6.21.2 Tensor Matrix Linear Map Context

Tensor Matrix Linear Map Context is the interpretive framework in which a second-order tensor, once a basis has been fixed for each of the vector spaces it acts between, is identified with a matrix that represents a linear map, so that every statement about the abstract tensor T ∈ V* ⊗ W can be translated into an equivalent statement about a concrete matrix acting on column vectors by multiplication. This context is what allows the purely algebraic object "second-order tensor" to be manipulated with the ordinary machinery of matrix algebra — multiplication, rank, determinant, eigenvalues — while keeping track of exactly which of those matrix properties are basis-independent features of the underlying linear map and which are artifacts of the chosen bases.

From Bilinear Object to Linear Map

The Two Faces of a Second-Order Tensor

A second-order tensor over vector spaces V and W can be read in two equivalent ways: as an element of a tensor product space, or as a linear map from one space to another. Concretely, a tensor T ∈ V* ⊗ W corresponds to a linear map T̂ : V → W defined by

where T = Σ φₖ ⊗ wₖ for covectors φₖ ∈ V* and vectors wₖ ∈ W. This correspondence is a linear isomorphism between the space V* ⊗ W and the space Hom(V, W) of linear maps from V to W, so nothing is lost or gained by switching between the two descriptions.

Why the Distinction Still Matters

Although the two descriptions are isomorphic, they carry different intuitions. The tensor-product description emphasizes bilinearity and the universal property, while the linear-map description emphasizes action on vectors and composition. The matrix linear map context adopts the second viewpoint deliberately, because it is the one that connects directly to bases, coordinates, and the operations of ordinary matrix arithmetic.

Constructing the Matrix Once Bases Are Fixed

Choosing Bases and Reading Off Entries

Fix a basis {eᵢ} of V with dual basis {εⁱ} of V*, and a basis {fⱼ} of W. Writing T = Σᵢⱼ Tⱼⁱ εⁱ ⊗ fⱼ, the scalars Tⱼⁱ are exactly the entries of the matrix representing T̂ in these bases: the j-th row, i-th column entry is the coefficient of fⱼ in the expansion of T̂(eᵢ).

Matrix-Vector Multiplication as the Concrete Realization of Application

If v = Σᵢ vⁱ eᵢ has coordinate column [v¹, ..., vⁿ]ᵀ, then the coordinates of T̂(v) in the basis {fⱼ} are given by ordinary matrix-vector multiplication:

This equation is the entire content of the matrix linear map context: it states that tensor application, an abstractly defined contraction of indices, becomes literally the row-by-column multiplication rule taught in elementary matrix algebra once coordinates are introduced.

Diagram of the Correspondence

What Depends on the Basis and What Does Not

Basis-Independent Properties

Certain properties of T̂ are visible directly in the matrix but do not depend on which bases were chosen to compute it: the rank of the matrix (the dimension of the image of T̂), whether T̂ is injective or surjective, and — when V = W — the trace, determinant, and eigenvalues of the matrix. These quantities are invariant under the change-of-basis transformation T ↦ P⁻¹TP (or T ↦ Q⁻¹TP for maps between different spaces with change-of-basis matrices P, Q), which is why they are treated as genuine properties of the linear map itself rather than of any particular matrix representation.

Basis-Dependent Artifacts

By contrast, the specific numerical entries Tⱼⁱ of the matrix, the notion of a "diagonal" entry when V ≠ W, and any statement about individual rows or columns are artifacts of the chosen bases and change under a basis transformation. The matrix linear map context requires distinguishing these two categories carefully: a computation performed on the matrix is only meaningful as a statement about the tensor if the quantity computed is one of the basis-independent invariants.

Composition and Tensor Contraction

Matrix Multiplication as Contraction

If T̂ : V → W and Ŝ : W → U are two linear maps represented by matrices T and S in compatible bases, the composite map Ŝ ∘ T̂ : V → U is represented by the ordinary matrix product ST. In tensor index notation, this product is the contraction

summing over the shared index j, which shows that matrix multiplication is nothing other than tensor index contraction written in a notation that suppresses the summed index.

Practical Significance

This identification is what justifies applying the entire toolkit of numerical linear algebra — matrix inversion, LU and eigenvalue decomposition, singular value decomposition — directly to second-order tensors, since any algorithm that operates correctly on matrices under change of basis operates correctly on the underlying tensor as long as the basis-dependent versus basis-independent distinction is respected when interpreting the output.