This is a glossary of tensor theory. For expositions of tensor theory from different points of view, see:
For some history of the abstract theory see also multilinear algebra.
Algebraic notation
This avoids the initial use of components, and is distinguished by the explicit use of the tensor product symbol.
- Tensor product
- If v and w are vectors in vector spaces V and W respectively, then
- is a tensor in
- That is, the ⊗ operation is a binary operation, but it takes values into a fresh space (it is in a strong sense external). The ⊗ operation is a bilinear map; but no other conditions are applied to it.
- Pure tensor
- A pure tensor of V ⊗ W is one that is of the form v ⊗ w.
- It could be written dyadically aibj, or more accurately aibj ei ⊗ fj, where the ei are a basis for V and the fj a basis for W. Therefore, unless V and W have the same dimension, the array of components need not be square. Such pure tensors are not generic: if both V and W have dimension greater than 1, there will be tensors that are not pure, and there will be non-linear conditions for a tensor to satisfy, to be pure. For more see Segre embedding.
- Tensor algebra
- In the tensor algebra T(V) of a vector space V, the operation becomes a normal (internal) binary operation. A consequence is that T(V) has infinite dimension unless V has dimension 0. The free algebra on a set X is for practical purposes the same as the tensor algebra on the vector space with X as basis.
- Hodge star operator
- Exterior power
- The wedge product is the anti-symmetric form of the ⊗ operation. The quotient space of T(V) on which it becomes an internal operation is the exterior algebra of V; it is a graded algebra, with the graded piece of weight k being called the k-th exterior power of V.
- Symmetric power, symmetric algebra
- This is the invariant way of constructing polynomial algebras.