(
S
(
G
D
2
i
-
C
2
i
)
+
D
5
i
),  i
=
1,2
,
3,  Φ
=
(
1
/
(
w
r
+
j
2
+
e
k
2
)
)
∈
C
s
×
s,  and e
k
=
1,  j
=
1
,
…
,
s,  k
=
1
,
…
,
s.Proof.
It follows from (28) that
(34)
A
X
=
C
⟺
(
R
A
-
1
)
*
(
Σ
A
*
0
)
U
*
X
=
C
⟺
(
Σ
A
*
0
)
U
*
X
=
(
R
A
)
*
C
,
X
B
=
D
⟺
X
U
(
Σ
B
,
0
)
R
B
-
1
=
D
⟺
X
U
(
Σ
B
,
0
)
=
D
R
B
.
Then,
(35)
∥
A
X
-
C
∥
2
+
∥
X
B
-
D
∥
2
=
∥
(
Σ
A
*
0
)
U
*
X
-
(
R
A
)
*
C
∥
2
+
∥
X
U
(
Σ
B
,
0
)
-
D
R
B
∥
2
=
∥
(
Σ
A
*
0
)
U
*
X
U
-
(
R
A
)
*
C
U
∥
2
+
∥
U
*
X
U
(
Σ
B
,
0
)
-
U
*
D
R
B
∥
2
.
Assume that
(36)
U
*
X
U
=
(
X
11
X
12
X
13
X
14
X
15
X
16
X
21
X
22
X
23
X
24
X
25
X
26
X
31
X
32
X
33
X
34
X
35
X
36
X
41
X
42
X
43
X
44
X
45
X
46
X
51
X
52
X
53
X
54
X
55
X
56
X
61
X
62
X
63
X
64
X
65
X
66
)
,
(37)
(
R
A
)
*
C
U
=
(
C
11
C
12
C
13
C
14
C
15
C
16
C
21
C
22
C
23
C
24
C
25
C
26
C
31
C
32
C
33
C
34
C
35
C
36
C
41
C
42
C
43
C
44
C
45
C
46
)
,
U
*
D
R
B
=
(
D
11
D
12
D
13
D
14
D
21
D
22
D
23
D
24
D
31
D
32
D
33
D
34
D
41
D
42
D
43
D
44
D
51
D
52
D
53
D
54
D
61
D
62
D
63
D
64
)
,
and then
(38)
∥
(
Σ
A
*
0
)
U
*
X
U
-
(
R
A
)
*
C
U
∥
2
+
∥
U
*
X
U
(
Σ
B
,
0
)
-
U
*
D
R
B
∥
2
=
∥
(
X
11
X
12
⋯
X
16
G
X
21
+
S
X
51
G
X
22
+
S
X
52
⋯
G
X
26
+
S
X
56
X
61
X
62
⋯
X
66
0
0
⋯
0
)
-
(
C
11
C
12
⋯
C
16
C
21
C
22
⋯
C
26
C
31
C
32
⋯
C
36
C
41
C
42
⋯
C
46
)
∥
2
+
∥
(
X
11
X
12
X
13
0
X
21
X
22
X
23
0
⋮
⋮
⋮
⋮
X
61
X
62
X
63
0
)
-
(
D
11
D
12
D
13
D
14
D
21
D
22
D
23
D
24
⋮
⋮
⋮
⋮
D
61
D
62
D
63
D
64
)
∥
2
=
∥
X
11
-
C
11
∥
2
+
∥
X
11
-
D
11
∥
2
+
∥
X
12
-
C
12
∥
2
+
∥
X
12
-
D
12
∥
2
+
∥
X
13
-
C
13
∥
2
+
∥
X
13
-
D
13
∥
2
+
∥
X
61
-
C
31
∥
2
+
∥
X
61
-
D
61
∥
2
+
∥
X
62
-
C
32
∥
2
+
∥
X
62
-
D
62
∥
2
+
∥
X
63
-
C
33
∥
2
+
∥
X
63
-
D
63
∥
2
+
∥
G
X
21
+
S
X
51
-
C
21
∥
2
+
∥
X
21
-
D
21
∥
2
+
∥
X
51
-
D
51
∥
2
+
∥
G
X
22
+
S
X
52
-
C
22
∥
2
+
∥
X
22
-
D
22
∥
2
+
∥
X
52
-
D
52
∥
2
+
∥
G
X
23
+
S
X
53
-
C
23
∥
2
+
∥
X
23
-
D
23
∥
2
+
∥
X
53
-
D
53
∥
2
+
∥
G
X
24
+
S
X
54
-
C
24
∥
2
+
∥
G
X
25
+
S
X
55
-
C
25
∥
2
+
∥
G
X
26
+
S
X
56
-
C
26
∥
2
+
∥
X
14
-
C
14
∥
2
+
∥
X
15
-
C
15
∥
2
+
∥
X
16
-
C
16
∥
2
+
∥
X
64
-
C
34
∥
2
+
∥
X
65
-
C
35
∥
2
+
∥
X
66
-
C
36
∥
2
+
∥
X
31
-
D
31
∥
2
+
∥
X
32
-
D
32
∥
2
+
∥
X
33
-
D
33
∥
2
+
∥
X
41
-
D
41
∥
2
+
∥
X
42
-
D
42
∥
2
+
∥
X
43
-
D
43
∥
2
.
By Lemmas 5, 7, and 8, a general form of the least-square solutions can be expressed as (31); for arbitrary X
34, X
35, X
36, X
44, X
45, and X
46, there exists a least-square solution in C
n
×
n of (1) which is *congruent to (32), and the part (c) follows from the definition of Frobenius norm.


## 4. An Algorithm and Numerical Examples

Based on the main results of this paper, we in this section propose an algorithm for finding the least-square solutions to the system (1). All the tests are performed by MATLAB 6.5 which has a machine precision of around 1
0
-
16.Algorithm 1.
(1) InputA
∈
C
m
×
nand B
∈
C
n
×
l, and compute U
∈
C
n
×
n,  R
A
-
1
∈
C
m
×
m,  R
B
-
1
∈
C
l
×
l,  Σ
A, Σ
B
∈
C
n
×
p, and G
,
S
∈
C
s
×
s by the CCD of matrix pair [
A
*
,
B
].
(2) InputC
∈
C
m
×
n, D
∈
C
n
×
l, and compute C
i
j
(
i
=
1,2
,
3,4
;
j
=
1,2
,
3,4
,
5,6
) and D
l
k
(
l
=
1,2
,
3,4
,
5,6
;
k
=
1,2
,
3,4
) according to (37).
(3) Compute the least-square solutions of the system (1) by (31).
(4) Compute the
*congruence class of the least-square and the minimum norm least-square solutions to the system (1) according to (32) and (33).Example 1.
Suppose(39)
A
=
[
-
1.625
0
-
0.6875
i
0.875
i
0.3438
0
-
2
0
-
0.5
i
0.875
i
0.25
0
-
0.75
0
-
0.125
i
0.25
i
0.0625
0
2.625
0
0.6875
i
-
0.875
i
-
0.3438
0
]
,
B
=
[
2
-
2
0
-
1
-
5
i
6
i
0
2
i
31
i
-
37
i
i
-
15
i
0
0
0
0
0
0
0
0
0
0
0
0
]
,
C
=
[
3
2
2
1
1
5
4
i
i
-
i
3
1
5
6
7
4
3
1
2
1
1
4
3
4
]
,
D
=
[
i
-
i
-
i
1
2
3
1
4
4
1
2
3
5
7
6
9
9
7
5
6
1
2
3
2
]
.
Applying Algorithm1, we obtain the following:
(40)
U
=
[
1
0
0
0
0
0
0
0
-
i
0
0
0
0
i
0
0
0
0
0
0
0
0
0
i
0
0
0
0
1
0
0
0
0
-
i
0
0
]
,
Σ
A
=
[
1
0
0
0
0.5
0
0
0
0
0
0
0
0
0.25
0
0
0
1
]
,
Σ
B
=
[
1
0
0
0
1
0
0
0
1
0
0
0
0
0
0
0
0
0
]
,
R
A
-
1
=
[
-
1.625
-
2
-
0.75
2.625
1.375
1
0.25
-
1.375
-
0.875
0
-
0.25
0.875
0.375
0
0.25
-
0.375
]
,
R
B
-
1
=
[
2
-
2
0
-
1
31
-
37
1
-
15
5
-
6
0
-
2
-
44
52
-
1
21
]
,
G
=
[
0.5
]
,
S
=
[
0.25
]
,
(
R
A
)
*
C
U
=
(
5
3
i
-
3
i
-
9
i
4
5
i
14
-
1
+
6
i
1
-
6
i
-
19
i
11
1
+
10
i
32
-
2
+
26
i
2
-
23
i
-
22
i
25
2
+
25
i
53
-
1
+
57
i
1
-
50
i
-
34
i
38
1
+
46
i
)
,
U
*
D
R
B
=
(
0.03
-
0.05
i
0.02
-
0.05
i
0.04
-
0.02
i
0.02
-
0.04
i
-
0.51
i
-
0.27
i
-
0.33
i
-
0.25
i
0.43
i
0.21
i
0.29
i
0.2
i
0.41
i
0.26
i
0.25
i
0.23
i
1.35
0.67
0.78
0.62
-
1.27
i
-
0.71
i
-
0.82
i
-
0.65
i
)
.
The least-square solutions to the system (1) are(41)
X
=
U
(
5.03
-
0.05
i
0.02
+
2.95
i
0.04
-
3.02
i
-
9
i
4
5
i
-
0.51
i
-
0.27
i
-
0.33
i
0
0
0
0.43
i
0.21
i
0.29
i
X
34
X
35
X
36
0.41
i
0.26
i
0.25
i
X
44
X
45
X
46
2.02
-
0.06
i
0.864
-
0.366
i
0.498
+
1.448
i
76
i
44
4
+
40
i
-
0.84
i
-
0.5
i
-
0.53
i
-
22
i
25
2
+
25
i
)
U
*
,where X
34, X
35, X
36, X
44, X
45, and X
46 are arbitrary.
For arbitraryX
34, X
35, X
36, X
44, X
45, and X
46, there exists a least-square solution in C
6
×
6 of (1) which is *congruent to(42)
Y
=
(
5.03
-
0.05
i
0.02
+
2.95
i
0.04
-
3.02
i
-
9
i
4
5
i
-
0.51
i
-
0.27
i
-
0.33
i
0
0
0
0.43
i
0.21
i
0.29
i
X
34
X
35
X
36
0.41
i
0.26
i
0.25
i
X
44
X
45
X
46
2.02
-
0.06
i
0.864
-
0.366
i
0.498
+
1.448
i
76
i
44
4
+
40
i
-
0.84
i
-
0.5
i
-
0.53
i
-
22
i
25
2
+
25
i
)
.There exists a minimum norm least-square solution in C
6
×
6 of (1) which is c
*ongruent to(43)
Y
=
(
5.03
-
0.05
i
0.02
+
2.95
i
0.04
-
3.02
i
-
9
i
4
5
i
-
0.51
i
-
0.27
i
-
0.33
i
0
0
0
0.43
i
0.21
i
0.29
i
0
0
0
0.41
i
0.26
i
0.25
i
0
0
0
2.02
-
0.06
i
0.864
-
0.366
i
0.498
+
1.448
i
76
i
44
4
+
40
i
-
0.84
i
-
0.5
i
-
0.53
i
-
22
i
25
2
+
25
i
)
.



---

*Source: 703529-2014-09-03.xml*