72 5 Symmetric Second Rank Tensors
of products of the tensor. Likewise, the expression for the invariantI 3 of a symmetric
traceless tensor, as given in (5.45), follows from (5.50) withμ=κ. The symmetric
traceless part of equation (5.50) leads to
Sμν SνλSλκ =1
2
SνλSλν Sμκ, (5.51)which is a remarkable relation for the triple product of an irreducible second rank
tensor. In matrix notation, this can be checked by writing the tensor in its principal
axes coordinate system.
5.2 Exercise:
Verify the Relation(5.51)for the Triple Product of a Symmetric Traceless Tensor
Hint: use the matrix notation
⎛
⎝a 00
0 b 0
00 c⎞
⎠,
withc=−(a+b), for the symmetric traceless tensor in its principal axes system.
Compute the expressions on both sides of (5.51) and compare.
5.6.2 Quadruple Products of Tensors.
Multiplication of (5.50)or(5.51) withSκμimplies
SκμSμν SνλSλκ =1
2
SνλSλν Sκμ Sμκ. (5.52)This relation says: the trace of the four fold product of a symmetric traceless sec-
ond rank tensor is equal to one half of the square of its norm squared. Similarly,
multiplication of (5.49) withSκμyields an expression for the fourth order product
SκμSμνSνλSλκin terms of the trace, of two fold and three fold products of the tensor.
Notice that the equations presented in Sects.5.5and5.6are specific for symmetric
second rank tensors in three dimensional space. This is obvious in the formulation of
the Hamilton-Cayley theorem (5.50), there are just three principal values in 3D. For
second rank tensors in 2D or in 4D, analogous, but different relations apply, which
give rise to different consequences.