Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 312767, 13 pages
doi:10.1155/2010/312767
Research Article
Further Results on the Reverse Order Law for
{1, 3}-Inverse and {1, 4}-Inverse of a Matrix Product
Deqiang Liu and Hu Yang
College of Mathematics and Statistics, Chongqing University, Chongqing 400030, China
Correspondence should be addressed to Deqiang Liu, ldq7705@163.com
Received 28 October 2009; Accepted 16 April 2010
Academic Editor: Panayiotis Siafarikas
Copyright q 2010 D. Liu and H. Yang. 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.
Both Djordjević 2007 and Takane et al. 2007 have studied the equivalent conditions for
B{1, 3}A{1, 3} ⊆ AB{1, 3} and B{1, 4}A{1, 4} ⊆ AB{1, 4}. In this note, we derive the
necessary and sufficient conditions for B{1, 3}A{1, 3} ⊇ AB{1, 3}, B{1, 4}A{1, 4} ⊇ AB{1, 4},
B{1, 3}A{1, 3} AB{1, 3} and B{1, 4}A{1, 4} AB{1, 4}.
1. Introduction
Let Cm×n denote the set of all m × n matrices over the complex field C. For A ∈ Cm×n , its range
space, null space, rank, and conjugate transpose will be denoted by RA, NA, rA, and
A∗ , respectively. The symbol dim RA denotes the dimension of RA. The n × n identity
matrix is denoted by In , and if the size is obvious from the context, then the subscript on In
can be neglected.
For a matrix A ∈ Cm×n , a generalized inverse X of A is a matrix which satisfies some
of the following four Penrose equations:
1 AXA A,
2 XAX X,
3 AX∗ AX,
4 XA∗ XA.
1.1
Let ∅ /
η ⊆ {1, 2, 3, 4}. Then Aη denotes the set of all matrices X which satisfy i for all i ∈ η.
Any matrix X ∈ Aη is called an η-inverse of A. One usually denotes any {1}-inverse of A
by A1 or A− , and any {1, 3}-inverse of A by A1,3 which is also called a least squares ginverses of A. Any {1, 4}-inverse of A is denoted by A1,4 which is also called a minimum
norm g-inverses of A. The unique {1, 2, 3, 4}-inverse of A is denoted by A† , which is called
the Moore-Penrose generalized inverse of A. General properties of the above generalized
inverses can be found in 1–3. The research in this area is active, especially about the {2}inverse and the reverse order law for generalized inverse; see 4–7.
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Journal of Inequalities and Applications
There are very good results for the reverse order law for {1}-inverse and {1, 2}inverse of two-matrix or multi-matrix products, and Liu and Yang 8 studied equivalent
conditions for B{1, 3, 4}A{1, 3, 4} ⊆ AB{1, 3, 4}, B{1, 3, 4}A{1, 3, 4} ⊇ AB{1, 3, 4}, and
B{1, 3, 4}A{1, 3, 4} AB{1, 3, 4}. Moreover, Wei and Guo 9 derived the reverse order
law for {1, 3}-inverse and {1, 4}-inverse of two-matrix products by using the product
singular value decomposition P-SVD. However, there is a fly in the ointment in Wei
and Guo’s results. That is, those results contain the information of subblock produced
by P-SVD. In other words, they are related to P-SVD. In order to overcome this
shortcoming, two methods are employed. One is operator theory; the other is maximal
and minimal rank of matrix expressions. Using these two different methods, both 6, 10
obtain
B{1, 3}A{1, 3} ⊆ AB{1, 3} ⇐⇒ RA∗ AB ⊆ RB,
1.2
B{1, 4}A{1, 4} ⊆ AB{1, 4} ⇐⇒ RBB∗ A∗ ⊆ RA∗ .
1.3
These results are our hope because there is no information of the P-SVD in them. Note
that RA∗ AB ⊆ RB and RBB∗ A∗ ⊆ RA∗ are equivalent to rB, A∗ AB rB and
rA∗ , BB∗ A∗ rA, respectively. Therefore, these results are only related to the range
space or the rank of A, A∗ , B, B∗ or their expressions. However, there are no analogs for
B{1, 3}A{1, 3} ⊇ AB{1, 3} and B{1, 4}A{1, 4} ⊇ AB{1, 4}. In this note, we derive the
necessary and sufficient conditions for them. And after this we present a new equivalent
conditions for B{1, 3}A{1, 3} AB{1, 3} and B{1, 4}A{1, 4} AB{1, 4}, and this results
are not related to P-SVD. To our knowledge, there is no article discussing these in the
literature.
In this note we will need the following two lemmas.
Lemma 1.1 see 11, 12. Let A ∈ Cm×n , B ∈ Cm×k , X ∈ Ck×l , C ∈ Cl×n and D ∈ Cl×k . Then
1 rA, B rA rB − dim RA ∩ RB;
1.4
2 rBX rX − dim NB ∩ RX;
1.5
3 r
C
A
rA r C I − A† A ;
1.6
4 max rA − BXC min rA, B, r
A
X
5 max r D − CA
A1,3
1,3
B min r
6 min r D − CA1,3 B r
A1,3
A∗ A A ∗ B
C
A∗ A A∗ B
C
C
D
D
r
1.7
;
− rA, r
B
D
B
;
1.8
⎜
⎟
− r ⎝ 0 B ⎠.
1.9
⎛
D
A 0
C D
⎞
Journal of Inequalities and Applications
3
Lemma 1.2 see 13. Let Ai,j ∈ Cmi ×nj 1 ≤ i, j ≤ 3 be given; X ∈ Cm1 ×n3 and Y ∈ Cm3 ×n1 are
two arbitrary matrices. Then
⎛
A11 A12
⎞
X
⎛
A12
⎞
⎟
⎟
⎜
⎜
min r ⎝A21 A22 A23 ⎠ rA21 , A22 , A23 r ⎝A22 ⎠
X,Y
Y
A32 A33
max r
A11 A12
A32
−r
A21 A22
A12
A22
A22 A23
−rA21 , A22 , r
A32 A33
−
A22
A32
− rA22 , A23 .
1.10
2. Main Results
In this section, we first give the minimal rank of D − B1,3 A1,3 with respect to any B1,3 and
A1,3 . Secondly, the necessary and sufficient conditions for the inclusion B{1, 3}A{1, 3} ⊇
AB{1, 3} are obtained by using our previous result. Finally, we also give the necessary
and sufficient conditions for B{1, 3}A{1, 3} AB{1, 3}, B{1, 4}A{1, 4} ⊇ AB{1, 4}, and
B{1, 4}A{1, 4} AB{1, 4}.
Lemma 2.1. Let A ∈ Cm×n , B ∈ Cn×k and D ∈ Ck×m . Then
min r D − B
B1,3 ,A1,3
1,3
A
1,3
r
B∗ BD
B∗
A∗
A∗ A
− min r
B∗
A
,r
BD
A∗
−r
D
A∗
n . 2.1
Proof. The expression of {1, 3}-inverses of A can be written as A1,3 A† FA V , where FA I − A† A and the matrix V is arbitrary; see 1. By combining this fact with elementary block
matrix operations, it follows that
r D − B 1,3 A1,3 r B† FB V A† FA V − D
r B† A† B† FA V FB V A† FB V FA V − D
⎛
0
⎜
⎜ 0
⎜
⎜
⎜ 0
r⎜
⎜−B†
⎜
⎜
⎜ In
⎝
V
0
0
0 In V
⎟
0 Im ⎟
⎟
⎟
0 In F A 0 ⎟
⎟ − k − m − 3n.
−D 0 0 0 ⎟
⎟
⎟
†
A In 0 0 ⎟
⎠
0 0 0 0
0 −Im 0
0
FB
0
Ik
⎞
2.2
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Journal of Inequalities and Applications
Applying 1.10 to 2.2 gives
min r D − B1,3 A1,3 r FB , B† A† − D, −B† FA
B1,3 ,A1,3
†
−D 0
max −r FB , B FA , r
−rFA −r
A† F A
FB −D
0
0
.
A† −FA
2.3
By using the elementary block matrix operations, the rank of the first partitioned matrix in
the right-hand side of 2.3 is simplified as follows:
r FB , B† A† − D, −B† FA
r
−B† FB −D
⎛
0
In
0
A† −FA
B†
0
0
−n
0
0
0
⎞
⎜ †
⎟
⎜B −B† Ik − B† B −D
0
0⎟
⎜
⎟ − n − r A† − r B †
r⎜
⎟
0
A† −In A† A A† ⎠
⎝ 0 In
0 0
0
0
0
A†
⎛ † † †
⎞
B B B B 0
0
0
⎜ †
⎟
⎜B 0 Ik −D
0
0⎟
⎜
⎟ − n − rA − rB
r⎜
0
−In A† ⎟
⎝ 0 In 0
⎠
0 0
0 −A† −A† A A†
†
B BD B†
r
k − rA − rB.
A† A† A
2.4
Using the formula rAB ≤ min{rA, rB} together with the fact that
B∗ B
0
0
A∗ A
B† B† ∗
B† BD
∗
B BD
A† A† A
∗
B BD B∗
0
A† A† 0
B†
∗
A∗
A∗ A
B∗
,
A∗ A∗ A
†
B BD B†
A†
2.5
A† A
means that
r
B†
A†
A† A
B† BD
r
B∗
A∗
A∗ A
B∗ BD
.
2.6
Journal of Inequalities and Applications
5
Substituting 2.6 into 2.4 yields
†
†
†
r FB , B A − D, −B FA r
B∗
A∗
A∗ A
B∗ BD
k − rA − rB.
2.7
Similarly, we obtain
†
r FB , B FA r
−D
0
B∗
k − rA − rB,
A
A∗
2.8
r
n − rA ,
D
A† −FA
BD
FB −D 0
r
n k − rA − rB.
r
0 A† −FA
A∗
r
It is always ture that RI − A† A NA. Therefore,
rFA r I − A† A n − rA.
2.9
Substituting 2.7–2.9 into 2.3 yields 2.1.
Theorem 2.2. Let A ∈ Cm×n and B ∈ Cn×k . Then the following statements are equivalent:
1 B{1, 3}A{1, 3} ⊇ AB{1, 3};
2 rA∗ AB, B rA rAB min{rA∗ , B, max{n rA − m, n rB − k}}.
Proof. We know that B{1, 3}A{1, 3} ⊇ AB{1, 3} is equivalent to saying that for an arbitrary
{1, 3}-inverse AB1,3 , there are {1, 3}-inverses A1,3 and B1,3 satisfying B1,3 A1,3 AB1,3 . That is,
B{1, 3}A{1, 3} ⊇ AB{1, 3} ⇐⇒ max
min r AB1,3 − B1,3 A1,3 0.
2.10
AB1,3 A1,3 ,B1,3
By using the formula 2.1, we get
min r AB1,3 − B1,3 A1,3
B1,3 ,A1,3
r
B∗
A∗
A∗ A
B∗ BAB1,3
− min r
B∗
A
, r
BAB1,3
A∗
−r
AB1,3
A∗
n .
2.11
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Journal of Inequalities and Applications
Using the formulas 1.9 and 1.8 together with elementary block matrix operations,
the maximal and minimal ranks of first partitioned matrix in the right-hand side of 2.11 are
as follows:
min r
B∗ BAB1,3
B∗
A∗
A∗ A
B∗
AB1,3
min
AB1,3
⎛
r
∗
0
A∗ A∗ A
∗
∗
∗
−B∗ B
−
AB
0
⎞
1,3
⎛
I, 0
⎞
⎛
AB
0
0
⎞
⎟
⎜
⎜ 0
I
0 ⎟
∗ ⎟
⎟
⎜
B ⎠ − r⎜ ∗
∗ ⎟
−B
B
0
B
⎠
⎝
0
A ∗ A∗ A
A∗ A∗ A
0
A∗ A∗ A
∗ ∗ B AA
B∗ BAB1,3 B∗
r
rA − rAB max r
.
B∗
A∗
A∗ A
AB1,3
B A AB B A
⎜
r ⎝ −B∗ B
0
0
I
⎜
∗ ⎟
B ⎠ r⎝ 0
0
2.12
Therefore, for an arbitrary {1, 3}-inverse AB1,3 ,
r
B∗ BAB1,3
B∗
A∗
A∗ A
r
B ∗ A∗ A
B∗
rA − rAB.
2.13
Using formulas 1.6 and 1.5, we get
r
BAB1,3
A∗
−r
AB1,3
A∗
r BAB1,3 I − AA† − r AB1,3 I − AA†
− dim NB ∩ R AB1,3
2.14
I − AA† .
Substituting 2.13 and 2.14 into 2.11 produces
min r AB
B1,3 ,A1,3
1,3
−B
1,3
A
1,3
r
B ∗ A∗ A
B∗
− min r
rA − rAB
B∗
A
1,3
, n − dim NB ∩ R AB
I − AA
.
†
2.15
Journal of Inequalities and Applications
7
Furthermore, we have
max
min r AB1,3 − B1,3 A1,3
AB1,3 B1,3 ,A1,3
⎛
r⎝
B ∗ A∗ A
B∗
⎧ ⎛ ⎞
⎫
⎨
⎬
B∗
⎠ rA − rAB − min r ⎝ ⎠, n − a ,
⎩
⎭
A
⎞
2.16
where a maxAB1,3 dim NB ∩ RAB1,3 I − AA† .
Next, we want to prove that a is equal to min{k − rB, m − rA}. First observe that
a ≤ min{k − rB, m − rA} since a ≤ dim NB k − rB and a ≤ maxAB1,3 rAB1,3 I −
AA† ≤ rI − AA† dim NA∗ m − rA. Therefore, a min{k − rB, m − rA} holds
if and only if there is a {1, 3}-inverse AB1,3 such that
dim NB ∩ R AB1,3 I − AA† min{k − rB, m − rA}.
2.17
Suppose that m − rA ≤ k − rB. Also note that rAB1,3 I − AA† ≤ m − rA
for arbitrary {1, 3}-inverses AB1,3 . Therefore, for some AB1,3 , 2.17 holds if and only if
there is a {1, 3}-inverse AB1,3 such that RAB1,3 I − AA† ⊆ NB and rAB1,3 I −
AA† m − rA hold—that is,
min r
B
AB1,3
I
1,3
AB
I − AA
†
−
0
C
0,
2.18
where C is any k × m matrix and rC m − rA. It follows from the formula 1.7
that maxX rI − B† BXI − AA† min{rI − B† B, rI − AA† } m − rA. Therefore,
there is a matrix X0 satisfying rI − B† BX0 I − AA† m − rA. Let C I −
B† BX0 I − AA† . It is always true that rC m − rA, BC 0, and B∗ A∗ I − AA† 0. Use these equations together with the formula 1.9 to conclude that 2.18 holds.
Therefore, if m − rA ≤ k − rB, then there is a {1, 3}-inverse AB1,3 such that 2.17
holds.
On the other hand, suppose that m−rA > k−rB. Also note that dim NB k−rB.
Therefore, for some AB1,3 2.17 holds if and only if there is a {1, 3}-inverse AB1,3 such
that NB RI − B† B ⊆ RAB1,3 I − AA† holds, that is,
min r I − B† B − AB1,3 I − AA† X 0,
AB1,3
2.19
8
Journal of Inequalities and Applications
where X is some m × k matrix. Use the formula 1.9 to find that
min r I − B† B − AB1,3 I − AA† X
AB1,3
B A AB B A I − AA X
r
∗
∗
∗
∗
I − B† B
I
I − AA† X
r
†
−r
r
I − AA X
†
I − B† B
⎛
AB
⎜
− r⎝ 0
I
0
⎞
⎟
I − AA† X ⎠
I − B† B
I − AA† X
I − B† B
r X ∗ I − AA† , I − B† B − r X ∗ I − AA† .
2.20
We know from 2.20 that 2.19 holds if and only if there is an m × k matrix X such that
RI − B† B ⊆ RX ∗ I − AA† . In fact, note that rI − B† B dim NB k − rB and
, Q1 , and
rI − A† A dim NA∗ m− rA, and let P1 , P2
Q2 be nonsingular matrices
Ik−rB 0
Im−rA 0
†
†
Q1 and I − A A P2
Q2 . Using this together with
such that I − B B P1
0
0
0
0
m − rA > k − rB means that if X ∗ P1 P2−1 , then RI − B† B ⊆ RX ∗ I − AA† . Therefore,
if m − rA > k − rB, then there is a {1, 3}-inverse AB1,3 such that 2.17 holds.
In summary, there is a {1, 3}-inverse AB1,3 such that 2.17 holds. That is, a min{k − rB, m − rA}. Apply this to 2.16 to obtain that
max
min r AB1,3 − B1,3 A1,3 rA∗ AB, B rA − rAB
AB1,3 B1,3 ,A1,3
− min{rA∗ , B, max{n rB − k, n rA−m}}.
2.21
Noting that 2.10 and letting the right-hand side in 2.21 be equal to zero, then the
equivalence between 1 and 2 follows immediately.
It is obvious that B{1, 3}A{1, 3} AB{1, 3} if and only if B{1, 3}A{1, 3} ⊆ AB{1, 3}
and B{1, 3}A{1, 3} ⊇ AB{1, 3}. Also note Theorem 2.2 and formula 1.2. It is easy to obtain
the following theorem.
Theorem 2.3. Let A ∈ Cm×n and B ∈ Cn×k . Then the following statements are equivalent:
1 B{1, 3}A{1, 3} AB{1, 3};
2 rB, A∗ AB rB and rA rB rAB min{rA∗ , B, max{n rB − k, n rA − m}}.
The following theorems can be obtained by applying Theorem 2.2 or Theorem 2.3 to the product
B∗ A∗ and using the fact that X ∈ D{1, 3} if and only if X ∗ ∈ D∗ {1, 4}.
Journal of Inequalities and Applications
9
Theorem 2.4. Let A ∈ Cm×n and B ∈ Cn×k . Then the following statements are equivalent:
1 B{1, 4}A{1, 4} ⊇ AB{1, 4};
2 rBB∗ A∗ , A∗ rB rAB min{rA∗ , B, max{n rA − m, n rB − k}}.
Theorem 2.5. Let A ∈ Cm×n and B ∈ Cn×k . Then the following statements are equivalent:
1 B{1, 4}A{1, 4} AB{1, 4};
2 rBB∗ A∗ , A∗ rA and rA rB rAB min{rA∗ , B, max{n rA −
m, n rB − k}}.
3. Examples
In this section, we give two examples. The first example comes from 14, and they verify
that B{1, 2, 3}A{1, 2, 3} ⊆ AB{1, 2, 3}. However, this example does not only satisfy this
result. In Example 3.1, we know that this example satisfies Theorems 2.3 and 2.5, and so we
have B{1, 3}A{1, 3} AB{1, 3} and B{1, 4}A{1, 4} AB{1, 4}. In this example, we will
verify these results. Secondly, we give another example which only satisfies B{1, 3}A{1, 3} ⊃
AB{1, 3} and B{1, 4}A{1, 4} ⊃ AB{1, 4}.
Example 3.1. Let
⎛
⎞
1 0 0
⎜
⎟
A ⎝0 1 1⎠,
⎛
⎞
1 0 0
⎜
⎟
B ⎝1 1 0⎠.
0 1 1
3.1
1 1 0
It is easy to obtain that
rB, A∗ AB rA∗ , BB∗ A∗ rB rA rB, A∗ 2.
3.2
From Theorems 2.3 and 2.5, we can conclude that
B{1, 3}A{1, 3} AB{1, 3},
B{1, 4}A{1, 4} AB{1, 4}.
Now we verify this statement. Since
⎧⎛
⎫
⎞
⎪
⎪
1
0
0
⎪
⎪
⎪
⎪
⎟
⎨⎜
⎬
⎟
⎜ a1
a
a
2
3
A{1, 3} ⎜
⎟ | a1 , a2 , a3 ∈ C ,
⎪
⎪
⎝
⎪
⎪
1
1⎠
⎪
⎪
⎩ −a1 −a2 ⎭
−a3 2
2
⎫
⎧⎛
⎞
1 0 0
⎪
⎪
⎪
⎪
⎪
⎪
⎟
⎬
⎨⎜
1
1
⎟
⎜
B{1, 3} ⎜−1
⎟ | a4 , a5 , a6 ∈ C ,
⎪
⎪
⎝
2 2⎠
⎪
⎪
⎪
⎪
⎭
⎩
a4 a5 a6
3.3
10
⎧⎛
1
⎪
⎪
⎪
⎨⎜
⎜
AB{1, 3} ⎜−1
⎪
⎝
⎪
⎪
⎩
a7
0
1
4
a8
Journal of Inequalities and Applications
⎫
⎞
0
⎪
⎪
⎪
⎬
1⎟
⎟
⎟ | a7 , a8 , a9 ∈ C ,
⎪
4⎠
⎪
⎪
⎭
a9
3.4
we easily find that
⎧⎛
1
⎪
⎪
⎪
⎨⎜
⎜
B{1, 3}A{1, 3} ⎜−1
⎪
⎝
⎪
⎪
⎩
a
0
1
4
b
⎫
⎞
0
⎪
⎪
⎪
⎬
1⎟
⎟
⎟ | ai ∈ C, i 1, 2, . . . , 6 ,
⎪
4⎠
⎪
⎪
⎭
c
3.5
where a a4 a1 a5 −a1 a6 , b a2 a5 −a2 a6 1/2a6 , and c a3 a5 −a3 a6 1/2a6 . It is obvious
that B{1, 3}A{1, 3} ⊆ AB{1, 3}. If a1 a2 0, a3 1, a4 a7 , a5 a8 a9 , and a6 2a8 ,
then we have a a7 , b a8 , and c a9 , that is, B{1, 3}A{1, 3} ⊇ AB{1, 3}. Therefore,
B{1, 3}A{1, 3} AB{1, 3}.
On the other hand, since
⎫
⎧⎛
⎞
−a1
1
a1
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎟
⎬
⎨⎜
1⎟
⎜
⎟
⎜
0
a
−a
2
2
| a1 , a2 , a3 ∈ C ,
A{1, 4} ⎜
⎟
2⎠
⎪
⎪
⎪
⎪
⎪
⎪⎝
⎪
⎪
⎭
⎩ 0 −a3 1
a3
2
⎫
⎧⎛
⎞
1 a4 −a4
⎪
⎪
⎪
⎪
⎬
⎨⎜
⎟
⎟
−1
a
1
−
a
,
a
,
a
∈
C
|
a
,
B{1, 4} ⎜
5
5⎠
4 5 6
⎝
⎪
⎪
⎪
⎪
⎭
⎩
0 a6 −a6
⎫
⎧⎛
⎞
−a7
1 a7
⎪
⎪
⎪
⎪
⎪
⎪
⎟
⎬
⎨⎜
⎟
⎜
1
AB{1, 4} ⎜−1 a8 −a8 ⎟ | a7 , a8 , a9 ∈ C ,
⎪
⎪
⎝
⎪
⎪
2⎠
⎪
⎪
⎭
⎩
−a9
0 a9
3.6
⎫
⎧⎛
⎞
1 d −d
⎪
⎪
⎪
⎪
⎬
⎨⎜
⎟
1
⎟
⎜
B{1, 4}A{1, 4} ⎝−1 e −e ⎠ | ai ∈ C, i 1, 2, . . . , 6 ,
⎪
⎪
⎪
⎪
2
⎭
⎩
0 f
−f
3.7
we easily see that
where d a1 − 1/2a4 a2 a4 a3 a4 , e 1/2 − a1 − a3 − 1/2a5 a2 a5 a3 a5 , and f a2 a6 a3 a6 − 1/2a6 . It is obvious that B{1, 4}A{1, 4} ⊆ AB{1, 4}. If a1 a7 , a2 a7 a8 a9 ,
a3 1/2 − a7 − a8 , a4 a5 0 and a6 1, then we have d a7 , e a8 , and f a9 , that is,
B{1, 4}A{1, 4} ⊇ AB{1, 4}. Therefore, B{1, 4}A{1, 4} AB{1, 4}.
Journal of Inequalities and Applications
11
Example 3.2. Let
⎛
⎞
1 0 0 0
⎜
⎟
A ⎝0 1 1 0⎠,
0 1 1 0
⎛
1
⎜
⎜1
B⎜
⎜0
⎝
0
⎞
0 0
⎟
1 0⎟
⎟.
1 0⎟
⎠
0 0
3.8
It is easy to obtain that
rA rB rAB 2,
rB, A∗ AB rA∗ , BB∗ A∗ rB, A∗ 3.
3.9
From Theorems 2.2 and 2.4, we can find that
B{1, 3}A{1, 3} ⊇ AB{1, 3},
B{1, 4}A{1, 4} ⊇ AB{1, 4}.
3.10
rB rA 2. Using Theorems
Furthermore, note that rB, A∗ AB rA∗ , BB∗ A∗ 3 /
2.3 and 2.5, we can conclude that
B{1, 3}A{1, 3} ⊃ AB{1, 3},
B{1, 4}A{1, 4} ⊃ AB{1, 4}.
3.11
Now we verify this statement. Since
⎫
⎧⎛
⎞
1
0
0
⎪
⎪
⎪
⎪
⎪
⎪⎜
⎪
⎪
⎟
⎪
⎪
⎪
⎪
⎟
⎜
a
a
a
1
2
3
⎪
⎪
⎟
⎬
⎨⎜
⎟
⎜
|
a
,
a
,
.
.
.
,
a
∈
C
A{1, 3} ⎜
,
⎟
1
2
6
⎜−a −a 1 −a 1 ⎟
⎪
⎪
⎪
⎪
1
2
3
⎟
⎜
⎪
⎪
⎪
⎪⎝
2
2⎠
⎪
⎪
⎪
⎪
⎪
⎪
⎭
⎩
a5
a6
a4
⎫
⎧⎛ 2 1
⎞
1
⎪
⎪
⎪
⎪
−
0
⎪
⎪⎜ 3 3
⎪
⎪
⎟
3
⎪
⎪
⎟
⎬
⎨⎜
⎟
⎜ 1 1 2
|
a
,
a
,
a
,
a
∈
C
,
B{1, 3} ⎜−
⎟
7
8
9
10
0 ⎟
⎪
⎪
⎪
⎪⎜
⎪
⎪
⎠
⎝
3
3
3
⎪
⎪
⎪
⎪
⎭
⎩
a7 a8 a9 a10
⎫
⎧⎛
⎞
1 0 0
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪⎜
⎟
⎬
⎨
⎜ 1 1 1 ⎟
⎟
⎜
|
a
,
a
,
a
∈
C
,
AB{1, 3} ⎜−
11
12
13
⎟
⎪
⎪
⎪
⎪
⎝ 2 4 4 ⎠
⎪
⎪
⎪
⎪
⎭
⎩
a11 a12 a13
3.12
12
Journal of Inequalities and Applications
we easily get that
⎧⎛
⎫
⎞
1 2
1 2
2 2
⎪
⎪
⎪
⎪
⎪
⎪
⎪⎜ 3 3 a1 − 6 3 a2 − 6 3 a3 ⎟
⎪
⎪
⎪
⎪
⎪
⎟
⎜
⎨⎜
⎬
⎟
⎟
⎜
1
1
1
1
1
1
| a1 , a2 , . . . , a10 ∈ C ,
B{1, 3}A{1, 3} ⎜
⎟
⎪
⎪
⎜− 3 − 3 a1 3 − 3 a2 3 − 3 a3 ⎟
⎪
⎪
⎪
⎪
⎪
⎪
⎠
⎝
⎪
⎪
⎪
⎪
⎩
⎭
a
b
c
3.13
where a a7 a1 a8 − a1 a9 a4 a10 , b 1/2a9 a2 a8 − a2 a9 a5 a10 , and c 1/2a9 a3 a8 −
a3 a9 a6 a10 . It is obvious that if a1 1/2, a2 1/4, a3 1/4, a4 a6 a8 0, a5 a12 − a13 ,
a7 2a13 a11 , a9 4a13 , and a10 1, then
⎛
⎞ ⎛
⎞
2 2
1 2
1 2
1 0 0
a
a
a
−
−
1
2
3
⎜ 3 3
6 3
6 3 ⎟
⎟
⎜
⎟ ⎜
⎟
⎜
⎟ ⎜
⎜
⎟
1
1
1
⎜ 1 1
⎟
⎟.
−
⎜− − a1 1 − 1 a2 1 − 1 a3 ⎟ ⎜
⎜
⎜ 3 3
2 4 4 ⎟
⎠
3 3
3 3 ⎟
⎝
⎠ ⎝
a11 a12 a13
a
b
c
3.14
That is, B{1, 3}A{1, 3} ⊇ AB{1, 3}. Furthermore, note that if a1 /
1/2, then there are some
B1,3 A1,3 which do not belong to AB{1, 3}. Therefore, B{1, 3}A{1, 3} ⊃ AB{1, 3}.
On the other hand, because
⎧⎛
1
⎪
⎪
⎪
⎪⎜
⎪
⎪
⎜
⎪
⎪⎜0
⎪
⎨⎜
⎜
A{1, 4} ⎜
⎜
⎪
⎪
⎪⎜0
⎪
⎜
⎪
⎪
⎪⎝
⎪
⎪
⎩
0
⎫
⎪
⎪
⎪
⎪
⎟
⎪
⎪
1⎟
⎪
⎪
a2 −a2 ⎟
⎪
⎟
⎬
2⎟
⎟ | a1 , a2 , a3 , a4 ∈ C ,
⎪
1⎟
⎪
⎪
a3 −a3 ⎟
⎪
⎟
⎪
⎪
2⎠
⎪
⎪
⎪
⎭
a4
−a4
a1
−a1
⎞
⎫
⎧⎛
⎞
a5 −a5 1 a5 − 1 a6
⎪
⎪
⎪
⎪
⎬
⎨⎜
⎟
⎟
a
−a
a
1
a
,
a
,
.
.
.
,
a
∈
C
|
a
,
B{1, 4} ⎜
7
7
7
8⎠
5 6
10
⎝
⎪
⎪
⎪
⎪
⎭
⎩
a9 −a9
a9 a10
⎫
⎧⎛
⎞
−a11
1 a11
⎪
⎪
⎪
⎪
⎪
⎪⎜
⎟
⎬
⎨
⎟
⎜ 1
1
AB{1, 4} ⎜− a12 −a12 ⎟ | a11 , a12 , a13 ∈ C ,
⎪
⎪
⎝ 2
⎪
⎪
2⎠
⎪
⎪
⎭
⎩
−a13
0 a13
3.15
we easily obtain that
⎧⎛
⎫
⎞
a5 d −d
⎪
⎪
⎪
⎪
⎨⎜
⎬
⎟
1
⎟
B{1, 4}A{1, 4} ⎜
,
a
,
.
.
.
,
a
∈
C
|
a
,
1 2
10
e
−e
a
⎝
⎠
7
⎪
⎪
⎪
⎪
2
⎩
⎭
−f
a9 f
3.16
Journal of Inequalities and Applications
13
where d a2 − a3 a1 a5 − a2 a5 a3 a5 a4 a6 , e a3 a1 a7 − a2 a7 a3 a7 a4 a8 , and f a1 a9 − a2 a9 a3 a9 a4 a10 . It is obvious that if a1 a11 , a2 a6 a8 a9 0, a3 a11 2a12 ,
a4 a13 , a5 a10 1 and a7 −1/2, then
⎛
⎞ ⎛
−a11
1 a11
a5 d −d
⎜
⎟ ⎜ 1
1
⎜
⎟ ⎜
a12 −a12 ⎜a7 e −e ⎟ ⎜−
⎝
2⎠ ⎝ 2
a9 f
−f
−a13
0 −a13
⎞
1⎟
⎟
⎟.
2⎠
3.17
That is, B{1, 4}A{1, 4} ⊇ AB{1, 4}. Furthermore, note that if a5 /
1, then there are some
B1,4 A1,4 which do not belong to AB{1, 4}. Therefore, B{1, 4}A{1, 4} ⊃ AB{1, 4}.
Acknowledgments
This work is supported by the Third Stage Training of “211 Project” Project no.: S-09110,
and supported by The Natural Science Foundation Project of CQ CSTC 2009BB6189.
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