6.22 Tensor Order Type Relation

Tensor Order Type Relation is the principle governing how the order of a tensor — the number of indices it carries — interacts with its type — the split of those indices into contravariant (upper) and covariant (lower) slots — to determine the full classification of a tensor as an element of a specific tensor product space. This relation clarifies that order alone is insufficient to specify a tensor's nature: two tensors of the same order can belong to entirely different spaces, transform differently under a change of basis, and support different operations, depending on how their indices are distributed between contravariant and covariant roles.

Order as a Count, Type as a Partition

Order Counts Indices Without Distinguishing Them

The order of a tensor is simply the total number of indices needed to write its components: a scalar has order 0, a vector or covector has order 1, and a tensor written Tⁱⱼ or Tᵢⱼₖ has order 2 or 3 respectively, regardless of which indices are upper and which are lower. Order is a coarse invariant — it says how many "slots" the tensor has but nothing about what kind of slots they are.

Type Refines Order by Specifying Variance

The type of a tensor, often written (p, q), records that it has p contravariant indices and q covariant indices, so that a (p, q)-type tensor has order p + q. The type is what fixes the tensor as an element of the specific space

so that the order p + q alone underdetermines this space whenever p + q ≥ 2, since different splits of the same total order give different, generally non-isomorphic-in-a-basis-free-sense spaces.

Why the Same Order Can Hide Different Types

Order Two, Three Different Types

An order-two tensor can be of type (2, 0), written Tⁱʲ, of type (0, 2), written Tᵢⱼ, or of type (1, 1), written Tⁱⱼ. All three have exactly two indices and are displayed as an equally sized array of numbers, but they belong to V ⊗ V, V* ⊗ V*, and V* ⊗ V (or V ⊗ V*) respectively — three different spaces with three different transformation laws under a change of basis.

Consequence for Transformation Behavior

Under a change of basis with transition matrix A, a (2,0) tensor transforms as T'ⁱʲ = AⁱₖAʲₗTᵏˡ, a (0,2) tensor transforms with the inverse-transpose in both slots, and a (1,1) tensor transforms with one factor of A and one of its inverse. Knowing only that a tensor has order two is not enough to know which of these transformation rules applies; the type must be specified separately.

Diagram of Order Versus Type

The Relation in the Reverse Direction

Type Always Determines Order

While order does not determine type, type always determines order by simple addition: a tensor of type (p, q) necessarily has order p + q, with no ambiguity. This asymmetry is the essential content of the tensor order type relation — type is the finer classification, order the coarser one obtained by forgetting the contravariant/covariant split and retaining only the total count.

Special Cases

Order zero admits only the type (0, 0), the scalars, since there are no indices to split. Order one admits exactly two types, (1, 0) (vectors) and (0, 1) (covectors), the smallest case where the order-type distinction has any content. From order two onward, the number of possible types for a given order n is n + 1, corresponding to the possible values of p from 0 to n with q = n − p.

Practical Role of the Relation

Guiding Correct Contraction and Composition

Knowing the type, not merely the order, of the tensors involved in an operation is what determines whether an index contraction is legitimate: only a contravariant index paired with a covariant index can be summed to produce a basis-independent result. The order type relation is therefore the classification that must be consulted before any tensor operation is performed, since order alone cannot certify that an operation is coordinate-free.

Guiding Classification in a Knowledge Hierarchy

Within a systematic treatment of tensor algebra, the order type relation is what justifies organizing tensors first by order and then, within each order, by type, since order provides the coarse grouping and type provides the finer distinctions that carry the actual algebraic and geometric meaning of each tensor.