5.9 Tensor Product Basis Structure

Tensor Product Basis Structure is the body of theory describing how bases of the individual factor spaces in a tensor product combine to produce a basis of the tensor product itself, how the resulting coordinate representation behaves under change of basis, and how this induced basis underlies the standard numerical representation of tensors as multidimensional arrays.

The Induced Basis Construction

Given finite-dimensional vector spaces V1, V2, …, Vn over a field F with dimensions d1, …, dn, and a chosen basis for each Vi, the tensor product V1 ⊗ V2 ⊗ ⋯ ⊗ Vn acquires an induced basis consisting of every tensor formed by selecting one basis vector from each factor,

ranging over every index tuple (k1, …, kn). This construction, and the fact that it genuinely produces a basis (linearly independent and spanning), is the central content of tensor product basis structure.

Dimension Multiplication

The induced basis directly explains the multiplicative dimension formula that distinguishes the tensor product from the direct sum of vector spaces.

The Dimension Formula

follows directly by counting the number of index tuples in the induced basis, and is the single most consequential structural fact distinguishing the tensor product from other combinations of vector spaces, such as the direct sum, whose dimension is instead the sum of the factor dimensions.

Contrast with Direct Sum Basis Structure

A basis of the direct sum V1 ⊕ ⋯ ⊕ Vn is simply the union (disjoint, appropriately embedded) of the individual factor bases, giving additive dimension; the tensor product's basis structure is combinatorial rather than additive, taking every possible combination across factors rather than simply collecting the factor bases side by side.

Coordinate Representation

The induced basis converts an abstract tensor into a concrete numerical object once coordinates are read off relative to it.

Multidimensional Array Form

Relative to the induced basis, any tensor t is represented by its array of coefficients tk1…kn, an n-dimensional array of shape d1 × d2 × ⋯ × dn. This array representation is the standard numerical form of a tensor used throughout computational linear algebra, and it exists only because the tensor product basis structure guarantees that such coefficients are uniquely determined once the factor bases are fixed.

Coordinates of Decomposable Tensors

A decomposable tensor v1 ⊗ ⋯ ⊗ vn has coordinates that factor as a product of the individual coordinates of each vi in its respective factor basis, tk1…kn = (v1)k1 (v2)k2 ⋯ (vn)kn, a special structural pattern within the general coordinate representation that is not shared by tensors of higher rank.

Behavior Under Change of Basis

Tensor product basis structure interacts with change of basis in a way that mirrors, but generalizes, the ordinary transformation rules for vectors and matrices.

Kronecker Product Transformation

If the basis of each factor Vi is changed via an invertible matrix Ai, the induced basis of the tensor product changes according to the Kronecker product A1 ⊗_K A2 ⊗_K ⋯ ⊗_K An of the individual matrices, reflecting the same combinatorial construction that built the induced basis in the first place.

Basis-Independent Meaning

Although the coordinate array of a tensor depends on the chosen factor bases, the abstract tensor itself does not; tensor product basis structure supplies the dictionary translating between the coordinate-dependent numerical representation used in computation and the coordinate-independent abstract object that representation stands for.

Structure Preserved Under Special Basis Choices

When the factor spaces carry additional structure, tensor product basis structure shows that this structure passes naturally to the induced basis.

Orthonormal Bases and Inner Products

If each factor space Vi carries an inner product and its selected basis is orthonormal with respect to it, the tensor product acquires a canonical inner product under which the induced basis is likewise orthonormal, obtained by declaring the inner product of two basis tensors to be the product of the corresponding factor inner products.

Operators Acting Factorwise

If a linear operator on the tensor product is built as a tensor product of operators on the individual factors, and each factor operator is diagonal with respect to its chosen basis, the induced basis of the tensor product simultaneously diagonalizes the combined operator, with eigenvalues given by the products of the individual factor eigenvalues — a structural fact used throughout the spectral theory of composite systems.

Illustrative Diagram

The grid represents the twelve basis tensors of a tensor product of a four-dimensional and a three-dimensional space, each cell corresponding to one pairing of a basis vector from each factor.