6.1.2 Tensor Type Classification Scope

Tensor Type Classification Scope explores how tensors are categorized by their type, defining their structure and operations within algebraic frameworks.

Tensor Type Classification Scope is the delimitation of the subject matter that belongs to the classification of tensors by their type, meaning the split of a tensor's indices into a definite number of upper (contravariant) positions and a definite number of lower (covariant) positions, written as the pair (p, q), together with the transformation behavior that each kind of index obeys under a change of basis. This scope statement fixes what counts as type classification — the covariant/contravariant split and its transformation consequences — and separates it from order classification, which counts indices without regard to their variance, and from rank as a numerical matrix invariant, which is excluded entirely.

What Falls Inside This Scope

The Definition of Type as an Index Signature

A tensor's type is the ordered pair (p, q) recording how many of its indices are contravariant (upper) and how many are covariant (lower). Writing a component as T_{j₁...j_q}^{i₁...i_p}, the type is read directly off the position of each index, independent of the numerical values the indices range over.

T has type (p, q) \Leftrightarrow T has p upper indices and q lower indices

Contravariant Index Transformation

Within scope is the rule that each upper index transforms with the inverse Jacobian of a change-of-basis map: if x' = J x, then a contravariant component transforms as

V^{i^{'}} = \sum_{j} J_{j}^{i^{'}} - 1 V^{j}

fixing contravariance as a transformation-based, not merely notational, property.

Covariant Index Transformation

Within scope is the complementary rule that each lower index transforms with the Jacobian directly (not its inverse):

W_{i^{'}} = \sum_{j} J_{i^{'}}^{j} W_{j}

so that vectors (type (1,0)) and covectors (type (0,1)) are distinguished by which of these two rules governs their single index.

Diagram of the Type Split

Named Special Cases of Type

Within scope are the standard names attached to particular (p, q) pairs: type (0,0) is a scalar, type (1,0) is a vector, type (0,1) is a covector or linear functional, type (1,1) is a linear operator once a specific vector space is fixed, and type (0,2) covers bilinear forms such as a metric tensor.

What Falls Outside This Scope

Order as a Bare Index Count

The total index count k = p + q, considered independently of how the indices split into upper and lower positions, belongs to tensor order classification and is only referenced here to note that a single order k may correspond to several distinct types, for example order 2 covers types (2,0), (1,1), and (0,2).

Rank as a Numerical Invariant of a Matrix

The dimension of the column space of a type-(1,1) tensor once represented as a matrix is a separate numerical invariant, excluded from this scope, since two tensors of identical type (1,1) can have different matrix rank.

Symmetry Properties of a Given Type

Whether a type-(0,2) or type-(2,0) tensor is symmetric or antisymmetric under exchange of its same-variance indices is a structural property layered on top of type and is classified separately.

Specific Geometric or Physical Tensors

The classification of named tensors from geometry or physics — the metric tensor, the Riemann curvature tensor, the stress tensor — as instances of a given type is outside this scope, which concerns only the abstract classification scheme itself, not its application to particular tensors.

Boundary Cases Addressed Within This Scope

Mixed Tensors of Equal Upper and Lower Count

A type (p, p) tensor is within scope as the general case of a "mixed" tensor with equal numbers of contravariant and covariant indices, distinguished from type (p, 0) or (0, q) tensors, which are called purely contravariant or purely covariant, respectively.

Type Behavior Under Tensor Product

type (A \otimes B) = (p_{A} + p_{B}, q_{A} + q_{B})

is within scope as the direct consequence of the type-signature definition applied to a tensor product, since the upper and lower index counts of each factor simply add.

Type Behavior Under Contraction

Within scope is the fact that contracting one upper index against one lower index lowers the type from (p, q) to (p-1, q-1), keeping the order reduction of two consistent with a matched removal of one index of each variance, which is the only form of contraction that is type-consistent.

Purpose of Fixing This Scope

Isolating Variance From Other Classification Axes

By confining this scope strictly to the covariant/contravariant split and its transformation law, type classification remains a clean, independent axis that can be combined with order classification and rank without redundant overlap between the three schemes.

Supplying the Vocabulary for Index Notation Elsewhere

Terms such as "contravariant index," "covariant index," and "mixed tensor," used throughout the wider discussion of tensor algebras, draw their precise meaning from the definitions fixed in this scope, ensuring consistent usage in every subsequent topic that refers to a tensor's type.