6.21.4 Tensor Matrix Coordinate Representation

Tensor Matrix Coordinate Representation is the specific numerical array obtained when a second-order tensor is expanded in a chosen pair of bases, together with the full record of which bases produced it and the rule by which that array must change if the bases are replaced. It is the mechanism that converts the abstract, basis-free tensor T into a concrete object — a matrix of numbers — that can be written down, computed with, and compared, while preserving enough bookkeeping to recover the original basis-free tensor from any one of its coordinate representations.

From Tensor to Coordinates

The Expansion That Produces the Matrix

Given a tensor T ∈ V* ⊗ W, a basis {eᵢ} of V with dual basis {εⁱ}, and a basis {fⱼ} of W, the coordinate representation is the array Tⱼⁱ defined by the expansion

This array, once bases are fixed, can be arranged as an ordinary dim(W) × dim(V) matrix, with the coordinate representation existing only relative to the specific pair of bases used in the expansion.

The Representation Is Not Unique

Because different bases produce different arrays for the same tensor T, "the coordinate representation" is always implicitly relative to a stated choice of basis; the phrase "the matrix of T" is shorthand that suppresses this dependency and should be understood as "the matrix of T in the basis currently under discussion."

The Transformation Rule That Ties Representations Together

Change of Basis Induces Change of Coordinates

If {e′ᵢ} is a new basis related to {eᵢ} by e′ᵢ = Σₖ Aₖᵢeₖ, and similarly {f′ⱼ} related to {fⱼ} by a matrix B, then the coordinate representation of the same tensor T in the new bases is related to the old one by

where C denotes the inverse of the transition matrix A acting on the contravariant index. This transformation rule is what makes "coordinate representation" more than an arbitrary array: it certifies that the array is tied to a specific, well-defined basis-free object, and it is the precise criterion by which two apparently different matrices can be recognized as representing the same tensor.

Diagram of Multiple Representations of One Tensor

Practical Aspects of Coordinate Representations

Standard Basis Convention

In most applied contexts, the coordinate representation is computed relative to the standard basis of Rⁿ, so that the tensor's matrix is simply the array of numbers one would type directly into a spreadsheet or numerical library, with the transformation rule invoked only when a basis change — such as a rotation of coordinate axes — is explicitly under consideration.

Coordinate Representation and Invariants

Any quantity computed from a coordinate representation that is unchanged when the transformation rule above is applied — trace, determinant, rank, eigenvalues for a (1,1) tensor — is a genuine property of the tensor T itself; any quantity that changes under the transformation is a property only of the particular coordinate representation. The coordinate representation is therefore the working surface on which computations are performed, while the transformation rule is the test that separates tensor-level facts from artifacts of the chosen coordinates.

Recovering the Tensor from Its Representation

Given the coordinate array Tⱼⁱ together with the explicit bases {eᵢ}, {εⁱ}, {fⱼ} used to produce it, the original tensor T is recovered exactly via the expansion formula above; the coordinate representation together with its basis data is therefore a complete and faithful encoding of the tensor, never a lossy approximation of it.