Mathematical Tools for Physics

12—Tensors 370

Theithcomponent of which is
Tjivj

If you write this as a square matrix times a column matrix, the only difference between this result and that of
Eq. ( 16 ) is that the matrix is transposed. This vector valued functionT ̃is called the transpose of the tensorT.
The nomenclature comes from the fact that in the matrix representation, the matrix of one equals the transpose
of the other’s matrix.
By an extension of the language, this applies to the other form of the tensor,T:
T ̃(~u, ~v) =T(~v, ~u)

Symmetries
Two of the common and important classifications of matrices, symmetric and antisymmetric, have their reflections
in tensors. A symmetric tensor is one that equals its transpose and an antisymmetric tensor is one that is the
negative of its transpose. It is easiest to see the significance of this when the tensor is written in the bilinear
functional form:
Tij=T(ˆei,ˆej).

This matrix will equal its transpose if and only if

T(~u, ~v) =T(~v, ~u)

for all~uand~v. Similarly, if for all~uand~v

T(~u, ~v) =−T(~v, ~u)

thenT=−T ̃. Notice that it doesn’t matter whether I speak ofTas a scalar-valued function of two variables or
as a vector-valued function of one; the symmetry properties are the same.
From these definitions, it is possible to take an arbitrary tensor and break it up into its symmetric part and
its antisymmetric part:

T=

1

2

(

T+T ̃

)

+

1

2

(

T−T ̃

)

=TS+TA (29)

TS(~u, ~v) =

1

2

[

T(~u, ~v) +T(~v, ~u)

]

TA(~u, ~v) =

1

2

[

T(~u, ~v)−T(~v, ~u)

]