Overview
A {% k %}-tensor on a
vector space
{% V %} is a multilinear function from the Cartesian product of {% k %} copies of {% V %} to the reals.
{% T : V \times V \times ... \times V \rightarrow \mathbb{R} %}
where {% T %} is linear in each argument.
Extensions
- Range is an arbitrary field, or vector space
- Domain is a {% k %} Cartesian product of {% k %} possibly different vector spaces.
Tensor Product
If {% T %} is a {% k %}-tensor over {% V %} and {% S %} is an {% m %} tensor over {% V %}, then the tensor product of {% T %} and {% S %} is defined as
{% T \otimes S (v_i,...,v_k, v_{k+1},... v_{k+m}) = T(v_i,...,v_k) \times S(v_{k+1},... v_{k+m}) %}
Mixed Tensors
A mixed tensor is a multilinear function over a vector space {% V %} and its dual space {% V^* %}
{% \displaystyle T : V_1 \times V_2 \times ... \times V_n \rightarrow \mathbb{R} %}
where {% V_i %} is either equal to {% V %} or {% V^* %}.
Tensor Components
Simliar to matrices, a set of numeric values can be computed and associated with a tensor, given a basis for each vector space in its domain.
As an example, consider a tensor defined as
{% T(V_1,V_2, V_3) %}
for {% V_1 = V^* %} and {% V_2 = V_3 = V %} then
{% T^i_{j,k} = T(e_{1i}, e_{2j}, e_{3k},) %}