1.2.39 Tensor Type Definition

Tensor Type Definition classifies tensors by their transformation rules, key to understanding their behavior in algebra and physics.

Tensor Type Definition is the characterization of a tensor by the ordered pair recording how many contravariant factors and how many covariant factors combine to form the tensor product space in which it lives. Written $(r, s)$ , the type of a tensor is the finest standard classification of its position within the tensor algebra of a vector space, refining the coarser notion of order by distinguishing which of its factors are copies of the space itself and which are copies of its dual.

Formal Definition

Let $V$ be a vector space over a field $F$ , with dual space $V^{*}$ . A tensor $T$ is said to have type $(r, s)$ if it is an element of the tensor product space

T_{s}^{r} (V) = \underset{r}{⏨} V \otimes \dots \otimes V \otimes \underset{s}{⏨} V^{*} \otimes \dots \otimes V^{*}

Here $r$ counts the number of factors equal to $V$ itself, called the contravariant factors, and $s$ counts the number of factors equal to $V^{*}$ , called the covariant factors. Equivalently, by the identification of tensor product spaces with spaces of multilinear maps, a tensor of type $(r, s)$ can be regarded as a multilinear map

T : \underset{r}{⏨} V^{*} \times \dots \times V^{*} \times \underset{s}{⏨} V \times \dots \times V \to F

taking $r$ covectors and $s$ vectors as arguments and returning a scalar.

Contravariant and Covariant Terminology

The terms contravariant and covariant originate from how the components of a tensor transform under a change of basis. A contravariant index, associated with a factor of $V$ , transforms with the inverse of the change-of-basis matrix, in the same manner as the coordinates of a vector, whereas a covariant index, associated with a factor of $V^{*}$ , transforms directly with the change-of-basis matrix, in the same manner as the coefficients of a linear functional. By convention, contravariant indices are written as superscripts and covariant indices as subscripts, so that a type $(r, s)$ tensor has components

T_{j_{1} \dots j_{s}}^{i^{1} \dots i^{r}}

with $r$ superscript indices and $s$ subscript indices, and this notational convention is chosen precisely to make each index's transformation behavior visible directly in its placement.

Type in Familiar Special Cases

Type (1, 0): Vectors

A tensor of type $(1, 0)$ is simply an element of $V$ itself, an ordinary vector, with a single contravariant index and no covariant indices.

Type (0, 1): Covectors

A tensor of type $(0, 1)$ is an element of the dual space $V^{*}$ , a covector or linear functional, with a single covariant index.

Type (0, 2): Bilinear Forms

A tensor of type $(0, 2)$ is a bilinear form on $V$ , taking two vector arguments and returning a scalar, with components $T_{i j}$ carrying two covariant indices. Symmetric type $(0, 2)$ tensors that are also positive-definite give rise to inner products, called metric tensors in geometric contexts.

Type (1, 1): Linear Maps

A tensor of type $(1, 1)$ is canonically identified with a linear map from $V$ to itself, since an element of $V \otimes V^{*}$ corresponds to a linear endomorphism whose matrix, relative to a basis, has components $T_{j}^{i}$ with one contravariant and one covariant index, matching the row and column indices of an ordinary matrix.

Type Versus Order and Rank

The order of a tensor, $r + s$ , is recoverable from the type but discards the split between contravariant and covariant factors, so distinct types can share the same order — a type $(2, 0)$ tensor and a type $(1, 1)$ tensor are both order two but have different transformation behavior and generally cannot be canonically identified without extra structure, such as a metric, on $V$ . Type is likewise independent of tensor rank, which measures the decomposability of a specific element rather than classifying the ambient space in which it lives; two tensors of the same type can have entirely different tensor ranks.

Role Within Tensor Algebra

Type is the primary label organizing the full tensor algebra of a vector space $V$ , which decomposes as a direct sum

\underset{r, s \geq 0}{\oplus} T_{s}^{r} (V)

over every possible type. Fundamental operations behave predictably with respect to type: the tensor product of a type $(r_{1}, s_{1})$ tensor and a type $(r_{2}, s_{2})$ tensor produces a tensor of type $(r_{1} + r_{2}, s_{1} + s_{2})$ , while contraction over one contravariant and one covariant index reduces a type $(r, s)$ tensor to type $(r - 1, s - 1)$ . Knowing the type of a tensor is therefore what determines which operations apply to it and how its components transform under a change of basis, making type the central classifying invariant of tensor algebra.