# Glossary of tensor theory

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.

 Contents

## Classical notation

Rank of a tensor

A tensor written in component form is an indexed array. The rank of a tensor is the number of indices required.

A dyadic tensor has rank two, and may be represented as a square matrix. The conventions aij, aij, and aij, do have different meanings, in that the first may represent a quadratic form, the second a linear transformation, and the distinction is important in contexts that require tensors that aren't orthogonal (see below). A dyad is a tensor such as aibj, product component-by-component of rank one tensors. In this case it represents a linear transformation, of rank one in the sense of linear algebra - a clashing terminology that can cause confusion.

Einstein notation This states that in a product of two indexed arrays, if an index letter in the first is repeated in the second, then the (default) interpretation is that the product is summed over all values of the index. For example if aij is a matrix, then under this convention aii is its trace. The Einstein convention is generally used in physics and engineering texts, to the extent that if summation is not applied it is normal to note that explicitly.

Covariant tensor, Contravariant tensor

The classical interpretation is by components. For example in the differential form aidxj the components ai are a covariant vector. That means all indices are lower; contravariant means all indices are upper.

This refers to any tensor with lower and upper indices.

Orthogonal tensor

In the presence of a tensor δij, there is no need to maintain the distinction of upper and lower indices. That is the case given a distinguished set of orthogonal co-ordinates. Orthogonal tensors are also called cartesian tensors

## 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

[itex]v \otimes w[itex]

is a tensor in

[itex]V \otimes W[itex].

That is, the [itex] \otimes [itex] operation is a binary operation, but it takes values in a fresh space (it is in a strong sense external). The [itex] \otimes [itex] operation is bilinear; but no other conditions are applied to it.

### Pure tensor

A pure tensor of [itex]V \otimes W[itex] is one that is of the form [itex]v \otimes w[itex].

It could be written dyadically aibj, or more accurately aibj ei[itex] \otimes [itex]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 > 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

[itex] \otimes [itex]

becomes a normal (internal) binary operation. This is at the cost of T(V) being of 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.

### Exterior power

The wedge product is the anti-symmetric form of the [itex] \otimes [itex] 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.

## Abstract algebra

This is an operation on fields, that does not always produce a field.

These may be worked out directly, or by a theory of Clifford modules.

These are the derived functors of the tensor product, and feature strongly in homological algebra. The name comes from the torsion subgroup in abelian group theory.

These are highly abstract approaches used in some parts of geometry.

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