Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 925092, 18 pages
doi:10.1155/2012/925092
Research Article
Zero Triple Product Determined Matrix Algebras
Hongmei Yao1, 2 and Baodong Zheng1
1
2
Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
College of Science, Harbin Engineering University, Harbin 150001, China
Correspondence should be addressed to Hongmei Yao, hongmeiyao@163.com
Received 9 August 2011; Accepted 20 December 2011
Academic Editor: Xianhua Tang
Copyright q 2012 H. Yao and B. Zheng. 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.
Let A be an algebra over a commutative unital ring C. We say that A is zero triple product
determined if for every C-module X and every trilinear map {·, ·, ·}, the following holds: if
{x, y, z} 0 whenever xyz 0, then there exists a C-linear operator T : A3 −→ X such that
{x, y, z} T xyz for all x, y, z ∈ A. If the ordinary triple product in the aforementioned definition
is replaced by Jordan triple product, then A is called zero Jordan triple product determined. This
paper mainly shows that matrix algebra Mn B, n ≥ 3, where B is any commutative unital algebra
even different from the above mentioned commutative unital algebra C, is always zero triple
2, is also zero Jordan
product determined, and Mn F, n ≥ 3, where F is any field with chF /
triple product determined.
1. Introduction
Over the last couple of years, several papers characterizing bilinear maps on algebras
through their action on elements whose certain product is zero have been written; see 1–6.
The philosophy in these papers is that certain classical problems concerning linear maps
that preserve zero product, Jordan product, commutativity, and so forth can be sometimes
effectively solved by considering bilinear maps that preserve certain zero product properties.
For example, in 1, in order to determine whether a linear map preserving zero product
resp., zero Jordan product, zero Lie product is “closed” to a homomorphism, the authors
introduced the definitions of zero product resp., zero Jordan product, zero Lie product
determined algebras. The core idea of these definitions is to answer the aforementioned
questions by determining the bilinear maps preserving zero product resp., zero Jordan
product, zero Lie product. Furthermore, as the main task, they gave the positive answer
that the matrix algebra Mn B of n × n matrices over a unital algebra B is zero product
determined, and under some further restrictions on B, Mn B is still zero Jordan resp., Lie
product determined.
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Meanwhile, in preserver problems, there have appeared many kinds of preserving
product forms, such as the forms in papers 7–10. Particularly, in 10, there appears a
definition of Jordan triple product. This definition has applications not only in Banach algebra
see 11 but also in generalized inverse of matrices see 12.
Inspired by the aforementioned, this gives rise to a question: whether can we
generalize the products in paper 1 to more generalized forms, such as triple product and
Jordan triple product? And immediately, it follows another interesting preserver problem:
whether a linear map preserving zero triple product resp., zero Jordan triple product is
still “closed” to a homomorphism? In order to answer the aforementioned questions, we
introduce the following definitions. Let C be a fixed commutative unital ring and A an
algebra over C. By A3 , we denote the C-linear span of all elements of the form xyz, where
x, y, z ∈ A. X is a C-module and we denote by {·, ·, ·} : A × A × A → X a C-trilinear map.
Consider the following conditions:
a for all x, y, z ∈ A such that xyz 0, we have {x, y, z} 0;
b there exists a C-linear map T : A3 → X such that {x, y, z} T xyz for all x, y, z ∈
A.
Trivially, b implies a. We call that A is a zero triple product determined algebra if
for every C-module X and every C-trilinear map {·, ·, ·}, a implies b. And we say that A
is a zero Jordan triple product determined algebra if we replace the above triple product by
Jordan triple product x ◦ y ◦ z xyz zyx.
From the previous definitions, it is interesting to examine whether the matrix algebra
Mn B of n × n matrices over a unital algebra B is still zero triple product resp., zero Jordan
triple product determined. If the unital algebra B has no further restrictions, this problem
will be difficult. Therefore, the purpose of this paper is to characterize the zero triple product
resp., zero Jordan triple product determined algebra under some additional restrictions on
the unital algebra B. In Section 2, we show that the answer is “yes” for the triple product if
B is any commutative unital algebra even different from C. The Jordan triple product case,
treated in Section 3, is more difficult; we only show that the matrix algebra Mn F, where F
is any field with chF / 2 chF stands for the characteristic of a field is also zero Jordan triple
product determined.
Finally, we end this section by giving an equivalent condition of b in the previous
definition which is more convenient to use:
m
b if xt , yt , zt ∈ A, t 1, . . . , m, are such that m
t1 xt yt zt 0, then
t1 {xt , yt , zt } 0.
2. Zero Triple Product Determined Matrix Algebras
In this part, we will consider the matrix algebra Mn B, where B is any commutative unital
algebra even different from the fixed commutative unital ring C mentioned in Section 1. By
beij , where b ∈ B, we denote the matrix whose i, j entry is b and all other entries are 0.
Theorem 2.1. Let B be a commutative unital algebra, then Mn B is a zero triple product determined
algebra for every n ≥ 3.
Proof. In order to prove that a implies b, we only prove that a implies the equivalent
condition b’. Set A Mn B, let X be a C-module, and let {·, ·, ·} be a C-trilinear map such
that for all x, y, z ∈ A, xyz 0 implies {x, y, z} 0. Throughout the proof, a, b, and c denote
arbitrary elements in B and i, j, k, l, r, and h denote arbitrary indices.
Journal of Applied Mathematics
3
k or r /
l, we get
First, since aeij bekl cerh 0 if j /
aeij , bekl , cerh 0
if j / k or r / l.
2.1
k, and consequently
Second, aeij abeik bejl − ekl celh 0, where j /
aeij , bejl , celh {abeik , ekl , celh },
2.2
replacing a by ab and b by 1 in 2.2, we have
aeij , bejl , celh abeij , ejl , celh ,
2.3
and using 2.3 and 2.2, this yields that
aeij , bejl , celh {abei1 , e1l , celh }.
2.4
k, we obtain
Similarly, from aeij bejl bcejk celh − ekh 0, where l /
aeij , bejl , celh aeij , bcej1 , e1h .
2.5
Finally, combining 2.5 and 2.4, it follows that
aeij , bejl , celh aeij , bcej1 , e1h {abcei1 , e11 , e1h }.
Let xt , yt , zt ∈ A be such that
m
t1
2.6
xt yt zt 0; then we only need to show
m
xt , yt , zt 0.
2.7
t1
Writing
xt n
n i1 j1
atij eij ,
yt n
n k1 l1
it follows, by examining the i, h entry of
m
t1
t
bkl
ekl ,
zt n
n t
crh
erh ,
r1 h1
2.8
xt yt zt , that for all i and h, we have
m n n
t
atij bjlt clh
0.
t1 j1 l1
2.9
Note that
⎧
⎫
m
m ⎨
n n n n
n
n
⎬
t
t
atij eij ,
bkl
ekl ,
crh
erh ,
xt , yt , zt ⎩ i1 j1
⎭
t1
t1
r1 h1
k1 l1
2.10
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by 2.1, this summation reduces to
n n n m
m n t
atij eij , bjlt ejl , clh
elh ,
xt , yt , zt t1
t1 i1 j1 l1 h1
2.11
and using 2.6 and 2.9, we obtain
n n n n m
m t
atij bjlt clh
ei1 , e11 , e1h
xt , yt , zt t1
t1 i1 j1 l1 h1
⎧
n m n ⎨
n n
i1 h1
⎩ t1
j1 l1
t
atij bjlt clh
ei1 , e11 , e1h
⎫
⎬
2.12
⎭
0.
Therefore, the result of this theorem holds.
3. Zero Jordan Triple Product Determined Matrix Algebras
In this part, we only consider the matrix algebra Mn F, where F is a field with chF /
2, and
C is still the fixed commutative unital ring mentioned in Section 1. Let beij , where b ∈ F, be
the matrix whose i, j entry is b and all other entries are 0.
Theorem 3.1. Let F be a field with chF / 2; then Mn F is a zero Jordan triple product determined
algebra for every n ≥ 3.
Proof. Set A Mn F, let X be a C-module, and let {·, ·, ·} be a C-trilinear map such that for
all x, y, z ∈ A, x ◦ y ◦ z 0 implies {x, y, z} 0. Let a, b, and c be arbitrary elements from F
and i, j, k, l, r, and h denote arbitrary indices.
First, for j / k, h / k or j / k, l / i or l / r, h / k or l / r, and l / i, we have aeij ◦ bekl ◦
cerh 0, and so
aeij , bekl , cerh 0
if j /
k, h /
k or j /
k, l /
i or l /
r, h /
k or l /
r, l /
i.
3.1
For any z ∈ A and i /
k, j /
k, we have aeij abeik ◦ bejk − ekk ◦ z 0 and z ◦ aeij abeik ◦
bejk − ekk 0, which implies
k, j /
k,
aeij , bejk , z {abeik , ekk , z} if i /
z, aeij , bejk {z, abeik , ekk } if i / k, j / k.
3.2
3.3
Similarly,
k, j /
k,
bejk , aeij , z {ekk , abeik , z} if i /
k, j z, bejk , aeij {z, ekk , abeik } if i /
/ k.
3.4
3.5
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5
For any z ∈ A and i /
k, we have aeik −eii ◦abeik bekk ◦z 0, aeik −ekk ◦abeik beii ◦z 0,
z ◦ aeik − eii ◦ abeik bekk 0, and z ◦ aeik − ekk ◦ abeik beii 0, and consequently,
{aeik , bekk , z} {eii , abeik , z}
if i /
k,
3.6
{aeik , beii , z} {ekk , abeik , z}
if i /
k,
3.7
k,
{z, aeik , bekk } {z, eii , abeik } if i /
3.8
{z, aeik , beii } {z, ekk , abeik } if i / k.
3.9
k,
{bekk , aeik , z} {abeik , eii , z} if i /
3.10
{beii , aeik , z} {abeik , ekk , z}
if i /
k,
3.11
k,
{z, bekk , aeik } {z, abeik , eii } if i /
3.12
k.
{z, beii , aeik } {z, abeik , ekk } if i /
3.13
Similarly,
Since aeih ◦ eii − ehh ◦ eii ehh 0 and eii − ehh ◦ eii ehh ◦ aeih 0, if h /
i, it follows that
{aeih , eii , eii } {aeih , ehh , ehh }
if i / h,
3.14
{eii , eii , aeih } {ehh , ehh , aeih }
if i /
h.
3.15
Using 3.8, 3.11, 3.15, and 3.14, we arrive at
h.
{eii , eii , aeih } {aeih , ehh , ehh } {ehh , ehh , aeih } {aeih , eii , eii } if i /
3.16
Using 3.7 and 3.12, we have
aeij , beji , ejj abeij , eji , ejj
if i / j.
3.17
if i /
j.
3.18
if i /
j.
3.19
Since aeij beji ◦ aeij − beji ◦ eii 0 if i /
j, then
aeij , beji , eii beji , aeij , eii
Using 3.18 and 3.17, it follows that
aeij , beji , eii abeji , eij , eii
Using 3.6, 3.8, 3.18, and 3.17, we obtain
eii , aeij , beji baeji , eij , eii eij , baeji , eii
if i /
j.
3.20
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Journal of Applied Mathematics
Similarly, using 3.10, 3.9, and 3.17, this yields
eii , aeji , beij abeji , eij eii
if i /
j.
3.21
1
a ◦ b ◦ ceii , eii , eii .
2
3.22
Further, we claim that
{aeii , beii , ceii } If i /
j, then 1/2abeii aeij − 1/2abejj ◦ beji − eii ejj ◦ ceii cejj 0; consequently,
1
1
abeii aeij − abejj , beji − eii ejj , ceii cejj
2
2
0,
if i / j.
3.23
Using 3.8–3.13 and 3.16, we get
aeij , beji , ceii cejj 1
abeii , eii , ceii
2
1
abejj , ejj , cejj ,
2
if i /
j.
3.24
From aeii beji − beij − aejj ◦ beii − aeij aeji − bejj ◦ ceii cejj 0, where i / j, it follows
that
aeii beji − beij − aejj , beii − aeij aeji − bejj , ceii cejj 0.
3.25
Using 3.3, 3.4, 3.6–3.13, and 3.16, we arrive at
{aeii , beii , ceii } aejj , bejj , cejj beij , aeji , ceii cejj beji , aeij , ceii cejj .
3.26
Then, by 3.24, we have
{abeii , eii , ceii } abejj , ejj , cejj {aeii , beii , ceii } aejj , bejj , cejj ,
where i / j.
3.27
Since n ≥ 3, we choose k such that k /
i, j; applying 3.27, we get
{aeii , beii , ceii } aejj , bejj , cejj {aekk , bekk , cekk } {aeii , beii , ceii }
{abeii , eii , ceii } abejj , ejj , cejj {abekk , ekk , cekk } {abeii , eii , ceii }
2{abeii , eii , ceii } {aekk , bekk , cekk } aejj , bejj , cejj ,
3.28
then
{aeii , beii , ceii } {abeii , eii ceii }.
3.29
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7
Similarly, by ceii cejj ◦ beji − eii ejj ◦ 1/2abeii aeij − 1/2abejj 0 and ceii cejj ◦
j, we get
beii − aeij aeji − bejj ◦ aeii beji − beij − aejj 0, where i /
{ceii , beii , aeii } {ceii , baeii , eii }.
3.30
Then, combining 3.29 and 3.30, we have {ceii , aeii , beii } {abceii , eii , eii }. Hence, the claim
3.22 holds.
Finally, we claim that
aeij , beji , ceij eii , eii , a ◦ b ◦ ceij ,
where i / j.
3.31
For i / j, we have −1/2 bceij b−1 ceji cejj ◦ −1/2 abeii 1/2 abejj aeji ◦ eii − ejj beij 0, and consequently {−1/2 bceij b−1 ceji cejj , −1/2 abeii 1/2 abejj aeji , eii −
ejj beij } 0. Using 3.8–3.13, 3.16, 3.17, and 3.19, this can be reduced to
1
1
bceij , aeji , beij .
ab2 ceij , eii , eii abceij , eji , ejj cejj , abejj , ejj 2
2
3.32
Replacing b by 1 and 2 in 3.32, respectively, we have
1
1
ceij , aeji , eij ,
cejj , aejj , ejj 2
2
4aceij , eii , eii 2aceij , eji , ejj cejj , aejj , ejj ceij , aeji , 2eij .
aceij , eii , eii aceij , eji , ejj 3.33
3.34
Computing 3.34–2 3.33, this yields
2aceij , eii , eii ceij , aeji , eij ,
3.35
then taking 3.35 into 3.33, we derive
aceij , eji , ejj 1
cejj , aejj , ejj ,
2
3.36
1
cejj , abejj , ejj ,
2
3.37
replacing a by ab in 3.36, we get
abceij , eji , ejj then taking 3.37 into 3.32, it follows that
1
2
ab ceij , eii , eii bceij , aeji , beij ,
2
3.38
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Journal of Applied Mathematics
and replacing b by b 1 in 3.38, and using 3.38, 3.35, and 3.16, we get
ceij , aeji , beij eii , eii , 2abceij eii , eii , a ◦ b ◦ ceij ,
where i /
j.
3.39
Therefore, 3.31 holds.
m
Let xt , yt , and zt be such that m
t1 xt ◦yt ◦zt 0; we have to prove that
t1 {xt , yt , zt } 0. Writing
xt n
n atij eij ,
yt i1 j1
n
n t
bkl
ekl ,
zt k1 l1
it follows, by examining the i, h entry of
m
t1
n
n r1 h1
t
crh
erh ,
xt ◦ yt ◦ zt 0, that for all i, h, we have
n n m t1 j1 l1
3.40
t
atij bjlt clh
cijt bjlt atlh 0.
3.41
First, by 3.1, we get
n n n n m
m t
t
atij eij , bki
eki , crk
erk
xt , yt , zt t1
3.42
t1 i1 j1 k /
j r1
n n n n m t
atij eij , bjlt ejl , clh
elh
3.43
n n n m t
atij eij , bjit eji , crj
erj .
3.44
t1 i1 j1 l1 h1
i
t1 i1 j1 r /
The summation 3.42 can be written as
n n n n m i k/
t1 i1 j /
j r1
n
n n m t
t
t
t
t
atij eij , bki
eki , crk
erk eki , crk
erk ,
aii eii , bki
t1 i1 k /
i r1
using 3.4, 3.5, and 3.21, the first summation can be rewritten as
n m n n n t
t
atij eij , bki
eki , crk
erk
i k/
t1 i1 j /
j r1
n n n n m i k/
t1 i1 j /
j r1
t
t
ejj , atij bki
ekj , crk
erk
3.45
Journal of Applied Mathematics
9
n n n n n m m n n t
t
t
t
ejj , atij bki
ejj , atij bki
ekj , crk
erk ekj , cjk
ejk
i k/
j
t1 i1 j /
j r /
i k/
t1 i1 j /
j
n n n m n n n m n t t
ejj , ejj , atij bki
crk erj t t
atij bki
cjk ekj , ejk , ejj ,
i k/
j
t1 i1 j /
j r /
i k/
t1 i1 j /
j
3.46
and using 3.8, 3.12, 3.9, 3.10, 3.4, 3.17, and 3.18, we split the second summation
in three parts:
n
n n m t1 i1 k /
i r1
n
n n m t t
t
t
t
eki , crk
erk erk
atii eii , bki
aii bki eki , ekk , crk
t1 i1 k /
i r1
n
n n n n
m m t t
t t
t
t
erk ekk
aii bki eki , ekk , crk
aii bki eki , ekk , ckk
t1 i1 k /
i r /
k
n
n n n n
m m t t
t
t
t
erk , err eki , ckk
ekk
aii bki eki , crk
eii , atii bki
t1 i1 k /
i r /
k
t1 i1 k /
i
t1 i1 k /
i
n
n n n n
m m t t
t t
t
t
erk , err eik , eii
aii bki eki , crk
aii bki eki , cik
t1 i1 k /
i r /
k,i
t1 i1 k /
i
n n
m t t
ckk eki
eii , eii , atii bki
3.47
t1 i1 k /
i
n
n n n n
m m t t t
t t
crk eri , err eii , atii bki
aii bki cik eki , eik , eii
t1 i1 k /
i r /
k,i
t1 i1 k /
i
n n
m t t
ckk eki
eii , eii , atii bki
t1 i1 k /
i
n
n n n n
m m t t
t t
crk eri cik eki , eii
eii , eii , atii bki
eik , atii bki
t1 i1 k /i r / k,i
t1 i1 k /i
m n n
t t
ckk eki .
eii , eii , atii bki
t1 i1 k /
i
Therefore, the summation 3.42 can be rewritten as
n n n n n m m n n t t
t t
ejj , ejj , atij bki
atij bki
crk erj cjk ekj , ejk , ejj
i k/
j
t1 i1 j /
j r /
n
n n n n
m m t t
t t
crk eri cik eki , eii
eii , eii , atii bki
eik , atii bki
t1 i1 k /
i r /
k,i
i k/
t1 i1 j /
j
n n
m t t
ckk eki .
eii , eii , atii bki
t1 i1 k /
i
t1 i1 k /
i
3.48
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Journal of Applied Mathematics
The summation 3.43 can be written as
n n n n n n m m n t
t
t
atij eij , bjlt ejl , clh
atij eij , bjj
elh ejj , cjh
ejh
i h1
t1 i1 j /
t1 i1 j1 l /
i,j h1
n n n m t
atij eij , bjit eji , cih
eih ,
3.49
t1 i1 j1 h1
using 3.2, 3.10, 3.8, 3.16, 3.17, 3.18, and 3.6, the first summation can be rewritten
as
n n n n n n n n m m t
t
atij eij , bjlt ejl , clh
atij bjlt eil , ell , clh
elh elh
t1 i1 j1 l /
i,j h1
n n
m t1 i1 j1 l /
i,j h1
n
n n n n m t
atij bjlt eil , ell , clh
atij bjlt eil , ell , cllt ell
elh t1 i1 j1 l /
i,j h /
l
n n
m n
t1 i1 j1 l /
i,j
n t
atij bjlt eil , clh
elh , ehh t1 i1 j1 l /
i,j h /
l
n n
m n
n t1 i1 j1 l /
i,j h /
l,i
n n n m eii , atij bjlt eil , cllt ell
t1 i1 j1 l /
i,j
t
atij bjlt eil , clh
elh , ehh n n n m atij bjlt eil , clit eli , eii
t1 i1 j1 l /
i,j
m n n n eii , eii , atij bjlt cllt eil
t1 i1 j1 l / i,j
n n n n n n n m m t
atij bjlt clh
atij bjlt clit eli , eil , eii
eih , ehh , ehh t1 i1 j1 l /
i,j h /
l,i
n n
m i l/
t1 i1 j /
i,j
n n
m n
eii , eii , atij bjlt cllt eil n
t1 i1 j1 l /
i,j h /
l,i,j
m n n
t1 i1 j1 l /
i,j
n n t1 i1 j i,j
/i l/
n n
m t1 i1 l /
i
eii , eii , atii bilt cllt eil
n n n m t
ehh , ehh , atij bjlt clh
ejj , ejj , atij bjlt cljt eij
eih i l/
t1 i1 j /
i,j
atij bjlt clit eli , eil , eii m n n
t1 i1 l /
i
t t t
aii bil cli eli , eil , eii
n n n n n
m m eii , eii , atij bjlt ctll eil eii , eii , atii bilt cllt eil ,
i l
t1 i1 j /
/ i,j
t1 i1 l /i
3.50
and using 3.6, 3.3, 3.8, 3.16, and 3.20, we rewrite the second summation as
n n n n m m n n t
t
t
t
atij eij , bjj
eii , atij bjj
ejj , cjh
ejh eij , cjh
ejh
t1 i1 j / i h1
t1 i1 j / i h1
n n n n m m n t
t
t
eii , atij bjj
eii , atij bjj
eij , cjh
ejh eij , cjit eji
i h/
t1 i1 j /
i
i
t1 i1 j /
Journal of Applied Mathematics
11
n n n n m m n t
t
t
t
eii , atij bjj
eii , atij bjj
eij , cjh
ejh eij , cjj
ejj
i h/
t1 i1 j /
i,j
i
t1 i1 j /
n n m t t
eij , atij bjj
cji eji , eii
i
t1 i1 j /
n n n n n m m t t
t t
eii , atij bjj
eii , eii , atij bjj
cjh eih , ehh cjj eij
i h/
t1 i1 j /
i,j
i
t1 i1 j /
n n m t t
eij , atij bjj
cji eji , eii
i
t1 i1 j /
n n n n m m n t t
t t
ehh , ehh , atij bjj
eii , eii , atij bjj
cjh eih cjj eij
i h/
t1 i1 j /
i,j
i
t1 i1 j /
n n m t t
eij , atij bjj
cji eji , eii ,
i
t1 i1 j /
3.51
and then using 3.3, 3.2, 3.8, 3.6, 3.20, 3.22, and 3.31, the third summation is split
in four parts:
n n n m t
atij eij , bjit eji , cih
eih
t1 i1 j1 h1
m m n n n n n t
atij eij , bjit eji , cih
atij eij , bjit eji , ciit eii
eih t1 i1 j1 h /
i
n n n m t1 i1 j1 h /
i,j
n n n m t1 i1 j1 h /
i,j
n n m t
atij eij , bjit eji , cih
atij eij , bjit eji , cijt eij
eih i
t1 i1 j /
n n n
m m t
atij eij , bjit eji , ciit eii aii eii , biit eii , ciit eii
i
t1 i1 j /
t1 i1 j1
t1 i1
n n m t
atij eij , bjit cih
eii , eii , atij ◦ bjit ◦ cijt eij
ejh , ehh i
t1 i1 j /
n n n m m 1 t
eii , atij eij , bjit ciit eji aii ◦ biit ◦ ciit eii , eii , eii
2
i
t1 i1 j /
t1 i1
n n n m t1 i1 j1 h /
i,j
n n m t
atij bjit cih
eii , eii , atij ◦ bjit ◦ cijt eij
eih , ehh , ehh i
t1 i1 j /
n n n m m 1 t
t t t
t
t
eij , aij bji cii eji , eii a ◦ b ◦ c eii , eii , eii .
2 ii ii ii
i
t1 i1 j /
t1 i1
3.52
12
Journal of Applied Mathematics
Hence, the summation 3.43 can be rewritten as
n n n
m n t1 i1 j1 l /
i,j h /
l,i,j
n n m n t
ehh , ehh , atij bjlt clh
ejj , ejj , atij bjlt cljt eij
eih i l/
t1 i1 j /
i,j
n n n n
m m n t t t
atij bjlt clit eli , eil , eii aii bil cli eli , eil , eii
i l
t1 i1 j /
/ i,j
t1 i1 l /i
n n n n n
m m eii , eii , atij bjlt cllt eil eii , eii , atii bilt cllt eil
i l/
t1 i1 j /
i,j
t1 i1 l /
i
m m n n n n n t t
t t
ehh , ehh , atij bjj
eii , eii , atij bjj
cjh eih cjj eij
t1 i1 j /i h
/ i,j
n n m i
t1 i1 j /
n m 1
t1 i1
2
3.53
n n n m t t
t
eij , atij bjj
ehh , ehh , atij bjit cih
cji eji , eii eih
t1 i1 j1 h /
i,j
n n m t1 i1 j /i
t1 i1 j /i
n n m eii , eii , atij ◦ bjit ◦ cijt eij eij , atij bjit ciit eji , eii
t1 i1 j /i
atii ◦ biit ◦ ciit eii , eii , eii .
Finally, using 3.5, 3.7, 3.8, 3.10, 3.12, 3.16, and 3.21, the summation 3.44 can be
written as
n n n n n
m m t
t
t
aii eii , biit eii , cri
atij eij , bjit eji , crj
erj eri
i r /
i
t1 i1 j /
i
t1 i1 r /
n n n n n
m m t
t
t
ejj , atij eij , bjit crj
eri eri , err
aii eii , biit cri
t1 i1 j /i r /i
t1 i1 r /i
n n n n n m m t
t
ejj , atij eij , bjit crj
ejj , atij eij , bjit cjj
eri eji
i r /
i,j
t1 i1 j /
i
t1 i1 j /
n n
m t t t
aii bii cri eri , err , err
t1 i1 r /i
n m m n n n n t
t
ejj , ejj , atij bjit crj
atij bjit cjj
erj eij , eji , ejj
i r /
i,j
t1 i1 j /
n n
m t
eri .
err , err , atii biit cri
i
t1 i1 r /
i
t1 i1 j /
3.54
Journal of Applied Mathematics
13
Consequently, rewriting the summation 3.44 as
n n n n m m n t
ejj , ejj , atij bjit crj
atij bjit ctjj eij , eji , ejj
erj i r /
i,j
t1 i1 j /
i
t1 i1 j /
n n
m t
eri ,
err , err , atii biit cri
3.55
t1 i1 r /i
therefore, we get
m
xt , yt , zt 3.47 3.52 3.54.
t1
3.56
Next, we only need to prove that 3.47 3.52 3.54 0.
Replacing the indices r, k, i, and j by i, j, l, and h, correspondingly, in the first
summation of 3.48, and rewriting the summation as
n n n n n n n m m n t t
ejj , ejj , atij bki
ehh , ehh , cijt bjlt atlh eih ,
crk erj i k/
j
t1 i1 j /
j r /
t1 i1 j1 l1 h /
i,j,l
3.57
then adding 3.57 and the first, the seventh, the tenth summations of 3.53, together, this
yields
n n n n m t
ehh , ehh , atij bjlt clh
cijt bjlt atlh eih .
t1 i1 j1 l1 h /
i,j,l
3.58
Replacing i and k by l and i, respectively, in the fifth summation of 3.48, then by 3.16, this
summation is further equal to
n n
m eii , eii , atll bilt ciit eil ,
t1 i1 l /
i
3.59
and replacing r, k, and i by i, j, and l, correspondingly, in the third summation of 3.48, then
using 3.16, the summation is further equal to
n n n m eii , eii , atll bjlt cijt eil .
i l/
t1 i1 j /
i,j
3.60
Then adding 3.59, 3.60, and the fifth and the sixth summations of 3.53, together, this
yields
n n n m t1 i1 j1 l /
i,j
eii , eii , atij bjlt cllt atll bjlt cijt eil .
3.61
14
Journal of Applied Mathematics
Replacing k by j in the fourth summation of 3.48, then adding the 12th summation of 3.53,
we get
n n m eij , atij bjit ciit atii bjit cijt eji , eii .
i
t1 i1 j /
3.62
Replacing i and r by j and i, correspondingly, in the third summation of 3.55, then adding
the 8th summation of 3.53, we have
n n m t t
t t
eii , eii , atij bjj
cjj atjj bjj
cij eij .
i
t1 i1 j /
3.63
Replacing r, j, and i by i, j, and l, respectively, in the first summation of 3.55, and rewriting
this summation as
n m n n ejj , ejj , atlj bjlt cijt eij ,
i l/
t1 i1 j /
i,j
3.64
then adding 3.64 and the second summation of 3.53, together, and by 3.16, we derive
n n m n eii , eii , atij bjlt cljt atlj bjlt cijt eij .
i l/
t1 i1 j /
i,j
3.65
Replacing i and l by j and i, respectively, in the fourth summation of 3.53, then adding the
second summation of 3.55, we arrive at
n n m i
t1 i1 j /
t
atij bjit cjj
atjj bjit cijt eij , eji , ejj .
3.66
Replacing i, j, and k by l, i, and j, respectively, in the second summation of 3.48, and
rewriting this summation as
n n m n i l/
t1 i1 j /
i,j
n n m t t
atli bjlt cijt eji , eij , eii atji bjj
cij eji , eij , eii ,
i
t1 i1 j /
3.67
then using 3.19 and 3.18, the aforementioned summation can be rewritten as
n n n m i l/
t1 i1 j /
i,j
n n m t t
atli bjlt cijt eji , eij , eii eij , atji bjj
cij eji , eii .
i
t1 i1 j /
3.68
Journal of Applied Mathematics
15
Adding the second summation of 3.68 and the ninth summation of 3.53, together, we get
n n m i
t1 i1 j /
t t
t t
eij , atij bjj
cji atji bjj
cij eji , eii .
3.69
By 3.36, we know that the first summation of 3.68 can be written as
n n n m 1
i l/
t1 i1 j /
i,j
2
atli bjlt cijt eii , eii , eii ,
3.70
similarly, using 3.36, the third summation of 3.53 can be written as
n n n m 1
i l/
t1 i1 j /
i,j
2
atij bjlt clit eii , eii , eii ,
3.71
and then adding 3.70 and 3.71, together, we have
n n m n 1
i l/
t1 i1 j /
i,j
2
atij bjlt clit
atli bjlt cijt
eii , eii , eii .
3.72
Therefore, we derive
3.47 3.52 3.54 3.52 3.60 3.62 3.64 the 11th summation of 47
3.61 3.65 3.68 3.71 the 13th summation of 3.52.
3.73
Our first goal is to show
3.52 3.60 3.62 3.64 the 11th summation of 3.52 0.
3.74
First, notice that by 3.16, the summation 3.61 can be rewritten as
n n n m ell , ell , atij bjlt cllt atll bjlt cijt eil .
t1 i /
l j /
l l1
3.75
Then replacing l by h, we have
n n n m t t
t t
ehh , ehh , atij bjh
chh athh bjh
cij eih .
t1 i /
h j /
h h1
3.76
16
Journal of Applied Mathematics
Second, notice that
3.64 the 11th summation of 3.52 n n n m eii , eii , atij bjlt cljt cijt bjlt atlj eij . 3.77
i l/
t1 i1 j /
j
Then, by 3.16, and replacing j by h, this yields
m n n
n t t
t t t
clh cih
bhl alh eih .
ehh , ehh , atih bhl
3.78
t1 i /
h l/
h h1
Third, using 3.16, and replacing j by h in the summation 3.63, we obtain
n
n m t
t
t t
chh
cih
bhh athh eih .
ehh , ehh , atih bhh
3.79
t1 i /
h h1
Finally, rewriting 3.58 as
n n n n m t
ehh , ehh , atij bjlt clh
cijt bjlt atlh eih ,
3.80
t1 i /
h j /
h l/
h h1
then, adding 3.76, 3.78, 3.79, and 3.80, together, it follows that
n n n n m t
ehh , ehh , atij bjlt clh
cijt bjlt atlh eih
t1 i / h j1 l1 h1
⎧
n ⎨
n n n m ⎩
i h h1
t1 j1 l1
ehh , ehh ,
/
t
atij bjlt clh
cijt bjlt atlh eih
⎫
⎬
3.81
,
⎭
and using 3.41, we get 3.74 holds.
Next, we claim that
3.61 3.65 3.68 3.71 the 13th summation of 3.52 0.
3.82
First, using 3.19 and 3.36, we rewrite the summations 3.62 and 3.69, respectively, as
n n m 1
i
t1 i1 j /
2
n n m 1
i
t1 i1 j /
2
atij bjit ciit atii bjit cijt eii , eii , eii ,
t t
atij bjj
cji
t t
atji bjj
cij
eii , eii , eii ,
3.83
Journal of Applied Mathematics
17
and then, adding 3.83 and 3.72, this yields
n n n m 1
2
t1 i1 j / i l1
atij bjlt clit atli bjlt cijt eii , eii , eii .
3.84
Similarly, by 3.36, the summation 3.66 can be rewritten as
n n m 1
i
t1 i1 j /
2
t
atij bjit cjj
atjj bjit cijt ejj , ejj , ejj .
3.85
Adding 3.85 and the 13th summation of 3.53, it implies
n n m 1
t1 i1 j1
2
t
atij bjit cjj
atjj bjit cijt ejj , ejj , ejj .
3.86
Replacing i and j by l and i, respectively, in the summation 3.86, then adding 3.84, and
by 3.41, we get
n n n m 1
t1 i1 j1 l1
2
atij bjlt clit atli bjlt cijt eii , eii , eii
⎧
n n
n ⎨
m 1
⎩ t1
i1
j1
2
l1
atij bjlt clit atli bjlt cijt eii , eii , eii
⎫
⎬
⎭
3.87
0.
Therefore, 3.82 holds.
Consequently, we get 3.47 3.52 3.54 0, and using 3.56, we finish the proof
of this theorem.
Acknowledgments
The authors show great thanks to the referee for his/her valuable comments, which greatly
improved the readability of the paper. The work was mainly supported by the National
Natural Science Foundation Grants of China Grant no. 10871056 and the Fundamental
Research Funds for the Central Universities Grant no. HEUCF20111132.
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