6 Tensor Order and Type

Tensor Order and Type is the classification of a tensor by two related but distinct pieces of data: its order, the total number of indices (equivalently, the number of tensor factors) it carries, and its type (p, q), the more refined count of how many of those factors come from the original vector space V (contravariant slots) as opposed to its dual space V* (covariant slots), so that a tensor of type (p, q) is precisely an element of V^{⊗p} ⊗ (V*)^{⊗q}. Order alone measures the total complexity of a tensor's index structure, while type further tracks how each individual index transforms under change of basis, and both together determine which vector space a given tensor actually belongs to.

Order: Counting Total Indices

Definition of Order

The order (sometimes called rank in this specific sense, though "rank" is also used for other tensor invariants) of a tensor is the number of indices needed to specify its components, equivalently the number n such that the tensor is an element of an n-fold tensor product of copies of V and/or V*.

Order Zero, One, and Two as Familiar Cases

Order zero corresponds to scalars (elements of the field F itself), order one corresponds to ordinary vectors (elements of V) or covectors (elements of V*), and order two corresponds to matrices, bilinear forms, or linear operators, depending on which combination of V and V* factors is involved — already illustrating that order alone does not pin down which specific space a tensor belongs to.

Type: Distinguishing Contravariant from Covariant Slots

The (p, q) Classification

A tensor of type (p, q) is an element of

with p contravariant (upper-index, "vector-like") slots contributed by V, and q covariant (lower-index, "covector-like") slots contributed by V*; the order of such a tensor is p + q, but the type additionally records the split between the two kinds of slots.

Why Order Alone Is Insufficient

Two tensors can share the same order while having different types — for instance, an order-two tensor might be of type (2,0) (an element of V ⊗ V), type (1,1) (an element of V ⊗ V*, matching linear operators on V), or type (0,2) (an element of V* ⊗ V*, matching bilinear forms) — and these three types of order-two tensors behave differently under change of basis and represent conceptually distinct objects.

Diagram of Type Classification

Transformation Behavior Determined by Type

Contravariant Components Transform with the Inverse

Under a change of basis on V given by an invertible matrix, components in a contravariant (upper-index) slot transform using the inverse of that matrix, matching how the coordinates of an ordinary vector in V transform when the basis itself changes.

Covariant Components Transform with the Matrix Directly

Components in a covariant (lower-index) slot transform using the change-of-basis matrix directly (not its inverse), matching how coordinates of a linear functional in V* transform, since the dual basis transforms contragredient to the original basis.

Mixed-Type Tensors Combine Both Rules

A tensor of general type (p,q) has its component array c^{i₁...iₙ}_{j₁...j_q} transform with p applications of the inverse change-of-basis matrix (one per upper index) and q applications of the change-of-basis matrix itself (one per lower index), combining both transformation rules according to exactly how many slots of each kind the tensor has.

Order and Type Under Tensor Product Operations

Order and Type Add Under Tensoring

If a tensor of type (p₁,q₁) is tensored with a tensor of type (p₂,q₂), the result has type (p₁+p₂, q₁+q₂) and order (p₁+q₁)+(p₂+q₂), since tensoring simply concatenates the list of factors from V and V* involved.

Contraction Reduces Both Order and Type

Contracting a tensor over one contravariant and one covariant index (pairing a V factor against a V* factor via the natural evaluation pairing) reduces both p and q by one and the order by two, since contraction removes exactly one factor of each kind from the tensor product.

Significance of Order and Type

A Finer Classification Than Order Alone

Because tensors of the same order can have structurally different behavior depending on their split between contravariant and covariant slots, type refines order into the more informative classification actually needed to determine which specific tensor product space a tensor belongs to and how its components transform.

Foundation for Classical and Physical Tensor Calculus

The order-and-type classification is the direct algebraic basis for the classical, index-based treatment of tensors used throughout differential geometry and physics, where specifying a tensor field's type (p,q) is the standard first step in describing its transformation law, its role in contraction with other tensors, and its geometric or physical interpretation.