1.2.38 Tensor Order Definition

Tensor order defines the number of indices needed to identify its components, key to understanding its structure and behavior in algebra.

Tensor Order Definition is the characterization of the total number of vector-space factors, counted together with their duals, that combine to form the tensor product space in which a tensor lives. The order of a tensor, also called its degree or, in some contexts, its rank in a sense distinct from tensor rank, records how many indices are needed to describe the tensor in components and is the most basic invariant used to classify tensors by structural complexity.

Formal Definition

A tensor of type $(r, s)$ on a vector space $V$ is an element of the tensor product space

\underset{r}{⏨} V \otimes \dots \otimes V \otimes \underset{s}{⏨} V^{*} \otimes \dots \otimes V^{*}

combining $r$ copies of $V$ , called contravariant factors, and $s$ copies of the dual space $V^{*}$ , called covariant factors. The order of such a tensor is defined as the total number of factors,

order (T) = r + s

so that a tensor of type $(r, s)$ has order $r + s$ , regardless of how that total is split between contravariant and covariant factors. The order determines the number of indices needed to write the tensor in components, $T_{j_{1} \dots j_{s}}^{i^{1} \dots i^{r}}$ , with each index ranging over a basis of the corresponding factor.

Order in Low-Dimensional Cases

Order Zero

A tensor of order zero has no factors at all and is, by convention, simply a scalar in the base field $F$ . Order-zero tensors require no indices to describe.

Order One

A tensor of order one is either a vector, an element of $V$ itself, if it has type $(1, 0)$ , or a covector, an element of the dual space $V^{*}$ , if it has type $(0, 1)$ . Order-one tensors require a single index, $v^{i}$ or $ω_{i}$ , to describe in components.

Order Two

A tensor of order two, regardless of type, is representable in components by a two-index array, which in finite dimensions can always be arranged as a matrix. A bilinear form has type $(0, 2)$ , a linear map on $V$ has type $(1, 1)$ , and both are order-two tensors, illustrating that order alone does not determine type: two tensors of the same order can have different distributions of contravariant and covariant factors.

Order Three and Higher

Tensors of order three or more require three or more indices and cannot in general be represented by a single two-dimensional array; they are instead described using multidimensional arrays or, more abstractly, purely in terms of the multilinear maps or tensor product elements they represent. The transition from order two to order three marks the boundary past which many familiar matrix-based tools, such as singular value decomposition and efficient rank computation, no longer directly apply.

Order Versus Related Notions

Order Versus Type

The order of a tensor records only the total count of factors, whereas the type $(r, s)$ records how that total splits between contravariant and covariant factors. Two tensors can share the same order while having different types, as with the bilinear-form and linear-map examples above, so order is a coarser invariant than type, though a simpler and more commonly cited one when full type information is unnecessary.

Order Versus Dimension

The order of a tensor is unrelated to the dimension of the underlying vector space $V$ . A three-dimensional vector space and a thousand-dimensional vector space each support tensors of every order, with the dimension of $V$ instead controlling how many values each individual index can take, and hence how many total components a tensor of a given order possesses: an order- $k$ tensor on an $n$ -dimensional space has $n^{k}$ components in a given basis.

Order Versus Tensor Rank

Order and tensor rank are independent measures of complexity. Order is fixed by the tensor product space in which a tensor lives and does not depend on the particular element chosen; tensor rank, by contrast, depends on the specific element and measures how many decomposable tensors are needed to sum to it. A tensor of high order can nonetheless have low tensor rank if it happens to be decomposable or close to it, and conversely a low-order tensor, such as an order-two matrix, can have rank up to the dimension of the space.

Role Within Tensor Algebra

Order provides the primary organizational axis of the tensor algebra associated with a vector space $V$ , which decomposes as a direct sum of tensor product spaces graded by order, one for each type $(r, s)$ . Operations such as the tensor product itself add orders, since combining a tensor of order $p$ with a tensor of order $q$ produces a tensor of order $p + q$ , while contraction reduces order by removing a matched contravariant and covariant factor pair. Because of this behavior under the basic operations of tensor algebra, order functions as a grading that keeps track of structural complexity as tensors are combined, decomposed, and manipulated.