6.7 Tensor Type Classification

Tensor Type Classification is the classification of tensors by the ordered pair (p, q), recording how many of a tensor's slots are contravariant (upper, p of them) and how many are covariant (lower, q of them), and attaching to each variance the specific transformation law — inverse-Jacobian for upper slots, direct Jacobian for lower slots — that governs how components change under a change of basis. It sits between order classification, which records only the total p + q, and valence classification, which records the full ordered arrangement of the p and q slots, making type the natural middle level of detail in the hierarchy of index-structure classification schemes.

The Core Definition

The Type Pair

For a tensor written with components T_{j₁...j_q}^{i₁...i_p}, the type is the ordered pair:

with p the number of upper indices and q the number of lower indices, related to order by p + q = k.

The Two Transformation Laws

Under a change of basis with Jacobian J, each upper index transforms by the inverse Jacobian, and each lower index transforms by the Jacobian directly:

with type precisely the label that tells which of these two rules governs each slot of a given tensor.

Named Tensors by Type

The Basic Named Types

Type (0,0) is a scalar; (1,0) is a vector; (0,1) is a covector; (1,1) is naturally identified with a linear operator once a specific space is fixed; (0,2) is a bilinear form, with a symmetric positive-definite example being a metric tensor.

Diagram of the Named-Type Grid

Type Arithmetic

Addition Under the Tensor Product

follows because forming the tensor product concatenates the upper indices of the two factors and separately concatenates their lower indices.

Matched Reduction Under Contraction

Contraction, to remain type-consistent, must pair one upper index against one lower index, reducing (p, q) to (p - 1, q - 1); this is the type-level refinement of the general fact that contraction always removes exactly two indices.

Type as an Ordered Pair and Its Covariant Part

Type Is Formally an Element of N × N

Type is defined precisely as an ordered pair of non-negative integers, with two tensors sharing a type exactly when their pairs are equal coordinate by coordinate, a formalization that supports indexing families of tensor spaces T^p_q(V) directly by their type.

The Covariant Part as a Distinguishable Sub-Structure

Within a fixed type (p, q), the q covariant slots form a distinguishable part of the tensor's domain, V^{⊗q}, governed uniformly by the direct-Jacobian rule and separable, in reasoning, from the p contravariant slots governed by the inverse-Jacobian rule.

Relating Type to Order and Valence

Type Refines Order

A single order k covers every type (p, q) with p + q = k; type adds the further information of how that total splits between the two variances, distinguishing, for instance, a bilinear form (0,2) from an operator (1,1), both of order 2.

Valence Refines Type

A single type (p, q) still admits (p+q)!/(p!q!) distinct arrangements of its upper and lower slots; valence classification resolves this remaining ambiguity by recording the exact left-to-right order of the slots, a level of detail type alone does not carry.

Why Type Classification Matters

Determining Transformation Behavior at a Glance

Knowing a tensor's type immediately reveals how each of its slots transforms under a change of basis, without needing to inspect specific components or actually carry out the transformation, making type the natural classification level for reasoning about coordinate changes.

Connecting Abstract Tensors to Familiar Linear-Algebraic Objects

Recognizing the standard identifications — vector as (1,0), covector as (0,1), operator as (1,1), bilinear form as (0,2) — lets type classification serve as a bridge, translating the abstract multilinear formalism of tensor algebra into the more familiar vocabulary of elementary linear algebra whenever that translation is useful.