1.2.54 Type p q Tensor Definition

A Type p q tensor is a multilinear map operating on p covariant and q contravariant indices in tensor algebra.

Type p q Tensor Definition is the formal specification of a tensor as a multilinear object built from p copies of a vector space and q copies of its dual space, classifying tensors by how many contravariant and covariant slots they possess. A tensor of type (p, q), also called a (p, q)-tensor or a tensor of rank p + q, is a multilinear map that takes p covectors (elements of the dual space) and q vectors as arguments and returns a scalar. Equivalently, it is an element of the tensor product space formed from p copies of the vector space V and q copies of its dual space V*. The integer p counts the contravariant indices, which transform like vector components under a change of basis, and the integer q counts the covariant indices, which transform like the components of linear functionals.

Formal Definition

The Multilinear Map Perspective

Let V be a finite-dimensional vector space over a field, and let V* denote its dual space, the space of linear functionals on V. A tensor of type (p, q) is a multilinear map

T : \times_{i = 1}^{p} V^{*} \times \times_{j = 1}^{q} V \to F

where F is the underlying field, p copies of V* supply the arguments that are dualized once more to interact with contravariant behavior, and q copies of V supply the arguments associated with covariant behavior. Multilinearity means the map is linear in each argument separately, holding all other arguments fixed.

The Tensor Product Perspective

Equivalently, a type (p, q) tensor is an element of the tensor product space

\otimes_{i = 1}^{p} V \otimes \otimes_{j = 1}^{q} V^{*}

This space is commonly denoted T^p_q(V). The two perspectives are related by the natural isomorphism between a finite-dimensional vector space and the dual of its dual, V ≅ V**, which allows tensors built from copies of V and V* to be reinterpreted as multilinear maps and vice versa.

Contravariant and Covariant Indices

The Role of p (Contravariant Order)

The number p specifies how many contravariant indices the tensor carries. Contravariant components, conventionally written with upper indices such as T^{i_1 i_2 ... i_p}, transform in the same way as the components of a vector under a change of basis. If the basis vectors are rescaled or rotated, contravariant components transform using the inverse of the transformation matrix, which is why they are said to transform "contrary" to the basis.

The Role of q (Covariant Order)

The number q specifies how many covariant indices the tensor carries. Covariant components, conventionally written with lower indices such as T_{j_1 j_2 ... j_q}, transform in the same way as the components of a linear functional. Under a change of basis, covariant components transform using the transformation matrix directly, following the same direction as the change applied to the basis vectors.

Mixed Index Notation

A general type (p, q) tensor is written in component form with p upper indices and q lower indices:

T_{j_{1} j_{2} \dots j_{q}}^{i^{1} i^{2} \dots i^{p}}

Each upper index ranges independently over the dimension of the vector space, and each lower index does the same, so the full component array has as many indices as p + q, giving the tensor its total rank.

Dimension and Component Count

Number of Independent Components

If V has finite dimension n, then the space T^p_q(V) of type (p, q) tensors has dimension n^(p+q). This follows because a basis for T^p_q(V) is formed by all possible tensor products of p basis vectors of V and q basis covectors of V*, and there are n choices for each of the p + q factors.

dim (T_{q}^{p}, (V)) = n^{p + q}

Basis Construction

Given a basis e_1, ..., e_n of V and its dual basis e^1, ..., e^n of V*, a basis for T^p_q(V) is given by all tensor products of the form e_{i_1} ⊗ ... ⊗ e_{i_p} ⊗ e^{j_1} ⊗ ... ⊗ e^{j_q}, where each index ranges from 1 to n. Any type (p, q) tensor can be expressed as a linear combination of these basis elements, weighted by its components.

Special Cases

Type (0, 0) Tensors

A type (0, 0) tensor has no contravariant and no covariant indices. It is simply a scalar, an element of the underlying field, unaffected by changes of basis.

Type (1, 0) Tensors

A type (1, 0) tensor is a contravariant vector, that is, an ordinary vector belonging to V itself. It carries a single upper index and transforms according to the standard rule for vector components.

Type (0, 1) Tensors

A type (0, 1) tensor is a covariant vector, also called a covector or a linear functional, belonging to V*. It carries a single lower index and transforms according to the standard rule for dual components.

Type (1, 1) Tensors

A type (1, 1) tensor can be identified with a linear map from V to V, since it takes one covector and one vector as arguments, which is the same data needed to represent a linear transformation acting on V. This identification is central to describing linear operators, such as those appearing in linear algebra, using tensor notation.

Type (0, 2) Tensors

A type (0, 2) tensor takes two vectors as arguments and returns a scalar, which is the structure used to represent bilinear forms, including inner products and metric tensors on a vector space.

Transformation Law Under Change of Basis

General Transformation Rule

When the basis of V changes, the components of a type (p, q) tensor transform according to a rule that applies the change-of-basis matrix once for each contravariant index and its inverse once for each covariant index. If A denotes the transformation matrix relating a new basis to an old one, and A^{-1} its inverse, the components transform as

{\tilde{T}}_{l_{1} \dots l_{q}}^{k^{1} \dots k^{p}} = A_{i_{1}}^{k^{1}} \dots A_{i_{p}}^{k^{p}} {(A^{- 1})}_{l_{1}}^{j^{1}} \dots {(A^{- 1})}_{l_{q}}^{j^{q}} T_{j_{1} \dots j_{q}}^{i^{1} \dots i^{p}}

using the Einstein summation convention, where repeated upper and lower indices are summed over.

Basis Independence of the Transformation Law

This transformation law is precisely what makes the object a well-defined tensor rather than an arbitrary array of numbers: any set of components that transforms according to this rule for every change of basis represents a single, basis-independent geometric object.

Diagrammatic Summary

The diagram represents a tensor T with p upper slots corresponding to contravariant indices, each drawing from the vector space V, and q lower slots corresponding to covariant indices, each drawing from the dual space V*. The total number of slots, p + q, gives the rank of the tensor.