4.18.5 Tensor Multilinear Form Tensor Type Relation

Tensor Multilinear Form Tensor Type Relation is the correspondence linking a scalar-valued multilinear form to a classical "type (p, q) tensor," identifying multilinear forms as exactly the tensors of purely covariant type (0, q), and showing how mixed tensors of general type (p, q) arise as multilinear maps that take both vectors and covectors as arguments. This relation connects the modern definition of tensors via multilinear maps to the classical physics-style indexing of tensors by upper and lower indices.

The Classical Type Notation

Covariant and Contravariant Slots

A tensor of type (p, q) on a finite-dimensional vector space V is classically described as an object with p upper (contravariant) indices and q lower (covariant) indices, transforming under change of basis so that upper indices contract against the inverse change-of-basis matrix and lower indices contract against the change-of-basis matrix itself. The pair (p, q) records how many of each kind of index the object carries.

Reinterpreting Indices as Argument Slots

Each lower index corresponds to an argument slot that accepts a vector from V, and each upper index corresponds to an argument slot that accepts a covector from V*. A type (p, q) tensor is thereby identified with a multilinear map

so that supplying p covectors and q vectors yields a scalar, matching the classical rule that a fully contracted tensor with all indices saturated produces a number.

Multilinear Forms as the Purely Covariant Case

Type (0, q) Tensors

A multilinear form f: V × ... × V → F of arity q, with no covector arguments, is exactly a type (0, q) tensor in the classical sense: it has q lower indices and no upper indices, and its component array T_{k₁...k_q} matches the classical array of a purely covariant tensor. This is the case already familiar from the study of scalar-valued multilinear forms, now placed within the broader family of type (p, q) tensors.

Type (p, 0) Tensors as Multilinear Forms on the Dual

By symmetry, a type (p, 0) tensor, purely contravariant, is a multilinear map V* × ... × V* → F, that is, a multilinear form defined on the dual space rather than on V itself. Since V** ≅ V in finite dimensions, a type (p, 0) tensor can equally be regarded as an element of V ⊗ ... ⊗ V, with the multilinear-form perspective and the elementary-tensor perspective describing the same object from two directions.

Mixed Tensors as Multilinear Forms With Two Kinds of Slots

General Type (p, q)

A general type (p, q) tensor is a multilinear map taking p covector arguments and q vector arguments, with the multilinearity condition holding separately in each of the p + q slots regardless of whether that slot accepts a vector or a covector. Its component array carries p upper indices and q lower indices, T^{i₁...i_p}_{j₁...j_q}, generalizing the purely covariant array of a multilinear form to one recording both kinds of index.

Component Array via Duality

Under the identification V ⊗ ... ⊗ V ⊗ V* ⊗ ... ⊗ V* ≅ (V* ⊗ ... ⊗ V* ⊗ V ⊗ ... ⊗ V)*, a type (p, q) tensor corresponds equally to an element of V^{⊗p} ⊗ (V*)^{⊗q}, so the multilinear-form description of type (p,q) tensors as maps out of (V*)^p × V^q and the elementary-tensor description as elements of V^{⊗p} ⊗ (V*)^{⊗q} are two faces of the same universal property, applied once with the roles of V and V* in the domain and once with them swapped into the tensor factors.

Examples Across Types

Type (0,1): Covectors

A type (0,1) tensor is a linear functional φ: V → F, an element of V*, matching the basic case of a multilinear form of arity one.

Type (1,0): Vectors

A type (1,0) tensor is a linear functional on V*, which by the canonical identification V ≅ V** corresponds to an ordinary vector v ∈ V, acting on covectors by v(φ) = φ(v).

Type (1,1): Linear Endomorphisms

A type (1,1) tensor is a multilinear map V* × V → F, (φ, v) ↦ φ(Av) for some linear map A: V → V, identifying the space of type (1,1) tensors with End(V), the space of linear endomorphisms of V; this is why an (1,1) tensor is classically presented as a matrix with one upper and one lower index, A^i_j, matching the standard matrix of a linear operator.

Type (0,2) and Type (2,0): Bilinear Forms and Their Duals

A type (0,2) tensor is a bilinear form on V; a type (2,0) tensor is a bilinear form on V*, equivalently, via the finite-dimensional identification of V ⊗ V with bilinear forms on V* × V*, an element of V ⊗ V. The metric tensor of Riemannian geometry is a type (0,2) symmetric tensor, while its inverse, used to raise indices, is the corresponding type (2,0) tensor.

Operations Respecting Tensor Type

Contraction Changes Type by (−1,−1)

Contracting one upper index against one lower index of a type (p,q) tensor, feeding the output of one covector slot's associated basis covector against a paired vector slot, produces a tensor of type (p-1, q-1); this operation corresponds, in the multilinear-form perspective, to evaluating the multilinear map on a specific pairing of a dual basis vector and its corresponding basis vector and summing over the shared index.

Tensor Product Adds Types Componentwise

The tensor product of a type (p,q) tensor and a type (p',q') tensor produces a type (p+p', q+q') tensor, matching the arity addition seen when combining two multilinear forms into one multilinear map on the concatenated list of argument slots.

Raising and Lowering Indices

Given a non-degenerate type (0,2) tensor g (a metric), contracting a vector slot of a tensor against g converts a lower index into an upper index, changing the tensor's type from (p,q) to (p+1,q-1); this operation, "raising an index," is the mechanism by which a purely covariant multilinear form and a mixed or purely contravariant tensor of the same total arity are identified once a metric is fixed, even though they are distinct type (p,q) tensors in the absence of such a choice.

Why the Relation Matters

Unifying Two Notations

The relation between multilinear forms and type (p,q) tensors reconciles the classical index-based description of tensors, developed for use in differential geometry and physics, with the coordinate-free definition of tensors via multilinear maps and universal properties, showing that both describe the same objects, one via transformation rules for arrays of numbers, the other via the abstract multilinearity condition those arrays are required to satisfy.

Generalizing Beyond Purely Covariant Forms

Recognizing multilinear forms as the special case (0,q) of the general type (p,q) classification clarifies that the theory of scalar-valued multilinear maps extends uniformly to tensors with mixed variance, with the same universal-property machinery, tensor products, contractions, symmetrization, applying without modification once vector and covector slots are both permitted as arguments.