Tensor Product - Tensor Product of Two Tensors

Tensor Product of Two Tensors

A tensor on V is an element of a vector space of the form

for non-negative integers r and s. There is a general formula for the components of a (tensor) product of two (or more) tensors. For example, if F and G are two covariant tensors of rank m and n (respectively) (i.e. F ∈ T_m0, and G ∈ T_n0), then the components of their tensor product are given by

In this example, it is assumed that there is a chosen basis B of the vector space V, and the basis on any tensor space T_sr is tacitly assumed to be the standard one (this basis is described in the article on Kronecker products). Thus, the components of the tensor product of two tensors are the ordinary product of the components of each tensor.

Note that in the tensor product, the factor F consumes the first rank(F) indices, and the factor G consumes the next rank(G) indices, so

The tensor may be naturally viewed as a module for the Lie algebra End(V) by means of the diagonal action: for simplicity let us assume r = s = 1, then, for each ,

where u* in End(V*) is the transpose of u, that is, in terms of the obvious pairing on V ⊗ V*,

.

There is a canonical isomorphism given by

Under this isomorphism, every u in End(V) may be first viewed as an endomorphism of and then viewed as an endomorphism of End(V). In fact it is the adjoint representation ad(u) of End(V) .