tensor calculus - Symmetry in terms of matrices

When we encounter a problem in physics which can be expressed in terms of matrices or tensors, why do we decompose the tensor in terms of its symmetric and antisymmetric or trace components? What is the physics motivation behind doing so?

Answer

OP is basically asking:

Why do we decompose (reducible) group representations in irreducible group representations?

Partial answer:

1. 
   

   To classify the (reducible) representation.

   
   



2. 
   

   Irreducible representations can not be further truncated without destroying the group symmetry.

   
   


3. 
   

   Because certain irreducible sub-representations of the given (reducible) representation may be forbidden by e.g. selection rules, other physical principles, etc, and this is always useful information.