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1 Preliminaries
Theorem 1.7.1 A second order tensor can be uniquely decomposed into the sum of
a symmetric tensor and an anti-symmetric tensor. The theorem is called the tensor
decomposition theorem.
Proof Existence: A second order tensor T can be written as
T =
1
2
(T + T c ) +
1
2
(T − T c )
(1.7.16)
where, T c is the conjugate tensor of T . Obviously, the first term on the right-handed
side is a symmetric tensor, the second term is an anti-symmetric tensor, this shows
that there exists the decomposition of a tensor.
Uniqueness: Let a second order tensor T have been decomposed into the sum of
a symmetric tensor S and an anti-symmetrical tensor A. It needs to prove that S and
A must have the expression identified by Eq. (1.7.16). T can also be written as
T = S + A
Taking the conjugate of the above expression, we have
T c = S c + A c = S − A
Do addition and subtraction of the above two expressions, we obtain
S =
1
2
(T + T c ), A =
1
2
(T − T c )
It shows that the decomposition approach of a tensor is unique. Quod erat
demonstrandum.
1.7.3 Algebraic Operations of Cartesian Tensors
(1) Additive operation
Let A i j = α li α m j A
lm and B i j = α li α m j B
lm be two same order (Here is a second
order) Cartesian tensors, do addition or subtraction of every component of the first
order tensor and the corresponding component of the second order tensor, the result
is made a new tensor with the same structure. For instance
C i j = A i j ± B i j = α li α m j (A
lm ± B
lm ) = α li α m j C
lm
(1.7.17)
Because Eq. (1.7.17) satisfies the definition of a second order tensor, so C i j is a
second order tensor. It is thus clear that the sum (or difference) of the two same order