Tensors for Physics

5.2 Principal Values 59

In matrix notation, (5.12) is equivalent to

Sμν=S ̄

⎛

⎝

100

010

001

⎞

⎠+^2

3

s

⎛

⎝

−^1200

0 − 210

001

⎞

⎠+q

⎛

⎝

100

0 − 10

000

⎞

⎠. (5.14)

For a symmetric traceless tensor one hasS(^1 )+S(^2 )+S(^3 )=0, i.e.S ̄=0. The
anisotropic part of the tensor is characterized by the linear combinationssandqof
its principal values. Forq=0 ands=0, the uniaxial case, discussed above, is
recovered. In the other special cases=0, butq=0, the symmetric traceless part
of the tensor is referred to as beingplanar biaxial. In general, a symmetric tensor,
in principle axis representation, is characterized by its three principal values or by
its trace and two parameters specifying its symmetric traceless part, as introduced
in (5.11). In principle, the labeling 1, 2 ,3 of the principle axes can be interchanged.
Preferentially, the 3-axis is associated either with the largest or the smallest eigen-
value.

5.2.5 Symmetric Dyadic Tensors

In general, the principal directions and values of a symmetric tensor can be found
by the methods used in Linear Algebra for the diagonalization of matrices. In many
problems of physics, the principal directions are obvious by symmetry considera-
tions, and then the principal values can be inferred from the eigenvalue equation. As
an example, the special case of a symmetric dyadic tensor constructed from two unit
vectorsuandvis considered. Thus we have

Sμν=

1

2

(uμvν+uνvμ). (5.15)

Lethbe a vector parallel to a principal direction. Then the eigenvalue equation
Sμνhν=Shμimplies that the principal valueSis determined byhμSμνhν=Shμhμ
and consequently
S=hμuμvνhν/hκhκ. (5.16)

First, the casevparallel touis considered. Then one principal direction is parallel to
uand the directions of the two other ones are not uniquely determined, but they lie
in the plane perpendicular tou. The principal values areS=1, forh=u, and two
principal values are 0, for two orthogonal directions which are perpendicular tou.In
short, the principal values are{ 1 , 0 , 0 }. Clearly, the tensor is uniaxial. The symmetry
direction can be identified with any one of the coordinate axes, frequently either the
1- or the 3-axis is chosen.