6.21.1 Tensor Matrix Two Index Form

Tensor Matrix Two Index Form is the representation of a second-order tensor as an array of scalars carrying exactly two indices, one running over the rows and one over the columns, so that the abstract object T is written concretely as Tⱼᵢ (or Tⁱʲ, or Tⁱⱼ, depending on the variance pattern) and displayed as a rectangular grid of numbers. This form is the bridge between the index notation used in tensor algebra and the row-and-column layout used in ordinary matrix algebra, and it is the form in which a second-order tensor is entered into computational software, printed on paper, or manipulated by hand.

The Two Indices and What They Track

Row Index and Column Index

In the two index form, the first index conventionally labels the row and the second labels the column, so that Tⱼᵢ denotes the entry located at row j, column i. When the tensor is written with the array

the subscript pair jᵢ is read left-to-right as (row, column), matching the convention used throughout matrix algebra.

Variance Encoded by Index Position

Unlike a bare matrix of numbers, the two index form of a tensor distinguishes upper indices from lower indices to record variance: a lower index transforms covariantly under a change of basis and an upper index transforms contravariantly. A purely covariant tensor is written Tᵢⱼ, a purely contravariant tensor Tⁱʲ, and a mixed tensor Tⁱⱼ or Tⱼⁱ. The two index form therefore carries more information than a plain matrix of numbers — it also records how those numbers are meant to transform.

Relation to the Abstract Tensor Object

Reading the Array as Coefficients in a Basis Expansion

Given bases {eᵢ} of V, {εʲ} dual basis of V*, {fₖ} of W, a mixed tensor T ∈ V* ⊗ W expands as

and the two index array Tₖⁱ is precisely the table of coefficients appearing in this expansion. The two index form is thus not an independent object but a coordinate presentation of the tensor, valid only relative to the bases used to produce it.

Distinguishing the Array from the Tensor Itself

Because the array of numbers changes when the basis changes according to the transformation rule appropriate to its variance pattern, the two index form must always be understood as "the tensor T expressed in basis {eᵢ}, {fₖ}" rather than as the tensor itself. Two different arrays, related by the correct transformation law, represent the same abstract tensor in different bases.

Diagram of the Index Layout

Operations Native to the Two Index Form

Transposition

Swapping the two indices, Tⱼᵢ ↦ Tᵢⱼ, defines the transpose. When both indices have the same variance, the transpose is again a tensor of the same type; when the tensor is mixed, index swapping is a purely formal array operation and does not correspond to a coordinate-free tensor operation unless a metric is used to raise or lower one of the indices first.

Trace and Contraction

When the tensor is of mixed type Tⁱⱼ, setting i = j and summing, Σᵢ Tⁱᵢ, produces the trace, a single scalar obtained directly from the two index array. This operation, called contraction, generalizes the matrix trace and is basis-independent precisely because it pairs one contravariant index with one covariant index.

Entrywise Access for Computation

The two index form is the layout expected by numerical libraries and spreadsheet-style tools: individual entries Tⱼᵢ are addressed directly by a pair of integers, rows and columns can be extracted, and standard matrix routines (multiplication, inversion, decomposition) operate on the array without any reference to the abstract tensor-product definition, provided the correct variance and transformation rules are respected when the results are reinterpreted as tensors.