Page 411 - Advanced Linear Algebra
P. 411
Tensor Products 395
)
1 A tensor !; ²= ³ is symmetric if
!~!
. The set of all symmetric tensors
for all permutations :
:; ²= ³ ~ ¸! ; ²= ³ ! ~ ! for all : ¹
is a subspace of ;²= ³ , called the symmetric tensor space of degree
over .
=
)
2 A tensor !; ²= ³ is antisymmetric if
!~²c ³ !
The set of all antisymmetric tensors
(; ²= ³ ~ ¸! ; ²= ³ ! ~ ²c ³ ! for all : ¹
is a subspace of ;²= ³ , called the antisymmetric tensor space or exterior
product space of degree over .
=
We can develop the theory of symmetric and antisymmetric tensors in tandem.
Accordingly, let us write (anti)symmetric to denote a tensor that is either
symmetric or antisymmetrtic.
taking to , an
!
Since for any Á ! . 4 , there is a permutation
(anti)symmetric tensor must have . 4 ² # ³ ~ . 4 and so
#
#~ : ²#³ ~ 4 8 ! ! 9
4 4 !. 4
# 4
Since is a permutation of . , it follows that is symmetric if and only if
²: ²#³³ ~ : ²#³
4
4
for all and this holds if and only if the coefficients : ! of : ²#³ are
4
equal, say ! 4 ~ for all ! . 4 . Hence, the symmetric tensors are precisely
the tensors of the form
#~ 8 ! 9 4
4 !. 4
The tensor is antisymmetric if and only if
#
²: ²#³³ ~ ²c ³ : ²#³ (14.4)
4
4
In this case, the coefficients of 4 :²#³ differ only by sign. Before examining
!
this more closely, we observe that 4 must be a set. For if 4 has an element
of multiplicity greater than , we can split . 4 into two disjoint parts:
